An adaptive amoeba algorithm for constrained shortest paths

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Self-adaptive differential evolution algorithm for numerical optimization

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Abstract—In this paper, we propose an extension of Self-adaptive Differential Evolution algorithm (SaDE) to solve optimization problems with constraints. In comparison with the original SaDE algorithm, the replacement criterion was modified for handling constraints. The performance of the proposed method is reported on the set of 24 benchmark problems provided by CEC2006 special session on constrained real parameter optimization.
2006 IEEE Congress on Evolutionary Computation Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006
Self-adaptive Differential Evolution Algorithm for Constrained Real-Parameter Optimization
“DE/rand/1”: Vi ,G = Xr ,G + F ⋅ Xr ,G − Xr G
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“DE/best/1”: Vi ,G = Xbest ,G + F ⋅ Xr ,G − X r G 1 2,

基于聚度的自适应动态混沌蚁群算法

2019，55（3）1引言旅行商问题（Traveling Salesman Problem ，TSP ）是数学领域中一个重要的问题。

假设一个商人从某地出发，要求选择一条最短路径，可以不重复地走完所有城市并回到起点。

TSP 问题是一类NP 完全组合优化问题。

目前，TSP 问题有很多求解算法，主要有贪婪算法、模拟退火[1]、遗传算法、粒子群算法[2]和蚁群算法等。

蚁群算法由于其正反馈和鲁棒性等特点，成为解决TSP 这类组合优化问题最有效的方法之一。

1996年，Marco Dorigo 提出了蚂蚁系统（AS ），该模基于聚度的自适应动态混沌蚁群算法刘明霞1，游晓明1，刘升21.上海工程技术大学电子电气工程学院，上海2016202.上海工程技术大学管理学院，上海201620摘要：针对蚁群算法收敛速度慢，容易陷入局部最优的问题，提出了一种基于聚度的自适应动态混沌蚁群算法（A_ACS ）。

在迭代前期利用聚度来衡量解的多样性，自适应调节局部信息素分布，同时引入混沌算子来增加种群多样性，避免算法陷入局部最优，从而提高解的精度；在迭代后期去掉混沌算子，减少混沌扰动性，来提高算法的收敛速度。

将A_ACS 用于TSP 问题，仿真结果表明，该算法较ACS 和MMAS 算法减少了搜索时间，并且提高了解的质量，其平衡了多样性与收敛性之间的矛盾，整体性能优于其他两种算法。

关键词：蚁群算法；聚度；混沌优化；自适应信息素更新；旅行商问题文献标志码：A中图分类号：TP18doi ：10.3778/j.issn.1002-8331.1809-0316刘明霞，游晓明，刘升.基于聚度的自适应动态混沌蚁群算法.计算机工程与应用，2019，55（3）：15-22.LIU Mingxia,YOU Xiaoming,LIU Sheng.Adaptive dynamic chaotic ant colony algorithm based on degree of puter Engineering and Applications,2019,55（3）：15-22.Adaptive Dynamic Chaotic Ant Colony Algorithm Based on Degree of AggregationLIU Mingxia 1,YOU Xiaoming 1,LIU Sheng 21.College of Electronic and Electrical Engineering,Shanghai University of Engineering Science,Shanghai 201620,China2.College of Management,Shanghai University of Engineering Science,Shanghai 201620,ChinaAbstract ：Aiming at the problem that the ant colony algorithm is slow in convergence and easily falls into a local opti-mum,an adaptive dynamic chaotic ant colony algorithm （A_ACS ）based on the degree of aggregation is proposed.In the early iterations,the degree of aggregation is used to measure the diversity of solutions and self-adaptively adjust the local pheromone distribution,and chaos operators are introduced to increase the diversity of the population to avoid the algo-rithm falling into a local optimum,thereby improving the accuracy of the solution.In the later iterations,the chaotic operator is removed to reduce the chaotic disturbance and increase the convergence speed of the algorithm.The A_ACS is used for the TSP problem.The simulation results show that the proposed algorithm reduces the search time and improves the quality of solution compared with the ACS and MMAS algorithm.It balances the contradiction between diversity and conver-gence,and the overall performance is better than the other two algorithms.Key words ：ant colony algorithm;degree of aggregation;chaotic optimization method;adaptive method of updating pheromone;traveling salesman problem基金项目：国家自然科学基金（No.61673258，No.61075115，No.61403249，No.61603242）。

基于AdaBoost的人脸识别

For a given face representation, we are interested in how to do classiﬁcation. When the face database becomes large, some traditional classiﬁcation methods may deteriorate rapidly or may not be working any more. One solution is to change the original complex problem into a set of smaller and simpler ones to solve. For this consideration, we develop a pairwise classiﬁcation framework to solve the multi-class face recognition problem. The motivation of pairwise comparisons also comes from the character discrimination experiments [11], which demonstrate that features useful to distinguish letter ’E’ from ’F’ may differ from those distinguishing ’E’ from ’R’. The pairwise architecture is to decompose the complex face recognition problem on a large database, into a simple discrimination be-

重合器改进的可靠性分配

Recloser Allocation for Improved Reliability of DG-Enhanced Distribution Networks Aleksandar Pregelj,Member,IEEE,Miroslav Begovic´,Fellow,IEEE,and Ajeet Rohatgi,Fellow,IEEEAbstract—The radial distribution feeder protection strategy is ﬁrst presented in this paper without consideration for distributed generation(DG).Then,the addition of DG across the feeder(con-strained in terms of power and/or energy capacity)is introduced in the model.If islanded operation of these DG sources is allowed on a feeder subjected to a disturbance,DG may reduce the number of interruptions and/or durations for customers residing within their protection zones,thus increasing the reliability of service.To that end,a procedure forﬁnding optimal positions for DG and protec-tion devices is presented for a feeder equipped with capacity-con-strained distributed generators,using a custom-tailored genetic al-gorithm,and the improvement in reliability is demonstrated on a test feeder.Tuning of the genetic algorithm parameters and an adaptive algorithm that eliminates the need for parameter tuning are the subject of a separate paper by the same authors.Index Terms—Dispersed storage and generation,power distri-bution planning,reliability.I.I NTRODUCTIONT HIS PAPER addresses the effects caused by the addition of distributed generators(DGs)constrained in terms of power and/or energy capacity on a non-radial distribution feeder.In a typical radial distribution feeder,overcurrent relays are only expected to detect the unidirectionalﬂow of current.In a loop (non-radial)DG-enhanced feeder,powerﬂow is not unidirec-tional,and conventional protection logic must be altered for the fault-detecting devices to successfully perform their func-tion[1].A faulted branch may be energized from both sides, and several protection devices may need to operate in order to completely interrupt the fault current.Various protection strate-gies,using local or SCADA measurements,may be utilized. Distributed generation and storage units may reduce the impact of faults on customers within their protection zones by creating islands of supply,thus increasing the reliability of service.How-ever,these units may be power and/or energy limited and may include renewable generators,whose output is dependent on the meteorological conditions,and may not be able to serve their local loads at all times.Most reliability assessment tools utilize algorithms designed for radial networks,which may only determine beneﬁts of using DG as a backup source[2].In this paper,the reliability algo-rithm designed for the general case of non-radial networks is presented and examined.It incorporates the impact of capacity-Manuscript received January10,2006.Paper no.TPWRS-00879-2005. A.Pregelj is with Baltimore Gas&Electric,Baltimore,MD21244USA (e-mail:pregelj@).M.Begovic´and A.Rohatgi are with the School of Electrical and Computer Engineering,Georgia Institute of Technology,Atlanta,GA30332USA(e-mail: miroslav@;rohatgi@).Digital Object Identiﬁer10.1109/TPWRS.2006.876649constrained DG units into the calculation of reliability indexes. The following related optimization tasks are investigated.•Optimize the placement of protection devices for a given DG allocation.•Optimize DG placement for a given allocation of protec-tion devices.•Optimize the placement of both protection devices and DGs.The optimization is performed with regard to commonly used reliability indexes,such as system average interruption duration index(SAIDI),system average interruption frequency index (SAIFI),and momentary average interruption event frequency index(MAIFIe).A genetic algorithm(GA)is presented here to solve all three problems.First,the results for the recloser placement problem are presented,assuming the DGs are al-ready deployed on the feeder.Then,the incremental reliability beneﬁts of DGs are quantiﬁed by simultaneously optimizing the allocation of both DGs and protection devices.The problem of optimal GA parameter tuning is treated in a separate paper by the same authors.II.P ROBLEM D ESCRIPTIONThe majority of existing distribution feeders are radial, meaning that theﬂow of power is always from the substation transformer downstream to the individual customers.This approach allows relatively simple coordination between pro-tective devices and signiﬁcant cost savings.The reliability modeling of such a feeder is therefore relatively easy.For a fault anywhere on the feeder,only one recloser operates—typ-ically,the one closest to the fault,to minimize the number of affected customers.If,on the other hand,a single or multiple loops are present on the feeder,the alternate current paths may be possible for a fault in different parts of the feeder.Depending on the location, a fault may be energized from both sides,and operation of more than one protection device is often necessary.When distributed generators are present on the feeder,the number and duration of outages to some customers may be re-duced.After the fault is isolated,generators in parts of the feeder not affected by the fault may be allowed to reconnect,allowing the portions of the feeder to operate as electric“islands.”As an example,Fig.1shows a radial feeder,equipped with a sub-station breaker and two reclosers.Assuming there is no DG at the end of the feeder,a fault anywhere on the line will lead to the opening of theﬁrst recloser upstream from the fault.For ex-ample,after a fault between reclosers1and2,recloser1oper-ates,leaving all customers downstream from it without service. If DG is present at the end of the feeder,recloser2may also0885-8950/$20.00©2006IEEEFig.1.Strategically placed reclosers and DGs on a feeder may reduce the number of customers affected by a fault.operate,allowing the portion of the feeder downstream from re-closer 2to operate as an island as long as the power balance is achievable.Typically,utilities use standardized indexes such as SAIFI and SAIDI,which measure the average accumulated duration and frequency of sustained interruptions per customer [3],[4].The SAIFI and the SAIDI are de ﬁned as follows:SAIFI (1)SAIDI(2)whereis the number of interrupted customers for each inter-ruptionevent,is the total number of customers,and is the restoration time for each interruption event.As the importance of temporary faults increases,more utilities are starting to use the MAIFIe index,which measures the number of momentary interruptions per customer.The MAIFIe is de ﬁned asMAIFIe(3)whereis the number of interrupting device operations.The recloser placement can be optimized with respect to any of these indexes.To include the effects of both sustained and momentary interruptions,a composite index may be used,as de ﬁned in thefollowing:SAIFISAIFISAIFISAIDISAIDISAIDIMAIFIeMAIFIeMAIFIe(4)whereand are weights for indexes SAIFI,SAIDI,and MAIFIe,respectively,and thesubscriptindicates the target value.III.C URRENT R EGULATIONSThe interconnection of distributed resources (both generators and energy storage devices)with electric power systems is reg-ulated by the IEEE 1547standard [5].The interconnection of photovoltaic DG systems connected through static (solid-state)inverters has been regulated by the IEEE 929and UL 1741tech-nical standards [6],[7].Several U.S.states have adopted inter-connection requirements based on these standards.The IEEE 1547establishes universally needed criteria and requirements for interconnection of distributed resources (DR)with the aggregate capacity not higher than 10MV A,intercon-nected to the electric power systems at typical primary and/or secondary distribution voltages.IEEE 1547includes require-ments applicable to all DG technologies,including synchronous machines,induction machines,and systems connected through power inverters.Existing distribution systems were not designed to incorpo-rate generation and storage devices.The complete protection and control strategy is designed under the assumption that the only source of electric power is the substation transformer.The interconnection requirements are therefore designed to limit the possible negative effects that DG may have on the operation of the existing devices.Thus,islanded operation of parts of the feeder is currently not allowed by the utilities —after a fault,DG has to disconnect and remain disconnected until the fault is cleared.The sequence of events after the fault,if islanding is not allowed,is as follows.•DG is tripped,and the fault is detected and isolated by one or more protection devices.•After the fault is cleared,the recloser reconnects the area to the rest of the feeder.•DG reconnects after normal operating voltage and fre-quency are established,with the appropriate time delay.On the other hand,if DG operation can be synchronized with the operation of the existing feeder protection devices,DG may be able to remain online for the duration of the fault and reduce the number and/or duration of outages to some of the customers.Note,however,that the islanded operation requires signi ﬁcant coordination of distributed generators with feeder protection de-vices,which is outside the scope of this paper.The DG system generally relies on the utility to provide its phase reference,and a phase error between the island and utility voltages can develop while the part of the system (equipped with DG)is islanding.If the utility attempts to reclose on the out-of-phase section of the grid,large surge currents could damage the DG system and the local load.To be able to operate in the island mode,DGs have to be able to serve the island load and therefore keep both the voltage and frequency within acceptable ranges.The subsequent work in this section quanti ﬁes the reliability bene ﬁts achieved by intentional islanding,which may be used to justify the nec-essary additional investments required to allow such operation.If the islanded operation is allowed,the sequence of events after the fault should be as follows.•DG is tripped,and the fault is detected and isolated by one or more protection devices.•DG reconnects,if it is not within the faulted zone.•After the fault is cleared,reclosers synchronize their re-closing operation with DG.The contribution to the fault current of a DG system con-nected to the utility using the solid-state inverter is typically very small.Inverters have no “inertia ”in their output and may respond immediately to the changes in the power system ’s op-erating conditions.The response time may be a fraction of the utility cycle due to the high frequency(1kHz)switching em-ployed by such inverters.Instead of detecting abnormal condi-tions by detecting large transient currents,they generally sensea short circuit by detecting the change in voltage (either magni-tude or frequency).Therefore,the fault currents ﬂowing through a particular recloser may not be large enough to trip it,and al-ternate methods for fault detection may be necessary.The fol-lowing study assumes that these methods are available and that for a fault anywhere on the feeder,the reclosers are coordinated in such a way that only the minimal number of neighboring reclosers operate and isolate the fault.The distributed gener-ation and storage systems deployed on the feeder are limited in power and energy capacity,which also limits their reliability contributions.Capacity-constrained generators are able to supply the load up to their rated power as long as the fuel is supplied.If the power output of the generator is suf ﬁcient to supply the local island load,the number and duration of faults for those loads will decrease.Both the SAIFI and SAIDI indexes will there-fore be lower,compared to the base case without DG.Energy limited generators (and storage systems)are able to sustain the load for a ﬁnite period.If a fault is cleared before the energy runs out,both SAIDI and SAIFI will be reduced.However,if the energy runs out before a fault is cleared,only SAIDI will be reduced.Renewable distributed generators,such as photo-voltaic and wind systems,create additional dif ﬁculties in quan-tifying their reliability bene ﬁts,due to the probabilistic nature of their input (sunshine,wind)and their power output.If an energy storage system is used in addition to renewable DG,the dura-tion of time and the load that the storage system can support in an island may increase due to renewable DG generation.How-ever,if renewable DGs with random energy input are the only distributed devices in an island,they may not be relied upon to support the local island load.The developed reliability model is capable of treating all three types of generators and may be used to quantify their impact on standardized reliability indexes.IV .C ALCULATING THE C OMPOSITE R ELIABILITYI NDEX OF A DG-E NHANCED F EEDERThis section details the calculations of the composite relia-bility index (4),which will be later used as a cost function in the optimization algorithm.It is assumed that only the minimal number of reclosers operate per fault,isolating the smallest pos-sible part of the feeder.The actual recloser control logic is out-side the scope of this paper.The protection devices placed on the feeder effectively di-vide the feeder into so-called reliability zones.In a radial feeder,placementof devices will result in the formationofzones.If the feeder is not radial,the number of zones may de-crease.For example,Fig.2shows the formation of ﬁve relia-bility zones after the allocation of ﬁve reclosers on a non-radial distribution feeder.For a fault anywhere within the zone,all customers within the zone will be disconnected,since there are no protection de-vices between them.Similarly,for a fault outside the zone,all customers may be either connected (if still connected to the sub-station or (if the zone operates as an island)disconnected.Con-sequently,all customers within the same reliability zone experi-ence the same number of outages and have the same reliability ofservice.Fig.2.Distribution feeder equipped with six capacity-constrained distributed generators and ﬁve reclosers and the corresponding reliability zones de ﬁned by dashed lines.Note that in general,reliability zones for different phases do not coincide,as some of the protection devices may be single-phase reclosers or fuses not present in other phases.The pro-cedure for classifying all feeder buses,branches,and genera-tors in the appropriate zones is relatively straightforward and is detailed in [8].Assuming the appropriate fault incidence rates (FIR)for all feeder branches and the fraction of permanent faults (FPF),the total annual number of permanent and momentary faults per each zone may be calculated.After a fault in a particular zone,all buses in that zone experi-ence an interruption.For each of the other zones,the maximum output of zone generators is compared with the load duration curve for all zone loads,and the number of permanent faults is reduced by the percentage of time that the zone generation exceeds zone load.Fig.3shows the load duration curve of a re-liability zone,and the maximum active power generation of its distributed generators.In this example,the zone generation (1MW)is higher than its load for slightly more than 50%of the time,indicating that for more than the half of the failures out-side this zone,its customers would not be disconnected.Let the load active power pro ﬁle be a discretefunction,where(5)Fig.3.Zone load duration curve and the maximum generation of distributed generators for a particular reliability zone.and is the interval between measurements.The percentage of time that DGs can supply the island is thensimplylength(6)where is the total island generation.In the case that all generators in the zone are energy limited,a check is performed to determine if they can maintain production for the duration of the fault.This is approximated by comparing their energy capacity with the energy curve,calculated as a run-ning sum of load power (obtained from the load pro ﬁle data)for the duration of the fault.Again,the number of permanent faults is reduced by the percentage of time that the energy capacity of DGs exceeds the energy curve.Therefore,for a fault withduration ,the energy curve will be a discretefunction,de ﬁnedas(7)whereis obtainedbyround (8)Finally,after the total numbers of interruptions and their du-rations for all zones are obtained,the composite reliability index is calculated according to (4).V .G ENETIC A LGORITHM FOR THE O PTIMAL A LLOCATION OF DG S AND P ROTECTION D EVICES IN A N ON -R ADIAL F EEDER The composite reliabilityindex depends strongly on the locations of protection devices and distributed generators.A re-location or addition of a single protection device changes the con ﬁguration of reliability zones,and the complete procedureexplained in the previous section needs to be repeated.The gen-eral optimization problem may thus be formulated as follows:For a given number of protectiondevices and distributed gen-erators ,obtain the following:—locations of protectiondevices:;—locations ofDGs:such that the composite reliabilityindex is minimized,sub-ject to normal operating constraints,where(9)(10)and•number of three-phase reclosers;•number of single-phase reclosers;•number of three-phase fuses;•number of single-phase fuses;•number of three-phase generators;•number of single-phase generators.The simultaneous optimization of both protection devices and DGs ’locations may be performed in the planning stage of the feeder design.Typically,either DGs or protection devices are al-ready present on the feeder,and the optimization problem scales back to ﬁnding the optimal locations of the other devices.This can be obtained by making the appropriatevectoror con-stant.Incremental studies,where the addition of a new DG or a protection device on a feeder already equipped with DGs and reclosers are considered,may be performed by making the ap-propriate elements ofvectorsand constant.The obvious problem is the choice of the optimization algo-rithm,due to the nature of the objective function (composite reli-ability index).Its inputs may only be discrete values indicating the types and locations of devices to be placed on the feeder.Furthermore,the objective function is neither continuous nor differentiable,with multiple local extreme points.Clearly,con-ventional gradient-based algorithms may not be used.Evolu-tionary algorithms,on the other hand,may overcome all of the problems mentioned above,as they base their decision solely on the value of the objective function.In this paper,a genetic algorithm is proposed,based on the algorithm presented in [9],[10]and used to solve all three optimization problems presented above.A GA searches the parameter space by mimicking the natural principles of reproductive evolution [11]–[13].It is a directed search algorithm,in which the search is performed on the set of all possible solutions to a problem.GA is capable of working with discrete data types and does not need any gradient infor-mation.Starting from an initial population of solutions,it ef-fectively implements the “survival of the ﬁttest ”strategy.Fitter solutions,with higher values of the objective function,are more likely to reproduce and/or survive to the next generation,thus improving the overall population.The population evolves using two genetic operators:mutation and crossover.Various tech-niques exist for selecting the solutions that will continue on to the next generation and/or be chosen for mutation and crossover.The GA terminates either after a pre-speci ﬁed number of gen-erations or after the population converges to a single solution.However,there is no evidence to support a claim that the goal of evolution,and thus the goal of a genetic algorithm,is to pro-duce the best possible solution.Therefore,in general,no theo-retical proof of global convergence exists for genetic algorithms.This problem is typically tackled by repeating the GA run with different (random)initial conditions.VI.P ARAMETERS OF A G ENETIC A LGORITHMIn the proposed application,the goal is to obtain the posi-tions of individual devices on the feeder,where these devices may be placed on a limited number of buses or branches.The locations where protection devices may be placed may be coded using discrete numbers between 1and ,where is the number of possible branches in the system where protection devices may be placed.Similarly,possible DG locations may be coded using discrete numbers ranging from 1to ,with being the number of possible buses where DGs may be placed.Although these values may also be coded using a binary representation,it is more natural and ef ﬁcient to use the discrete representation.Thus,let a single solution be de ﬁned as a speci ﬁc allocation of individual DGs and protection devices on the feeder,i.e.,the thsolution in the population isa-dimensional row vector of discretenumbers:(11)where is the number of possible branches in the system where reclosers may be placed (branches are numbered consequently from 1to),is the number of possible candidate buses forthe allocation ofDGs,is the number of protection devices to be placed on the feeder,andis the number of DGs.The population is then de ﬁned simply as a group of solutions,i.e.,(12)where is the population size.For example,for the case withonly two protection devices(),two distributed generators(),and with population consisting of ﬁve solutions(),the population isa matrix and may looklike(13)where the ﬁrst solution represents the situation with protection devices at buses 2and 28,and generators at buses 31and 44,etc.As the objective function is the composite reliability index,the vector of solutions,representing the “ﬁtness ”of solutions in the population,alsohaselements(14)Although the optimal convergence is not guaranteed,typi-cally,GAs perform very well for a variety of applications.How-ever,there are several parameters that may signi ﬁcantly affectthe convergence properties of the genetic algorithm and are typi-cally tuned in an application-speci ﬁc way.The crossover proba-bility determines the rate at which new solutions are introduced into the population.A large crossover probability yields more new solutions at each generation but may also lead to the van-ishing of ﬁt solutions before their genetic material has been ex-ploited.On the other hand,a small crossover probability may not introduce suf ﬁcient changes,and the algorithm may settle in a local minimum.The probability of mutation allows the algorithm to avoid set-ting in a local minimum by introducing random genetic changes into the current population.A low mutation probability may lead to a premature convergence to a local minimum.Conversely,too high of a mutation rate may negate the bene ﬁts of a directed search obtained by the crossover operator and may transform the algorithm into a general random search algorithm.Population size also signi ﬁcantly in ﬂuences the overall op-eration of the genetic algorithm.A small population size may not provide a suf ﬁcient sample size over the space of the so-lutions and may not retain genetic variety as the algorithm pro-gresses from generation to generation.On the other hand,a large population may require a prohibitively large number of func-tion evaluations,slowing down the algorithm.The population size should therefore be commensurate to the complexity of the problem,i.e.,it should depend on the size of the search space spanned by the optimization problem.The maximum number of generations determines the termi-nation criterion of the algorithm and should also be carefully chosen.Setting the maximum number of generations to an ex-tremely high value may slow the algorithm considerably.Natu-rally,setting it to a too low value may not allow the algorithm to search the entire parameter space and could lead to suboptimal results.The in ﬂuence of these parameters on the algorithm conver-gence properties is investigated and presented in terms of com-parative analysis in a companion paper [14].In addition,an adaptive genetic algorithm is proposed that eliminates the need for tuning mutation and crossover probabilities.VII.T EST R ESULTSTo validate the algorithm,the optimal recloser positions were ﬁrst located for three radial distribution feeders,obtained from a U.S.utility.These feeders represent three common types of distribution feeders:urban (high customer density,relatively small length),suburban (longer length,smaller density),and rural (widespread con ﬁguration,low and uneven customer den-sity).The results were con ﬁrmed using a commercial grade re-liability analysis program.As expected,the addition of protec-tion devices increases overall feeder reliability,decreasing the composite reliability index.The composite reliability index cal-culated for several protection philosophies is shown in Table I.The performance of the genetic algorithm is further demon-strated on the 69-segment,eight-lateral three-phase distribution feeder,based on [15].Total nominal feeder load is 3.8MW,and the load duration curve is also obtained from the utility data.AllTABLE IV ALUES OF C OMPOSITE R ELIABILITY I NDEX FOR T HREE D ISTRIBUTIONF EEDERS FOR V ARIOUS P ROTECTION PHILOSOPHIES TABLE IIP ERFORMANCE OF THE G A :P ERCENTAGE OF S UCCESSFUL R UNS AND THE A VERAGE N UMBER OF G ENERATIONS N EEDED TO O BTAIN THE SOLUTIONbranches and loads at the feeder are assumed to be three-phase.The feeder is not radial,with capacity constrained DGs scattered throughout the feeder.The considered ranges for the probabilities of crossover and mutation are 50%–90%and 10%–70%,respectively.Initial re-sults showed that high mutation rates may signi ﬁcantly improve the performance of the algorithm,most likely due to the large number of local extrema,which is why extremely high muta-tion rates (up to 70%)were considered.To determine the effec-tiveness of the algorithm,for each combinationofand ,the GA run was repeated 20times with different initial popula-tions,generated at random.The percentage of successful runs (in which the optimal solution is found)was recorded,as well as the average number of generations needed to obtain the op-timal solution.If a solution is not found in 100generations,a run is deemed unsuccessful.The population size is ﬁxed to 100solutions.Table II summarizes results obtained for the case when ﬁve reclosers were optimally placed on the feeder.Similar results were obtained for different number of reclosers.Note that the primary criterion that describes the operation of the algorithmis its success rate —aandcombination that yields a lower success rate but converges faster (in fewer generations)is infe-rior to the combination that produces a higher success rate.The performance of the algorithm is generally very good,and the algorithm is typically able to obtain the optimal solution.The presented results do not reveal thebestand combi-nation for each feeder but may be used to determine the rangesforand that yield satisfactory convergence properties of the algorithm.To avoid the problems associated with obtainingoptimaland,an adaptive algorithm,based on [16],is also developed.Increase in mutationprobabilityincreases the success ratio of the algorithm.It also decreases the number of gener-ations needed to reach the solution.Although Table II suggests that the algorithm performs best with extremely high mutation rates,these rates may prevent the convergence of the algorithm.Therefore,high mutation rates should only be used if the number of generations is set in advance.TABLE IIIC OMPOSITE I NDEX FOR V ARIOUS DG S IZES AND R ECLOSER P LACEMENT STRATEGIESThe previous analysis shows some of the dif ﬁculties in deter-mining the optimal parameters that yield a satisfactory combina-tion of convergence properties and provide a reasonable degree of certainty that the obtained solution is optimal.As there are no guarantees that the speci ﬁcally tuned GA will perform satis-factorily for a large varieties of feeder con ﬁgurations and sizes,the following conclusions are provided that may be helpful in determining the algorithm parameters for a speci ﬁc case.•The nature of the problem indicates that the algorithms per-form better for relatively high probabilities of crossover and mutation.The suggested ranges for the probability of crossover and mutation are 70%–90%,and 30%–50%,re-spectively,based on simulations presented in this paper.•The performance of the adaptive algorithm is very good,with convergence properties comparable to the results ob-tained with the best combinationofand ,although with a slightly lower success rate.•The algorithm may produce suboptimal solutions if it is ter-minated prematurely.To avoid premature termination,the algorithm should be either allowed to proceed for a large number of generations,or the relationship between the best and the average ﬁtness in the population should be moni-tored,and its convergence should be used as a sign that the solution has been reached.A high probability of mutation should not be used if the latter technique is employed.•As GA may produce suboptimal results,it should be tested multiple times,with different initial conditions.The suc-cess ratio and the number of generations needed to reach the solution may be used as the pointers for suitability of applied parameters.The following ﬁgures and tables attest to the performance of the algorithm.First,we look at the feeder with six DGs located as shown in Fig.2.Table III shows the best reliability indexes obtained when up to ﬁve reclosers are optimally placed on the feeder.The size of each DG was scaled from 0to 1MW,to show incremental reliability bene ﬁts obtained by islanded operation in parts of the feeder.Note that the optimal recloser positions depend on both the number of reclosers and the size of DGs.Finally,the algorithm was used to simultaneously optimize the locations of both reclosers and DGs,and the results are sum-marized in Table IV.As expected,the reliability index decreases even further.The bus numbering used to show DG and recloser positions in Table IV is consistent with bus numbering in Fig.2.Fig.4shows qualitatively the improvement in composite re-liabilityindex ,as the number of reclosers and sizes of DGs increase.Asexpected,is a non-increasing function of both。

纹理物体缺陷的视觉检测算法研究--优秀毕业论文

摘要
在竞争激烈的工业自动化生产过程中，机器视觉对产品质量的把关起着举足轻重的作用，机器视觉在缺陷检测技术方面的应用也逐渐普遍起来。与常规的检测技术相比，自动化的视觉检测系统更加经济、快捷、高效与安全。纹理物体在工业生产中广泛存在，像用于半导体装配和封装底板和发光二极管，现代化电子系统中的印制电路板，以及纺织行业中的布匹和织物等都可认为是含有纹理特征的物体。本论文主要致力于纹理物体的缺陷检测技术研究，为纹理物体的自动化检测提供高效而可靠的检测算法。纹理是描述图像内容的重要特征，纹理分析也已经被成功的应用与纹理分割和纹理分类当中。本研究提出了一种基于纹理分析技术和参考比较方式的缺陷检测算法。这种算法能容忍物体变形引起的图像配准误差，对纹理的影响也具有鲁棒性。本算法旨在为检测出的缺陷区域提供丰富而重要的物理意义，如缺陷区域的大小、形状、亮度对比度及空间分布等。同时，在参考图像可行的情况下，本算法可用于同质纹理物体和非同质纹理物体的检测，对非纹理物体的检测也可取得不错的效果。在整个检测过程中，我们采用了可调控金字塔的纹理分析和重构技术。与传统的小波纹理分析技术不同，我们在小波域中加入处理物体变形和纹理影响的容忍度控制算法，来实现容忍物体变形和对纹理影响鲁棒的目的。最后可调控金字塔的重构保证了缺陷区域物理意义恢复的准确性。实验阶段，我们检测了一系列具有实际应用价值的图像。实验结果表明本文提出的纹理物体缺陷检测算法具有高效性和易于实现性。关键字: 缺陷检测；纹理；物体变形；可调控金字塔；重构
Keywords: defect detection, texture, object distortion, steerable pyramid, reconstruction
II

International Journal of Approximate Reasoning

Dynamic importance sampling inBayesian networks based on probability treesq Serafı´n Moral a ,Antonio Salmero ´n b,*a Department Computer Science and Artiﬁcial Intelligence,University of Granada,Avda.Andalucı´a 38,18071Granada,Spain b Department Statistics and Applied Mathematics,University of Almerı´a,La Can ˜ada de San Urbano s/n,04120Almerı´a,Spain Received 1February 2004;received in revised form 1April 2004;accepted 1May 2004Available online 17September 2004AbstractIn this paper we introduce a new dynamic importance sampling propagation algorithm for Bayesian networks.Importance sampling is based on using an auxiliary sampling distribution from which a set of conﬁgurations of the variables in the network is drawn,and the perform-ance of the algorithm depends on the variance of the weights associated with the simulated conﬁgurations.The basic idea of dynamic importance sampling is to use the simulation of a conﬁguration to modify the sampling distribution in order to improve its quality and so reduc-ing the variance of the future weights.The paper shows that this can be achieved with a low computational eﬀort.The experiments carried out show that the ﬁnal results can be very good even in the case that the initial sampling distribution is far away from the optimum.Ó2004Elsevier Inc.All rights reserved.Keywords:Bayesian networks;Probability propagation;Approximate algorithms;Importance sampling;Probability trees0888-613X/$-see front matter Ó2004Elsevier Inc.All rights reserved.doi:10.1016/j.ijar.2004.05.005qThis work has been supported by the Spanish Ministry of Science and Technology,project Elvira II (TIC2001-2973-C05-01and 02).*Corresponding author.Tel.:+34950015669;fax:+34950015167.E-mail addresses:smc@decsai.ugr.es (S.Moral),antonio.salmeron@ual.es (A.Salmero´n).International Journal of Approximate Reasoning38(2005)245–261/locate/ijar246S.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–2611.IntroductionIn this paper we propose a new propagation algorithm for computing marginal conditional probabilities in Bayesian networks.It is well known that this problem is NP-hard even if only approximate values are required[7].It means that it is always possible toﬁnd examples in which polynomial approximate algorithms pro-vide poor results,especially if the distributions contain extreme probabilities:there is a polynomial approximate algorithm if all the probabilities are strictly greater than zero[8],but its performance quickly deteriorates when the probabilities approach to zero.There exist several deterministic approximate algorithms[1–5,13,16,20,21]as well as algorithms based on Monte Carlo simulation.The two main approaches are: Gibbs sampling[12,15]and importance sampling[6,8,10,11,18,19,22].A class of these simulation procedures is composed by the importance sampling algorithms based on approximate pre-computation[11,18,19].These methods per-formﬁrst a fast but non-exact propagation,consisting of a node removal process [23].In this way,an approximateÔa posterioriÕdistribution is obtained.In the second stage a sample is drawn using the approximate distribution and the probabilities are estimated according to the importance sampling methodology[17].In this paper we start oﬀwith the algorithm based on approximate pre-computa-tion developed in[18].One of the particularities of that algorithm is the use of prob-ability trees to represent and approximate probabilistic potentials.Probability trees have the ability of approximating in an asymmetrical way,concentrating more re-sources(more branching)where they are more necessary:higher values with more variability(see[18]for a deeper discussion on these issues).However,as pointed out in[5],one of the problems of the approximate algorithms in Bayesian networks is that sometimes theﬁnal quality of an approximate potential will depend on all the potentials,including those which are not needed to remove the variable when per-forming exact propagation.Imagine that weﬁnd that,after deleting variable Z, the result is a potential that depends on variable X,and weﬁnd that this dependence is meaningful(i.e.the values of the potential are high and diﬀerent for the diﬀerent cases of X).If there is another potential not considered at this stage,in which all the cases of X except one have assigned a probability equal to zero,then the discrimina-tion on X we have done when deleting Z is completely useless,sinceﬁnally only one value of X will be possible.This is an extreme situation,but it illustrates that even if the approximation is carried out locally,the quality of theﬁnal result will depend on the global factors.There are algorithms that take into account this fact,as Markov Chain Monte Carlo,the Penniless propagation method presented in[5],and the Adaptive Importance Sampling(AIS-BN)given in[6].In this work,we improve the algorithm proposed in[18]allowing to modify the approximate potentials(the sampling distribution)taking as basis the samples ob-tained during the simulation.If samples with very small weights are drawn,the algo-rithm detects the part of the sampling distribution(which is represented as an approximate probability tree)that is responsible of this fact,and it is updated in such a way that the same problem will not occur in the next simulations.Actually,this is away of using the samples to obtain the necessary information to improve the quality of the approximations taking into account other potentials in the problem.Trees are very appropriate for this task,as they allow to concentrate more eﬀorts in the most necessary parts,i.e.in the conﬁgurations that were more frequently obtained in past simulations and for which the approximation was not good.The rest of the paper is organised as follows:in Section2it is described how prob-ability propagation can be carried out using the importance sampling technique.The new algorithm,called dynamic importance sampling,is described in Section3.In Sec-tion4the performance of the new algorithm is evaluated according to the results of some experiments carried out in large networks with very poor initial approxima-tions.The paper ends with conclusions in Section5.2.Importance sampling in Bayesian networksThroughout this paper,we will consider a Bayesian network in which X={X1,...,X n}is the set of variables and each variable X i takes values on aﬁnite set X i.If I is a set of indices,we will write X I for the set{X i j i2I},and X I will denote the Cartesian product·i2I X i.Given x2X I and J I,x J is the element of X J ob-tained from x by dropping the coordinates not in J.A potential f deﬁned on X I is a mapping f:X I!Rþ0,where Rþis the set of non-negative real numbers.Probabilistic information will always be represented by means of potentials,as in[14].The set of indices of the variables on which a potential f is deﬁned will be denoted as dom(f).The conditional distribution of each variable X i,i=1,...,n,given its parents in the network,X pa(i),is denoted by a potential p i(x i j x pa(i))for all x i2X i and x pa(i)2 X pa(i).If N={1,...,n},the joint probability distribution for the n-dimensional ran-dom variable X can be expressed aspðxÞ¼Yi2N piðx i j x paðiÞÞ8x2X N:ð1ÞAn observation is the knowledge about the exact value X i=e i of a variable.The set of observations will be denoted by e,and called the evidence set.E will be the set of indi-ces of the variables observed.The goal of probability propagation is to calculate theÔa posterioriÕprobabilityfunction pðx0k j eÞ,for all x0k2X k,for every non-observed variable X k,k2N n E.No-tice thatpðx0k j eÞ¼pðx0k;eÞ8x0k2X kand,since pðeÞ¼Px0k2X kpðx0k;eÞ,we can calculate the posterior probability if we com-pute the value pðx0k ;eÞfor every x0k2X k and normalise afterwards.Let H={p i(x i j x pa(i))j i=1,...,n}be the set of conditional potentials.Then,pðx0k ;eÞcan be expressed asS.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–261247pðx0k ;eÞ¼Xx2X Nx E¼ex k¼x0kYi2Npiðx i j x paðiÞÞ¼Xx2X Nx E¼ex k¼x0kYf2Hfðx domðfÞÞ8x0k2X k:ð2ÞIf the observations are incorporated by restricting potentials in H to the observed values,i.e.by transforming each potential f2H into a potential f e deﬁned on domðfÞn E as f e(x)=f(y),where ydomðfÞn E¼x,and y i=e i,for all i2E,then we have,pðx0k ;eÞ¼Xx2X Nx k¼x0kYf e2Hf eðx domðfeÞÞ¼Xx2X NgðxÞ8x0k2X k;ð3ÞwheregðxÞ¼Qf e2Hf eðx domðfeÞÞif x k¼x0k; 0otherwise:Thus,probability propagation conveys the estimation of the value of the sum in(3), and here is where the importance sampling technique is used.Importance sampling is well known as a variance reduction technique for estimating integrals by means of Monte Carlo methods(see,for instance,[17]),consisting of transform-ing the sum in(3)into an expected value that can be estimated as a sample mean. To achieve this,consider a probability function pÃ:X N![0,1],verifying that pÃ(x)>0for every point x2X N such that g(x)>0.Then formula(3)can be written aspðx0k ;eÞ¼Xx2X N;gðxÞ>0gðxÞpÃðxÞpÃðxÞ¼EgðXÃÞpÃðXÃÞ8x0k2X k;ð4Þwhere XÃis a random variable with distribution pÃ(from now on,pÃwill be called thesampling distribution).Then,if f xðjÞg mj¼1is a sample of size m drawn from pÃ,for eachx0k2X k,^pðx0k ;eÞ¼1mX mj¼1gðxðjÞÞpÃðxðjÞÞð5Þis an unbiased estimator of pðx0k;eÞwith varianceVarð^pðx0k ;eÞÞ¼1mXx2X Ng2ðxÞpÃðxÞ!Àp2ðx0k;eÞ!:ð6ÞThe value w j=g(x(j))/pÃ(x(j))is called the weight of conﬁguration x(j).Minimising the error of an unbiased estimator is equivalent to minimising its var-iance.As formulated above,importance sampling requires a diﬀerent sample to esti-mate each one of the values x0k of X k.However,in[18]it was shown that it is possibleto use a single sample(i.e.a single set of conﬁgurations of the variables X N n E)toestimate the probability for all the values x0k .In such case,the minimum variance isreached when the sampling distribution,pÃ(x),is proportional to g(x).In such case, 248S.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–261the weights are equal to p (e )for all the conﬁgurations and the variance of the esti-mation of the conditional probability for each x 0k 2X k is:Var ð^p ðx 0k j e ÞÞ¼1mðp ðx 0k j e Þð1Àp ðx 0k j e ÞÞ:This provides very good estimations depending on the value of m (analogously to the estimation of binomial probabilities from a sample),but it has the diﬃculty that it is necessary to handle p (x j e ),the distribution for which we want to compute the mar-ginals.Thus,in practical situations the best we can do is to obtain a sampling distri-bution as close as possible to the optimal one.Once p Ãis selected,p ðx 0k ;e Þfor each value x 0k of each variable X k ,k 2N n E can be estimated with the following algorithm:Importance Sampling(1)For j :=1to m (sample size)(a)Generate a conﬁguration x (j )2X N using p Ã.(b)Calculate the weight:w j :¼Q f 2H f e ðx ðj Þdom ðf e ÞÞp Ãðx ðj ÞÞ:ð7Þ(2)For each x 0k 2X k ,k 2N n E ,compute ^pðx 0k ;e Þas the sum of the weights in for-mula (7)corresponding to conﬁgurations containing x 0k divided by m .(3)Normalise the values ^p ðx 0k ;e Þin order to obtain ^p ðx 0k j e Þ.The sampling distribution for each variable can be obtained through a process of eliminating variables in the set of potentials H .An elimination order r is considered and variables are deleted according to such order:X r (1),...,X r (n ).The deletion of a variable X r (i )consists of marginalising it out from the combina-tion of all the functions in H which are deﬁned for that variable.More precisely,the steps are as follows:•Let H r (i )={f 2H j r (i )2dom(f )}.•Calculate f r ði Þ¼Q f 2H r ði Þf and f 0r ði Þdeﬁned on dom(f r (i ))n {r (i )},by f 0r ði Þðy Þ¼P x r ði Þf r ði Þðy ;x r ði ÞÞfor all y 2dom(f r (i ))n {r (i )},x r (i )2X r (i ).•Transform H into H n H r ði Þ[f f 0r ði Þg .Simulation is carried out in an order contrary to the one in which variables are deleted.To obtain a value for X r (i ),we will use the function f r (i )obtained in the dele-tion of this variable.This potential is deﬁned for the values of variable X r (i )and other variables already sampled.The potential f r (i )is restricted to the already ob-tained values of variables in dom(f r (i ))n {r (i )},giving rise to a function which depends only on X r (i ).Finally,a value for this variable is obtained with probability propor-tional to the values of this potential.If all the computations are exact,it was proved in [11]that we are really sampling with the optimal probability p Ã(x )=p (x j e ).S.Moral,A.Salmero ´n /Internat.J.Approx.Reason.38(2005)245–261249However,the result of the combinations in the process of obtaining the sampling distributions may require a large amount of space to be stored,and therefore approximations are usually employed,either using probability tables[11]or proba-bility trees[18]to represent the distributions.Instead of computing the exact poten-tials we calculate approximate ones with much fewer values.Then the deletion algorithm is faster and the potentials need less space.The price to pay is that the sampling distribution is not the optimal one and the accuracy of the estimations will depend on the quality of the approximations.The way in which a probabilistic potential can be approximated by a probability tree is illustrated in1.In[11]an alternative procedure to compute the sampling distribution was used. Instead of restricting f r(i)to the values of the variables already sampled,all the func-tions in H r(i)are restricted,resulting in a set of functions depending only on X r(i). The sampling distribution is then computed by multiplying all these functions.If the computations are exact,then both distributions are the same,as restriction and combination commute.When the combinations are not exact,generally the op-tion of restricting f r(i)is faster and the restriction of functions in H r(i)is more accu-rate,as there is no need to approximate the result of the combination of functions depending only on one variable,X r(i).3.Dynamic importance samplingDynamic importance sampling follows the same general structure as our previous importance sampling algorithms but with the diﬀerence that sampling distributions can change each time a new conﬁguration x(j)is simulated.The algorithm follows the option of restricting the functions in H r(i)before combining them when computing the sampling distribution for X r(i).Any conﬁguration of valuesðxðjÞrð1Þ;...;xðjÞrðnÞÞ,is simulated in reverse order,as in theoriginal importance sampling algorithm:Starting with xðjÞrðnÞandﬁnishing with xðjÞrð1Þ.Assume that we have already simulated the values c j i¼ðxðjÞrðnÞ;...;xðjÞrðiþ1ÞÞand thatwe are going to simulate a value xðjÞrðiÞfor X r(i).Let us denote by fc jithe result ofrestricting potential f to the values of c j i,and let f0rðiÞbe the function that was com-puted when removing variable X r(i)in the elimination algorithm(i.e.the result ofsumming the combination of the potentials containing X r(i)over all the possible val-ues of that variable).The procedure to simulate xðjÞrðiÞmakes some additional computations in order toassess the quality of the sampling distribution.More precisely the following elements are computed:•ðH rðiÞÞc ji ¼f fc jij f2H rðiÞg:The result of restricting all the functions in H r(i)to thevalues already simulated.•q r(i):The result of the combination of all the functions inðH rðiÞÞc ji .This functioncan be represented as a vector depending only on variable X r(i).•xðjÞrðiÞ:The simulated value for X r(i)which is obtained by drawing a value with aprobability proportional to the values of vector q r(i).b rðiÞ¼Px rðiÞqrðiÞðx rðiÞÞ:Thenormalisation value of vector q r(i).•a r(i):The value of potential f0rðiÞwhen instantiated for the cases in c j i.The dynamic algorithm we propose is based on the next theorem,which states that,if no approximations have been made,then b r(i)must be equal to a r(i).Theorem1.Let a r(i)and b r(i)be as deﬁned above.If during the elimination process all the trees have been computed exactly(i.e.none of them has been pruned),then it holds thata rðiÞ¼b rðiÞ:Proof.b r(i)is obtained by restricting the potentials in H r(i)to c j i¼ðxðjÞrðnÞ;...;xðjÞrðiþ1ÞÞ,combining them afterwards,and summing out the variable X r(i).On the other hand,a r(i)is the result of combining the potentials in H r(i),summing out X r(i)from the combined potential,and restricting the result to c j i.f0 rðiÞis computed by combining the potentials in H r(i)and then summing out X r(i).It means that the computations of a r(i)and b r(i)involve the same operations but in a different order:The restriction to conﬁguration c j i is done at the beginning for b r(i) and at the end for a r(i).Nevertheless,if all the computations are exact the results should be the same,since combination and restriction trivially commute for exact trees.hHowever,combination and restriction do not commute if the potentials involved have been previously pruned,since one of the pruned values may correspond to con-ﬁguration c j i.b r(i)is the correct value,since in this case the restriction is evaluated before com-bining the potentials,and thus,no approximation is made when computing it.Whilst,a r(i)is the value that can be found in potential f0rðiÞ,which is combined,and eventually pruned,before being evaluated for c j i.Potential f0rðiÞis the one thathas been used to compute the sampling probabilities of variables XðjÞrðnÞ;...;XðjÞrðiþ1Þ.Therefore,if b r(i)and a r(i)are very diﬀerent,it means that conﬁguration c j i has been S.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–261251drawn with a probability of occurrence far away from its actual value.The worst sit-uation is met when a r(i)is much greater than b r(i).For example,assume an extreme scenario in which b r(i)is equal to zero and a r(i)is large.Then we would be obtaining, with high probability,a conﬁguration that should never be drawn(its real probabil-ity is zero).1This fact would produce negative consequences,because the weights of all these conﬁgurations would be zero and therefore they would be completely useless.If instead of zero values,the exact probability were very small,there would be a similar scenario,but now the weights would be very small,and the real impact of these conﬁgurations in theﬁnal estimation would not be signiﬁcant.Summing up, we would be doing a lot of work with very little reward.Dynamic importance sampling computes the minimum of the values a r(i)/b r(i)and b r(i)/a r(i),considering that this minimum is equal to one if a r(i)=0.If this value is lessthan a given threshold,then potential f0rðiÞis updated to the exact value b r(i)for thegiven conﬁguration c j i¼ðxðjÞrðnÞ;...;xðjÞrðiþ1ÞÞ.This potential will be used in the nextsimulations,and thus c j i will be drawn with a more accurate probability in the future. If,for example,b r(i)is zero,it will be impossible to obtain it again.Updating the potential does not simply mean to change the value a r(i)by the new value b r(i).The reason is that we should do it only for conﬁguration c j i and a single value on a tree aﬀects to more than one conﬁguration(if the branch corresponding to that conﬁguration has been pruned and some variables do not appear)and then we may be changing the values of other conﬁgurations diﬀerent to c j i.If b r(i)=0,we could even introduce zeros where the real exact value is positive,thus violating the basic property of importance sampling which says that any possible conﬁguration must have a chance to be drawn.For instance,assume that the branches in a tree corresponding to conﬁgurations c1and c2lead to leaves labeled with numbers0 and0.1respectively.Now consider that the tree is pruned replacing both branches by a single number,for instance,0.05.In this case,if during the simulation it is found out that conﬁguration c1should be labeled with0,if we just replaced the value0.05 by0we would be introducing a false zero for conﬁguration c2.In order to avoid the insertion of false zeroes,we must branch the tree represent-ing f0rðiÞin such a way that we do not change its value for conﬁgurations for whichb r(i)is not necessarily the actual value.Therefore,the basic problem is to determine a subset of variables{X r(n),...,X r(i+1)},for which we have to branch the node of thetree associated with f0rðiÞso that only those leaves corresponding to the values of thesevariables in c j i are changed to the new value.Theﬁrst step is to consider the subset of active variables,A r(i)associated withpotential f0rðiÞ.This set represents the variables for which f0rðiÞshould be deﬁned ifcomputations are exact,but potentials are represented by probability trees which are pruned without error when possible(a node such that all its children are leaves with the same value is replaced by a single leaf with that value).1If we had stored in f0rðiÞthe exact value(zero),then,as this value is used to simulate the values of(X r(n),...,X r(i+1)),the probability of this conﬁguration should have been zero.252S.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–261This set is computed during the variable elimination phase.Initially,A r(i)is the un-ion of the domains of all the potentials in H r(i)minus X r(i),which is the set of variablesof potential f0rðiÞif we would have applied a deletion algorithm with potentials repre-sented by probability tables.But this set can be further reduced:If a variable,say X j,can be pruned without error from f0rðiÞ(i.e.for every conﬁguration of the other vari-ables,f0rðiÞis constant on the values of X r(i))and all the potentials in H r(i)containingthis variable have been calculated in an exact way(all the previous computations have only involved pruning without error)then X j can be removed from A r(i).Though this may seem atﬁrst glance a situation diﬃcult to appear in practice,it happens for all the variables for which there are not observed descendants[18].All these variables can be deleted in an exact way by pruning the result to the constant tree with value1.0and this provides an important initial simpliﬁcation.Taking A r(i)as basis,we consider the tree representing f0rðiÞand follow the pathcorresponding to conﬁguration c j i(selecting for each variable in a node the child cor-responding to the value in the conﬁguration)until we reach a leaf.Let L be the label of that leaf and B r(i)be the set of all the variables in A r(i)which are not in the branch of the tree leading to leaf L.The updating is carried out according to the following recursive procedure:Procedure Update(L,a r(i),b r(i),B r(i))1.If B r(i)=;,2.Assign value b r(i)to leaf L3.Else4.Select a variable Y2B r(i)5.Remove Y from B r(i)6.Branch L by Y7.For each possible value y of Y8.If y is not the value of Y in c j i9.Make the child corresponding to y be a leaf with value a r(i)10.Else11.Let L y be the child corresponding to value y12.Update(L y,a r(i),b r(i),B r(i))In this algorithm,branching a node by a variable Y consists of transforming it into an interior node with a child for each one of the values of the variable.The idea is tobranch as necessary in order to be possible to change the value of f0rðiÞonly for thevalues of active variables A r(i)in conﬁguration c j i,leaving the values of this potential unchanged in other cases.Imagine the case of Fig.2,in which we have arrived to the leaf in the left with a value of a r(i)=0.4.Assume also that the variables in B r(i)are X, Y and Z,each one of them taking values in{0,1}and that the values of these variables in the current conﬁguration are1,0and1respectively.Finally,consider that we have to update the value of this conﬁguration in the tree to the new value b r(i)=0.6.The result is the tree in the right side of Fig.2.Observe that the order in which variables are selected in Step4is not relevant,since at the end all the variables in B r(i)are in-cluded and the sizes of the trees resulting from diﬀerent orders are the same.S.Moral,A.Salmero´n/Internat.J.Approx.Reason.38(2005)245–261253It must be pointed out that,unlike standard importance sampling,in the dynamic algorithm that we propose,the conﬁgurations in the sample are not independent,since the sampling distribution used to draw a conﬁguration may be modiﬁed according to the conﬁgurations previously simulated.However,the resulting estima-tor remains unbiased,as stated in the next theorem.Theorem 2.Let X k be a non-observed variable and e a set of observations.Then,for each x 0k 2X k ,the dynamic importance sampling estimator of p ðx 0k ;e Þ,denoted as ^p ðx 0k ;e Þ,is unbiased.Proof.Assume that the sampling distribution,p Ã,has been updated l times,and let p Ãi ,i =1,...,l ,denote the l sampling distributions actually used in the simulation process.Given a sample S ={x (1),...,x (m )},let S i ,i =1,...,l ,denote the elements in S drawn from p Ãi .Then,according to Eq.(5)^p ðx 0k ;e Þ¼1m X m j ¼1g ðx ðj ÞÞp Ãðx ðj ÞÞ¼1m X l i ¼1X x 2S i g ðx Þp i ðx Þ:According to Eq.(4),for a ﬁxed p Ãi ,E ½g ðx Þ=p Ãi ðx Þ ¼p ðx 0k ;e Þ,which means thatg ðx Þ=p Ãi ðx Þis an unbiased estimator of p ðx 0k ;e Þ.Therefore,^p ðx 0k ;e Þis the average of m unbiased estimators of p ðx 0k ;e Þ,and thus ^p ðx 0k ;e Þis an unbiased estimator of p ðx 0k ;e Þ.hThough all the cases in the sample are not independent,this does not imply that the ﬁnal variance is higher than when using independent samples.We must take into account that the dependence lies in the selection of the distribution to sample succes-sive conﬁgurations,but once this distribution is ﬁxed,then the conﬁguration is inde-pendent of the previous ones.In order to show that this reasoning is correct,we are going to simplify the scenario by considering a simple change of distribution instead of several distributions.This result can be easily extended to the generalcase.254S.Moral,A.Salmero´n /Internat.J.Approx.Reason.38(2005)245–261。

AMM第644-650卷目录PART 4

Table of ContentsPreface liii PART 4An Improved Scheme of One-Time Password Identity Authentication Based on theS/KEY SystemJ.Y. Li, H. Shi, Y.Q. Deng, J. Gong and Y. Guan (2763)The New Key-Stream Generator Based on the OFB Mode of AESH. Shi, J.W. Lu, Y.F. Ji, C. Wu, J. Gong and Y.Q. Deng (2768)Collaboration Research on Web 2.0B. Wu and C.Y. Zhang (2772)An Online E-Payment System Applying to Auto Insurance Based on Proxy Blind Signature L.M. Sha and S.Z. Yang (2776)Network Security Situation Awareness Based on Phishing DetectionJ.Y. Zhang, C.G. Song and X. Jin (2784)Supermarket Trolley Positioning System Based on ZigBeeZ. Zhang, X.P. Tao, L. Zeng and C. Wang (2788)Study of Web Service Discovery Algorithm Based on SemanticL. Zhao and W. Zhang (2793)Research on Usage Intention of Community Information SystemW.P. Li, J. Yang, K.S. Kim and W. Sun (2797)Discussion on the Application of Networking Technology in Intelligent Campus ConstructionA. Wang and X.Q. Zhang (2804)Design of Pesticide Safety Evaluation of SoftwareX.H. Zhang and Y. Lin (2808)The Key Technology and Application of the Internet of ThingsC.M. Li, R. Wang and L. Huang (2812)A Combined Method for Chinese Micro-Blogging Topic TrackingX. Zhang, B. Shang, L.L. Dong and Y.J. Zhu (2816)A Software Design Model Based on Big DataZ.L. He, X.H. Xiao and Y.H. He (2821)Research on Security of P2P TechnologyL.H. Wang (2826)Research of Network Information Platform Construction of ERP System in Manufacturing J.H. Zhang (2830)Optimization of Clustering Algorithm in Ad Hoc NetworkQ. Yu and P. Zong (2834)Research and Improvement of Dynamic Source Routing Protocol Based on Ad HocP. Zong and J. Qin (2838)Safety Strategy of Campus Network Realize Based on Core SwitchY.Y. Lu, Y. Yang and B. Zang (2842)Complex Opinion Network Correlation ClusteringF.Y. Wang, S. Qiu and Q. Li (2846)The Application of Database Technology in Network Management SystemG.L. Cheng and M.Z. Li (2850)Research on the SDN-Based Architecture of Space-Sky Information NetworkD.M. Yuan and R.W. Ren (2854)Study on the Campus Website ConstructionC. Liu (2857)Research on QoS Guarantee Technology for Intercom System Based on SIPZ. Li, Q.Y. Yang, Y.C. Zhou and H. Ren (2863)Applied Research for Campus Student Credit Management System under the Cloud Storage Y.J. Kang and L. Ma (2868)Assess on E-Commerce Transaction Based on Web TechnologyK. Xiao (2872)NTP DRDoS Attack Vulnerability and MitigationA. Alfraih Abdulaziz Nasser and W.B. Chen (2875)Binary Tree Model-Based Mobile Ad Hoc Network Dynamic Address AllocationMechanism ResearchJ.L. Liu and L. Zhu (2881)A High-Throughout Design of CAVLC Decoder for H.264/AVCY. Wang and X.Q. Su (2886)A Distributed Comprehensive-QoS Multicast Routing Algorithm on WSNsW.J. Xiao and S. Zhong (2890)Mobile Game Development with Flash as the EditorH.T. Zhang, Q.J. Sun and Y.C. Liu (2898)Design and Implementation of Service Traffic Awareness System in LTE NetworkJ. Wang, Z.Z. Zhang and Y.L. Luo (2902)A Hadoop-Based Performance Optimization of Network Stream Input FormatX.P. Wang, J.T. Luo, W. Gao and Y. Liu (2906)Dynamic Non-Cooperative Structured Deep Web SelectionS. Deng (2911)Multimedia Technology of Digital Tourism Based on Android SystemJ. Zhang (2915)Vulnerability Assessment of Information System Based on Weighted Directional Graph andComplex Network TechnologyY.Z. Li (2920)Key Technologies Analysis on Management System Data WarehouseX.F. Yang (2925)Research on Action Design System Based on TechnologyL. Xu, W. Lei and W.M. Xu (2929)Numerical Analysis and Performance Test Based on Multi-Media Internet Architecture J.P. Fan (2934)The Architecture and Implementation for International Trade Settlement Software DesignG.J. Zhang (2939)A Web Services Security Policy Description ModelH. Zeng, Y.W. Zhao and D.F. Ma (2943)The Simulation Platform in City Traffic Environment Based on TinyOS for WirelessSensor NetworksT.J. Ren, H.X. Lv, Z.Q. Wang, Y.R. Chen and Y.L. Liu (2947)Automatic Threat Assessment of Malware Based on Behavior AnalysisJ.G. Jiang, X.J. Ma, X.L. Qiu, M. Yu and C. Liu (2952)Design of Wireless Sensor Networks Border Router Based on IPv6D.W. Xu, L.L. Deng and S. Ren (2957)Research of Network Virtualization in Data CenterX.L. Tan, W.B. Wang and Y.Q. Yao (2961)The Software Support Analysis of Information SystemH. Wang, X.Y. Li, X.N. Wang and W.N. Liu (2965)Study on a Novel GIS-Based Routing StrategyY.W. Wang and D.Y. Ji (2969)Development of a New Routing Protocol Based on GPSR for Wireless Sensor NetworksC.F. Xing, L. Yang and Q.L. Han (2973)The Study of Network Information Security Based on Information Filtering Technology L.L. Wei and W. Yang (2978)The Implementation of Cloud Storage System Based on OpenStack SwiftZ.Y. Duan and Y.Z. Cao (2981)Research and Design on Multilevel Secure Database Inference ControlH.Y. Zhao, R.G. Liu and X.G. Liu (2985)Research on Secure Model in WiFi/WiMAX Mixed Networks Based on Pre-Authentication Z.T. Ni, Y. Zhong and L. Huang (2988)Research on Database Front-End Applications Exploration Based on PowerbuilderL. Huang, Y. Wang and Z.T. Ni (2992)Research on Dynamic Self-Adaptive Network Security Model Based on Mobile Agent K.Q. Fan (2996)Research on the Implementation Methods of Security Management of Distributed DatabaseApplication SystemN. Zheng and Y. Gao (3000)Research on Shared Information Platform for Expressway Management Information SystemC. Wang and Y.L. Li (3004)Research on Ad Hoc Network Security Protection Model Based on Mobile AgentQ. An, Y.J. Luo, H.Y. Zhao and J. Zhao (3008)Research on Information Resources Sharing Patterns Based on Cloud ComputingW.J. Yang, Y.J. Luo, H.Y. Zhao and X.T. Li (3012)Research on the Model of Personalized Recommendation System Based on Multi Agent Q. Wang, J.Z. Ping, L.L. Yu and Z.J. Wang (3016)Research of Music Retrieval System Based on Emotional Music TemplateX. Wang (3020)A New Trust Model in P2P NetworkX.L. Li, L.C. Wu, J.J. Xiang, H.L. Ma and F. Liao (3024)Research on Relay Node Placement Based on Hybrid Greedy Optimization Algorithm inWireless Sensor NetworksH. Xu and H. Zhang (3028)Security Research of the Mobile E-Government TerminalS.Y. Guan, Y. Fan and H.L. Lv (3032)Analysis of College Students' Online Business in ChinaA. Abuduaini and N. Aishanjiang (3036)Thought about the Construction of Digital Employment Information Service System of RuralMigrant Workers in the West Area of JilinX.L. Wang (3040)Design and Implementation on Sina Micro-Blog Client Based on the Android SystemC.Y. Shi (3045)On SOA Community Informationization Foundation Database Generic Interface Design Y.B. Zhou (3049)Research on Storage Strategy of Unstructured Small Files in HDFSL.T. Wu, T.N. Wang and H.R. Hu (3053)The Building of the Database of Art Resources Research for Academy of Fine ArtsM. Zhao (3057)Research and Implementation of Auxiliary Teaching System Based on C/S ModelY.J. Cong (3061)The Improvement of the Public Service System Based on Web TechnologyR. Qian (3065)Design of Extended Event Service Model Based on CORBAC.X. Zhao (3069)Modeling and Analysis of NOTAM Distribution Services Based on Petri NetJ. Hu, X.Y. Song and J.Z. Sun (3073)Research of the Database Access Technology FrameworkX.D. Zhang, Z.M. Teng and D.W. Zhao (3077)Dynamic Visual Elements in the Digital Media DesignT. Sun (3081)Design Research on the Resident Electronic Health Recorder SystermN. Liu (3085)The Web Development Technology Research of Cross Platform Mobile Application S. Sun and S.X. Cao (3090)The Universal Middleware Architecture Based on the Web of ThingsR. Zhang and P. Zhang (3094)Development of Travel Reservation System for Mobile PlatformL.J. Sun (3099)Research on Statistics Based Multi-Priority MAC Protocol for Ad Hoc NetworksP. Wang, H. Li, B.L. Ye, C. Chen and Y.B. Wang (3103)LEACH-EO: A More Energy-Efficient LEACH Protocol for WSNJ. Zhang, H. Yan, Y. Cui, H. Rong and J.P. Wang (3108)Authorization Management System of Micro Video Based on FFmpegQ. Guo, Y.G. Xu and S.X. Cao (3112)Research on Network Video Data Acquisition and Analysis Based on Big DataH. Ji and S.X. Cao (3116)The Design and Implementation of Community Property Management SystemQ.H. Wu and H. Zhao (3120)The Integration of Sports Information in Personalized NetworkH. Zhao, Q.H. Wu and J.B. Zhao (3124)Situation and Development Strategies of Sports Entertainment Groups inNetwork EnvironmentJ.B. Zhao, H.J. Ji and H. Zhao (3129)Reverse Engineering OWL 2 Ontologies to UML ModelsW.J. Li (3133)The Study on Educational iOS and Android Application Program of Sports Skillsand KnowledgeN. Liu and D.Y. Yang (3137)Research on Development of Books Interview PlatformH.M. Zhang and N. Li (3141)The Development of Information Integration System for Oil Production Equipment M.T. Wang (3145)The Design and Implementation of Oil Production Equipment Data Management System M.T. Wang (3149)The Development on Information Collection System Based Internet of ThingsC.W. Luo, X.W. Yin and C.D. Ni (3153)Design and Implementation of Computer Equipment Management System Based onOracle DatabaseX.M. Jiang (3157)Agricultural Products Traceability System Design Based on IOTX. Qian, D. Wang and W.M. Luo (3160)Design and Implementation of the Hospital Information Management System Based onthe M. Li and S.Y. Yang (3166)Professional Software Analysis and Comparison for Graphic DesignR.H. Wang (3170)A Network Security Risk Computation Approach Based on Attack GraphsC. Wang (3174)Application of Computer Network of Virtual Reality in Design of ArtJ. You (3178)Cryptographic Protocol Verification Based on the Extension RuleH. Lin (3181)The Frame Study of Translation System Design Based on Database ManagementL.H. Liu, F.J. Meng, Y. Lei, Y. Sun, J.Q. Mu, Z.L. Zhu, Y. Yan, Y.H. Zhang, L. Sun and Z. Lv (3185)Efficiency Analysis of Command Networks with Cross-Level of Different GroupsL.F. Yu, J.B. Wu, J. Liu, B.X. Xiu and W.M. Zhang (3189)Study and Practice Based on Network TechnologyW.H. Zhao, D.P. Xu, H.Y. Gong and Y. Li (3195)Application Research on Virtual Reality TechnologyY. Li, H.Y. Gong, D.P. Xu and W.H. Zhao (3199)Network Topology Discovery Algorithm Based on OSPF Link State AdvertisementZ.J. Shen and Y.S. Ge (3203)A Cloud-Based Mobile Telemedicine Consultation System Based on iPadH. Wang, T.H. Li and F. Wu (3208)Design of Network System Security System of Digital LibraryS. Liu (3212)Information Management System of Metrological Evaluation Based on SSH Framework S. Zhang, J.M. Zhu, Y.H. Qin and L.L. Qu (3216)Design and Implementation for the Upper Computer Software of the Two-DimensionalTurntable System Based on MFCY.N. Xiang Li and X.J. Yang (3220)Available Storage Space Sensitive Replica Placement Strategy of HDFSW.T. Zhao, Y. Ding and X.H. Zhang (3224)LBSN-Based Personalized Routes RecommendationL.C. Zhu, Z.J. Li and S.X. Jiang (3230)An Intelligent Human-Computer Collaborative Method for Creative DesignJ.W. Wang (3235)Research of Cloud Manufacturing Technology in the Development of Digital IntelligentProduction Control PlatformG.L. Feng (3241)A Remote and Unified Software Automated Deployment PlatformJ.P. Zhao, X.Y. Liu, H.H. Wu, X.L. Chen, L. Yang and D.H. Zhang (3245)Design and Implementation of Multi Granularity Access Framework Based on AOP Q. Wang and Y.B. Wang (3251)A Smart Grid Data Global Placement Strategy Based on Cloud ComputingY.K. Li, D. Xin and H.G. Liu (3256)Design and Implementation of Binary Utilities GeneratorJ.Q. Shen, J. Wu, Z.F. Zhang and H.Q. Ren (3260)SERP: A Simple Energy-Hops-Based Routing Protocol in WSNsM.X. Li, X.C. Zhou, X.H. Fan and S. Wang (3266)Human Error Simulation of Manufacturing Cell Based on Human-Machine Integrated ModelD.F. Zhao, X.D. Zhang, C.J. Gong and C.C. Wang (3270)A Methodology for the Exploration of 802.11BH. Yang (3275)Research on Data Aggregation Application Based on MashupM.Y. Cai and B. Pan (3279)Research on ASIC Firewall Based on State Detection TechnologyS.Q. Wang and H.Y. Chen (3283)Detection Technology for Hostile Attacks to Campus Wireless NetworkL. Ma and H.X. Yang (3287)Research on the Detection Method of the Malicious Attacks on Campus NetworkJ.L. Wang (3291)Detecting Overlapping Communities with MDS and Local Expansion FCML. Li, Z.M. Xia, S.H. Li, L. Pan and Z.H. Huang (3295)Comparison Analysis of RESTful and SOAP-WSDL Applied in the Image Management System R.Y. He (3300)Study on Digital Content Representation from Direct Label Graph to RDF/OWL Language into Semantic WebK.A.L. Khamis, L. Zhong and H.Z. Song (3304)Research on Multilevel Secure Database Inference ChannelH.Y. Zhao, J. Meng and X. Zhang (3310)Design and Realization of IOT-Based Video Monitoring SystemJ. Yin and C.H. Li (3314)Storage Model Based on Oracle InterMedia for Surveillance VideoB. Sun, W.S. Luo, L.B. Du and Q. Lu (3318)A Noise-Optimal Integrator for High-Precision SC Sigma Delta ModulatorsX.L. Wang and Y.W. Zhang (3322)Research on Vehicle Networking Transfer Channel Based on MAC Safety Information System Y. Zhou, T.J. Ren, Z.Q. Wang and Y.L. Liu (3329)Application of Neural Network in Network Intrusion DetectionZ. Yang and H. Du (3334)The Study of the Ontology and Context Verification Based Intrusion Detection ModelG.F. Guo (3338)Environmental Monitoring System Designing: A Internet of Things ApproachG.H. Wu, F. Liu, J.X. Li and W. Wang (3342)Remote Inspection System Algorithm Research of Wireless Base StationM.D. Bai and Y. Dong (3346)Application Research of Visual Simulation Technology in the Field of Marine EngineRoom SimulatorH.S. Shen, J.D. Zhang, Y.B. Li and F. Han (3350)Application of Function Point EstimatingC.H. Zhou (3357)Design and Research of Computer System High Confidence Fault TolerantJ.Q. Qi (3361)FMPC: A Fast Multi-Dimensional Packet Classification AlgorithmZ.H. Guo, L. Li, Q. Wang, M. Lin and R. Pan (3365)Code Protection Technology on iOSJ.T. Weng, Q. Mu, X.Y. Liao, Y.Z. Li, Q.X. Zhang and Y. Tan (3371)Distribution of Database in Cloud Based on Associated MatrixL.Y. Yao and W. Yang (3375)Finite Element Numerical Simulation Research on Fractured Horizontal Well’s Productivity M.X. Liu, J.H. Li and L. Zhang (3379)Research of Augmented Reality for Children’s Books on the Basis of ArtoolkitsW.G. Yang (3383)Research and Practice of Cloud Computing Based on HadoopA.S. Lu, J.J. Cai, W. Jin and L. Wang (3387)Review on Application of Virtual Reality in the Physical SimulationZ.J. Cai (3390)Overhead Analysis of Loop Parallelization with OpenMP DirectivesL.Y. Xiang, Z.Y. Fang, Y. Wang, G.N. Qu and Z. Chen (3394)Design and Implementation of Cloud Management SystemH.Y. Yu, T.X. Yang and X. Fu (3398)Study on Replacing-Element Imagery Design of Dongba Characters Combined withChinese CharactersW.J. Song, Y.F. Yang and R.P. Xu (3402)Research on Multilayer Security Audit Research Based on Attack Graph in Cloud Computing L.B. Wen (3408)An Efficient Secure Multichannel Traffic Management Scheme in 2.4 GHz Home AutomationNetwork with IoT FunctionsM. Wei and P. Wang (3412)Chapter 5: Electronics and Microelectronics, Embedded and IntegratedSystems, Power and Energy, Electric and Magnetic SystemsA Study on Reconfiguring On-Chip Cache with Non-Volatile MemoryM.Q. Wang, J.T. Diao, N. Li, X. Wang and K. Bu (3421)A Circuit Model of the MemcapacitorW. Wang, H. Xu, Y.W. Hou and H.J. Liu (3426)Titanium Oxide Memristor Based Digital Encoder CircuitY.W. Hou, X. Xu, W. Wang, X.B. Tian and H.J. Liu (3430)Application of Digital Coordinate Transformation to a New Posthole Broadband SeismometerC.Y. Peng, B. Xue and J.S. Yang (3434)Research and Design of Asynchronous FIFO Based on FPGAB.Q. Liu, M.Z. Liu, G. Yang, X.B. Mao and H.L. Li (3440)Wind Power Allocation Based on Predictive Power CorrectionZ.H. Feng, T.J. Jia, X.M. Xiao and F.J. Zhang (3445)The Application of LED Lighting in Museum Exhibition HallY. Zheng, L.W. Huang, M.M. Wang, H.Q. Chen and L.Z. Zhang (3449)A High Reliable Communication Method for the Terminal of a Networked and DistributedPower Supply SystemE. Lu, B. Huang, S.S. Li and Y. Yang (3453)A New Method of Cross-Correlation by Magnetic Dipole for Estimating MagnetizationDirection under the Influence of Remanent MagnetizationL. Shi, L.H. Guo and F.Y. Guo (3459)The Condition Assessment of Distribution Transformer Based on Improved AnalyticHierarchy ProcessX.P. Meng, L. Li, H. Wang and X. Ji (3463)Effects of Different Sample Resistances on the Insulator Leakage Current Acquisition Results P.C. Miao and Z.N. Xu (3468)Research and Design of Video Acquisition System Based on FPGAB.L. Liu, B.Q. Liu, L. Pan and L. Wang (3472)High Rate Data Transmission System Based on OFDM for Well LoggingQ.S. Gu, W. Chen and R.Q. Wu (3476)Application of On-Line Ultrasonic and UHF Partial Discharge Detection in 1000kV GISF. Chen, H.Z. Tang and H.G. Li (3480)Synchronization of Hyperchaotic Memristor-Based Chua’s CircuitsH.L. Huang, Y. Peng and J.J. Huang (3485)Prediction of Nuclear Power Project Cost Based on Improved Non-Equidistant GM (1,1) Modeland Exponential Regression ModelB.Y. Liu, D.X. Niu, J.P. Qiu, H. Xu and Y. Wang (3489)Design Principle and Applicative Value of Photovoltaic SystemM. Chen, L. Chen, X. Tang and Y.H. Hu (3493)Research of Signal Integrality in PCB Design for ARM9 Core BoardL.Y. Su (3497)Formal Analysis of Memory Leak DefectsW. Zhang, Z.Y. Ma, Q.L. Lu, L. Wan and D.W. Liu (3501)Research and Application of a Solution to Acquire Electric Energy DataS.Y. Long, X. Lu and Y.S. He (3505)The Transient Characteristics Analysis of Doubly-Fed Induction Generator during theAsymmetric Voltage SagJ.H. Zhang, H.R. Shen, L. Ding and C.L. Dai (3509)Optimal Substation Locating and Sizing Based on Improved Weighted Voronoi Diagram andGenetic AlgorithmP.Q. Miao, Q.B. Meng, Y. Li and Q.B. Tu (3515)Optimal Design of Transmission Line Route Based on GISY. Wang, W. Huang, Y. Li, Q.B. Tu and X. Tain (3522)A Control Method to Suppress Resonance and Improve Current Quality for Inverter Systemwith LCL FilterY.P. Lin, S. Tao, X.N. Xiao and M.R. Wu (3528)Charge and Discharge Protection Circuit of a Mobile Power Supply ImprovementS.G. Li, L.Z. Wang, Z.Z. Gao, X.J. Zhang and L.N. Zhao (3534)Risk Assessment of Power Transformer Life Cycle Cost Based on ExtensibleMatter-Element ModelG.H. Qiao, X.J. Guo, L.M. Wu, H.L. Liu, Z. Ren and N. Bai (3538)A New Method for Power System Transient Stability Assessment: Application of AdaptiveCombined ClassifiersS. Wei, B. Wang, D.C. Liu and J.H. Luo (3542)Influence of Moisture on the Space Charge Migration and Electric Field Behavior inOil-Paper InsulationJ. Fu, J. Hao, H. Yin, G.L. Wu and Q. Wang (3548)Intelligent Four-Probe Resistivity Meter Based on MCUJ.Z. Huo (3552)Routing Selection for Communication of Power System Wide-Area Protection ConsideringBackup PathX.W. Sun (3556)A Novel Broadband Vibration Energy HarvesterY. Liu, X.Y. He, S. Liu, Y. Wu and Y. Ou (3560)Equivalent Circuit of a Planar Transformer Used for TWTAB. Zhao and G. Wang (3564)A Reconfigurable Radix-r FFT Hardware Structure DesignY.X. Zhang, H.P. Zhao and J.Y. Yu (3568)One Kind of Band-Gap Voltage Reference Source with Piecewise High-Order Temperature Compensation and Power Supply Rejection RatioZ.D. Li and L. Xie (3575)Evaluation of Power System Black-Start Schemes Based on Improved DEA/AHPEvaluation ModelC.G. Shi and T. Liu (3579)A TDC Based BIST Scheme for Operational AmplifierJ. Yuan and W. Wang (3583)Design and Simulation of Arrayed Waveguide Grating for Miniature Raman Spectrometer Y.C. Xu, Q.N. Wang and W.Z. Zhu (3588)The Electromagnetic Parameters' Impact of λ/4 Type Dielectric Absorber onAbsorbing PropertiesH.C. Zhao, W.J. Hao, Y.Y. Yi, Y.F. Dong and X.D. Yu (3593)Ballistic Effect and Application in Circuit Design of Wide Band-Gap Semiconductor X.X. Liang, Z.Q. Cheng and M.S. Jia (3597)Design of Programmable DC Power Supply Based on ARMX. Wang, H. Chen and R. Wang (3601)Research on System Integration Technology for Operation & Maintenance Automation Systemof CSG EHV Power Transmission CompanyZ.Z. Zhou, X.Y. Chen and M. Sun (3605)Adaptive Sliding Mode Output Synchronization Tracking for Hyper-Chaotic Lü System Basedon Adaptive PWL FiltersY.B. Zhao, X.Z. Zhang and X.Y. Sun (3610)Steam Generator Water Level Intelligent Control of PWR Nuclear Power Plant Based on FeedWater RegulationX.H. Yang, J. Yang, Y.N. Wang and Y. Xue (3616)Research on Application of all Time Apparent Resistivity Translation Algorithm for LargeFixed Loop TEMF.L. Li and X. Zhu (3620)The ECG Data Storage System Design Based on SD Card and Reliability AnalysisY.L. Zhu and Y.D. Wang (3625)Danger Classes Detection System Design of High Voltage Transmission Conductor Galloping Y.F. Wang and L.L. Liu (3631)UHF RFID Reader DesignD.W. Xu, L.L. Deng and S. Ren (3635)A New Type of Lead Sealing for Electric Energy Meter Packaged on the SpotX.Z. Hou, H. Yan, L. Feng and D. Wei (3639)Design Based on PLC Programming Control De-Dust SystemG.Q. Wang (3643)Circuit Design for an Intelligent Dustbin Controlled by GesturesH.C. Zhou (3647)A New Perspective on ECL CircuitsR.B. Hu, S.T. Zhou, G.B. Chen, D.B. Fu and X.Y. Zhang (3651)Practicability Discussion and Verification of Using FPGA to NAND FlashM. Yang, K. Xu and X.F. Zhang (3655)Research & Development of Three-Phase Full - Controlled Bridge Rectification CircuitsExperiment Device Based on TC787Y. She (3659)A 14-Bit Pipeline ADC Behavior Model Using Verilog-A for SOCW. Liao, L.C. Lei and X.D. Zhou (3663)Applicability Analysis of PTN Technology in Henan Electric Power Transmission Network Y. Yang, J.X. Lv, S.W. Wang, P.L. Cai and W.C. Li (3667)The Researches on the Construction of the Management in Integrated Electric PowerCommunication Network SystemY. Yang, J.X. Lv, L. Sheng, Y. Sun and X.C. Zhao (3671)。

A sampling-based algorithm for multi-robot visibility-based pursuit-evasion

A Sampling-Based Algorithm for Multi-Robot Visibility-BasedPursuit-EvasionNicholas M.Stifﬂer Jason M.O’KaneAbstract—We introduce a probabilistically complete algo-rithm for solving a visibility-based pursuit-evasion problem in two-dimensional polygonal environments with multiple pur-suers.The inputs for our algorithm are an environment and the initial positions of the pursuers.The output is a joint strategy for the pursuers that guarantees that the evader has been captured.We create a Sample-Generated Pursuit-Evasion Graph(SG-PEG)that utilizes an abstract sample generator to search the pursuers’joint conﬁguration space for a pursuer solution strategy that captures the evaders.We implemented our algorithm in simulation and provide results.I.I NTRODUCTIONThere are many variants of the pursuit-evasion problem. The common theme amongst them is that one group of agents,the“pursuers”,attempts to track members of another group,the“evaders”.This paper considers a speciﬁc variant of the pursuit-evasion problem called visibility-based pursuit-evasion, which requires the pursuer(s)to systematically search an environment to locate the evaders,ensuring that all evaders will be found by the pursuers in aﬁnite time.The speciﬁc problem we consider is a visibility-based pursuit-evasion problem that utilizes a team of pursuers.The pursuers move through a polygonal environment seeking to locate an unknown number of evaders,which move at aﬁnite but unbounded speed.The pursuers have an omni-directional ﬁeld-of-view that extends to the environment boundary.The goal is toﬁnd a joint strategy for the pursuers that ensures that all of the evaders are seen.The visibility-based pursuit-evasion problem has an extra layer of complexity beyond the standard motion planning problem because of its capture guarantee.It is not enough to simply select a standard motion planner and attempt to generate a path for each pursuer through the environment. To guarantee that the pursuer strategy does indeed capture an evader if one exists,the planner must also reason about the regions of the environment that are not currently in the pursuers’visualﬁeld-of-view and how these regions interact with one another as the pursuers move within the environment.Two dominant threads of research involve the number of deployable pursuers available to solve the visibility-based pursuit-evasion ing only a single pursuer, there are results that yield complete[4],randomized[8], and optimal[22]solutions,as well as many other variants N.M.Stifﬂer and J.M.O’Kane are with the Department of Computer Science and Engineering,University of South Carolina,301Main St., Columbia,SC29208,USA.{stifﬂen,jokane}@ Fig.1:A pursuer strategy generated by our algorithm.Filled circles represent the pursuers’initial positions and open circles represent their goal positions.discussed in Section II.A consequence of using only a single pursuer is that these algorithms are only applicable when the environment can be represented as a simply-connected polygon.The authors considered the multiple pursuer visibility-based pursuit-evasion problem[23]in the past.In that work,we introduced a centralized algorithm for computing a pursuer solution strategy.The general idea is to create a Cylindrical Algebraic Decomposition(CAD)of the pursuers’joint conﬁguration space by using polynomials that capture where critical changes to the regions of the environment hidden from the pursuers occur.Then we compute the adjacency graph for the CAD and construct a Pursuit Evasion Graph(PEG)induced by the adjacency graph.A search through the PEG can produce one of the following outcomes: the search can reach a vertex where the pursuers’motions up to this point ensure that the evader has been captured,or the search terminates withoutﬁnding a solution and produces a statement recognizing that no solution exists.The drawback of the technique is the computational complexity required to construct the CAD and perform the adjacency test,which is doubly exponential in the number of pursuers.This paper differs from that work in that we no longer discretize the conﬁguration space and maintain a CAD nor compute the adjacency graph.The main contribution of this work is a probabilisti-cally complete algorithm for multiple pursuer visibility-based pursuit-evasion that generates a solution strategy for the2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2014) September 14-18, 2014, Chicago, IL, USApursuers to execute(Figure1)through the joint conﬁguration space.Our algorithm creates a graph that maintains the pur-suers’information state,and utilizes a sample generator that we treat as a“black box”to reason about unexplored areas in the pursuers’joint conﬁguration space.Our algorithm has some similarity to the Probabilistic Roadmap(PRM) algorithm[10],but differs in that our algorithm maintains information concerning the areas of the environment where the evader might be.The need for this additional information complicates both the update operations for the graph and the selection of samples.The remainder of this paper is structured as follows.In Section II we discuss related work to our problem.Section III contains a formal problem statement.A formal deﬁnition for the area not visible to the pursuers,called shadows,appears in Section IV.This paper makes several new contributions: 1)We introduce a graph that maintains a representation ofthe reachable parts of the pursuers’joint information space and provide details about its construction(Sec-tion V).2)We introduce an algorithm that uses this graph to searchfor a pursuer solution strategy(Section VI).3)We present simulation results(Section VII)that showour algorithm’s ability to generate solution strategies for various sample generators.Discussion and concluding remarks appear in Section VIII.II.R ELATED W ORKThe pursuit-evasion problem was originally posed in the context of differential games[5],[7].The lion and man game and the homicidal chauffeur are two such differential games. In the lion and man game,a lion tries to capture a man who is trying to escape[15],[21].In game theory,the homicidal chauffeur is a pursuit-evasion problem which pits a slowly moving but highly maneuverable runner against the driver of a vehicle,which is faster but less maneuverable,who is attempting to run him over[7],[19].Theﬁrst recognized instance of pursuit-evasion on a graph is the Parsons problem[17].The idea behind the Parsons problem,also known as the edge-searching problem,is to determine a sequence of moves for the pursuers that can detect all intruders in a graph using the least number of pursuers.A move consists of either placing or removing a pursuer on a vertex,or sliding it along an edge.A vertex is considered guarded as long as it has at least one pursuer on it,and any evader located therein or attempting to pass through will be detected.A sliding move detects any evader on an edge.The visibility-based pursuit-evasion problem was proposed by Suzuki and Yamashita[24]as a geometric formulation of the graph-based problem and can be viewed as an extension of the watchman route problem[1],in which the objective is to compute the shortest path that a guard should take to patrol an entire area populated with obstacles,given only a map of the area.A.Single PursuerThe capture condition for the general visibility-based pursuit-evasion problem[4]is deﬁned as having an evader lie within a pursuer’s capture region.There has been substantial research focused how the visibility-based pursuit-evasion problem changes when a robot has different capture regions. The k-searcher is a pursuer with k visibility beams[14], [24],the∞-searcher is a pursuer with omnidirectionalﬁeld of view[4],[16],and theφ-searcher is a pursuer whoseﬁeld-of-view[3]is limited to an angleφ∈(0,2π].Note that all of these approaches consider evaders with unbounded speed. Others have studied scenarios where there are additional constraints,such as the case of curved environments[13],an unknown environment[20],a maximum bounded speed for the pursuer[26],or constraints similar to those of a typical bug algorithm[18].B.Multiple PursuerAs a result of the problem complexity,there is a wide range of literature with differing techniques attempting to solve the multi-robot visibility-based pursuit-evasion prob-lem.One technique organizes the pursuers into teams,whose joint sensing capability are a set of moving lines,each of which is spanned between obstacles.By using these teams of robots as sweep lines,the authors guarantee de-tection of the evaders[12].Other researchers have used a mixed integer linear programming approach to solve a multi-pursuer visibility-based pursuit-evasion problem[25]. Another approach involves maintaining complete coverage of the frontier[2].There are other variants of the pursuit-evasion problem where the pursuers are teams of unmanned aerial vehicles[11].III.P ROBLEM S TATEMENTPortions of this section appear in the authors’prior work[23]and are included here for completeness.A.Representing the environment,evaders,and pursuers1)The environment:The environment is a polygonal free space,deﬁned as a closed and bounded set F⊂R2,with a polygonal boundary∂F.The environment is composed of m vertices.2)The evader:The evader is modeled as a point in F that can translate within the environment.Let e(t)∈F denote the position of the evader at time t≥0.The path e is a continuous function e:[0,∞)→F,in which the evader is capable of moving arbitrarily fast(i.e.aﬁnite,unbounded speed)within F.Note that,by assuming that there is a single evader,we have not sacriﬁced any generality.If the pursuers can guarantee the capture of a single evader,then the same strategy can locate multiple evaders,or conﬁrm that no evaders exist.3)The pursuers:A collection of n identical pursuers cooperatively move to locate the evader.We assume that the pursuers know F,and that they are centrally coordinated. Therefore,from a given collection of starting positions,the pursuers’motions can be described by a continuous functionFig.2:An environment with two pursuers and three shadows. p:[0,∞)→F n,so that p(t)∈F n denotes the joint conﬁguration of the pursuers at time t≥0.The function p,which our algorithm generates,is called a joint motion strategy for the pursuers.We use the notation p i(t)∈F to refer to the position of pursuer i at time t.Likewise,x i(t) and y i(t)denote the horizontal and vertical coordinates of p i(t).Without loss of generality,we assume that the pursuers move with maximum speed1.Each pursuer carries a sensor that can detect the evader. The sensor is omnidirectional and has unlimited range,but cannot see through obstacles.For any point q∈F,let V(q) denote the visibility region at point q,which consists of the set of all points in F that are visible from point q.That is, V(q)contains every point that can be connected to q by a line segment in F.Note that V(q)is a closed set.B.Capture conditionsThe pursuers’goal is to guarantee the capture of the evader for any continuous evader trajectory.Deﬁnition A joint motion strategy is a solution strategy if, for any continuous evader trajectory e:[0,∞)→F, there exists some time t and some pursuer i such that e(t)∈V p i(t) .IV.S HADOWSThe key difﬁculty in locating our evader is that the pursuers can not,in general,see the entire environment at once.This section contains some deﬁnitions for describing and reasoning about the portion of the environment that is not visible to the pursuers at any particular time.Deﬁnition The portion of the environment not visible to the pursuers at time t is called the shadow region S(t),and deﬁned asS(t)=F− i=1,...,n V p i(t) .Note that the shadow region may contain zero or more nonempty path-connected components,as seen in Figure2. Deﬁnition A shadow is a maximal path connected compo-nent of the shadow region.Notice that S(t)is the union of the shadows at time t.The important idea is that the evader,if it has not been captured, is always contained in exactly one shadow,in which it can move freely.As the pursuers move,the shadows can change in any of four ways,called shadow events.•Appear:A new shadow can appear,when a previously visible part of the environment becomes hidden.•Disappear:An existing shadow can disappear,when one or more pursuers move to locations from which that region is visible.•Split:A shadow can split into multiple shadows,when the pursuers move so that a given shadow is no longer path-connected.•Merge:Multiple existing shadows can merge into a single shadow,when previously disconnected shadows become path-connected.These events were originally enumerated in the context of the single-pursuer version of this problem[4]and examined more generally by Yu and LaValle[28].A.Shadow LabelsFor our pursuit-evasion problem,the crucial piece of information about each shadow is whether or not the evader might be hiding within it.Deﬁnition A shadow s is called clear at time t if,based on the pursuers’motions up to time t,it is not possible for the evader to be within s without having been captured.A shadow is called contaminated if it is not clear.That is,a contaminated shadow is one in which the evader may be hiding.Notice that,since the evader can move arbitrarily quickly, the pursuers cannot draw any more detailed conclusion about each shadow than its clear/contaminated status;if any part of a shadow might contain the evader,then the entire shadow is contaminated.Therefore,our algorithm tracks the clear/contaminated status of each shadow.Each time a shadow event occurs, the labels can be updated based on worst case reasoning.•Appear:New shadows are formed from regions that had just been visible,so they are assigned a clear label.•Disappear:When a shadow disappears,its label is discarded.•Split:When a shadow splits,the new shadows inherit the same label as the original.•Merge:When shadows merge,the new shadow is as-signed the worst label of any of the original shadows’labels.That is,a shadow formed by a merge event is labeled clear if and only if all of the original shadows were also clear.Notice in particular that,if all of shadows are clear,then we can be certain the evader has been seen at some point. The result of this reasoning is that we can connect the shadow labels to our goal ofﬁnding a solution strategy.A pursuer strategy is a solution strategy if and only if,after its execution,all of the shadows are clear.bel DominanceThe following provides some insight to the hierarchy of preferable shadow rmally,we prefer one shadowlabel to another if in addition to having the same shadows labelled as cleared,there are additional shadows in the label that are also labelled as cleared.This allows us to say that one shadow label dominates another shadow label.Deﬁnition Given two shadow labels corresponding to a shadow region S,we say that a label l dominates a label l′if the following condition holds:∀s∈S If l′s=clear then l s=clear This relation is useful because our algorithm discards any shadow labels that are dominated by another shadow label reachable at the same pursuer conﬁguration.V.S AMPLE-G ENERATED P URSUIT-E VASION G RAPH This section introduces the primary data structure used in our algorithm.We begin by describing the graph’s structure and also elaborate on a non-trivial graph operation.A.Graph StructureThe Sample-Generated Pursuit-Evasion Graph(SG-PEG) is a rooted directed graph whose vertices represent joint pursuer conﬁgurations.A vertex in the SG-PEG contains1)a joint pursuer conﬁguration(denoted jpc),and2)the set of non-dominated shadow labels reachable byfollowing a path from the root,through the graph,to that conﬁguration.For an edge to exist between any two vertices in the SG-PEG there must be a line segment in F n that connects the joint pursuer conﬁguration at the source vertex with the joint pursuer conﬁguration at the target vertex.Given an arc of the SG-PEG,e=(x,y),the edge stores a mapping from the reachable shadow labels in x to the corresponding shadow labels in y.The operations available to a SG-PEG graph are A DD-V ERTEX and A DD E DGE.These operations differ from the corresponding operations on a standard graph because of the book-keeping needed to keep track of the reachable shadow labels.The A DD V ERTEX operation is trivial,but details concerning the A DD E DGE operation appear in the next section.B.Edge CreationWhen a new connection is established between a source and target vertex in the SG-PEG,the source’s reachable shadow labels are used to update the target’s reachable labels (Algorithm1).In this section we discuss the shadow label update criterion,the update label subroutine,and the process of adding a new reachable label to a vertex.1)Computing a New Label:In the authors’prior work [23],we provided a family of polynomials that capture where critical changes can occur to the region of the environment hidden from the pursuers.Although complete,the quantity and complexity of the polynomials(there are O n3m3 polynomials,where n corresponds to the number of pursuersand m corresponds to the number of environment vertices)in this family makes the task of analytically identifying where Algorithm1A DD E DGE(v,v′)Input:a source vertex v and a target vertex v′1:for each label in v’s reachable set do2:updated←C OMPUTE L ABEL(v.jpc,label,v′.jpc) 3:A DD R EACHABLE(v′,updated)these changes occur computationally expensive.Instead,we update the shadow labels numerically.The general idea is that if we partition the line segment connecting any two joint pursuer conﬁgurations in F n into a collection of evenly spaced joint pursuer conﬁgurations we can incrementally track the shadow changes.To ensure that all of the shadow events are captured there must be at least one sample capable of capturing each successive shadow event while traversing along the segment.The computation of a new shadow label(Algorithm2) takes as input two joint pursuer conﬁgurations,a source and target,and a shadow label corresponding to the shadow region at the source conﬁguration.The output is the shadow label that results from the pursuers moving from the source conﬁguration to the target conﬁguration given the initial shadow label.Figure3illustrates this process.Initially,there are two contaminated shadows.As the pursuers move to the target conﬁguration,a shadow appears as the pursuers move to the right(a cleared shadow).As the pursuers reach the target conﬁguration the central shadow disappears.We begin by partitioning(Algorithm2line2)the segment connecting the source and target conﬁgurations in F n into aﬁnite collection of evenly spaced joint pursuer conﬁgura-tions.We then loop through this collection of joint pursuer conﬁgurations,updating the shadow label along the way, returning theﬁnal label of the sequence.The process of computing the new shadow labels for our discretized segments appears in Algorithm2lines4-11.The process starts by computing the shadow regions of both the source and target conﬁgurations.We initialize the label corresponding to the target conﬁguration as all cleared.We check all of the shadows in the shadow region of the goal conﬁguration for an intersection with contaminated shadows belonging to the shadow region of the source conﬁguration. If an intersection with a contaminated shadow occurs then the corresponding shadow in the target conﬁguration is also labelled as contaminated.2)Adding a Reachable Label:Theﬁnal step involves adding the newly computed shadow label to the target vertex (Algorithm3).It may also be the case that the individual shadows of the new label are all cleared,in which case a solution has been found.If the target vertex contains a shadow label in its set of reachable labels that dominates the new shadow label,then the new label does not contribute any new information and we return.Similarly,if there are labels in the vertex’s set of reachable labels that are dominated by the new shadow label then those labels are removed.If the new shadow label is not dominated and is not a solution strategy then we add the new shadow label to the vertex’sBefore During AfterFig.3:An illustration of the update step.Initially there are two contaminated shadows(red).After running the U PDATE method,there is a cleared shadow(green)and a contaminated shadow(red).Algorithm2C OMPUTE L ABEL(p,l,p′)Input:a starting conﬁguration p,starting label l,anda goal conﬁguration p′1:label←l2:<p1,...,p k>←D ISCRETIZE(p,p′)3:for each p i,p i+1where i<k do4:oldshadows←S HADOW R EGION(p)5:newshadows←S HADOW R EGION(p′)6:newlabel←0···0⊲initially all cleared 7:for each s′in newshadows do8:for each s in oldshadows do9:if label s=1and s′intersects s then 10:newlabel s′←111:label←newlabel12:return labelAlgorithm3A DD R EACHABLE(v,l)Input:a SG-PEG vertex v and a label l1:function A DD R EACHABLE(v,l)2:if v contains a label that dominates l then return 3:add l to v as a reachable label4:delete labels in v dominated by l5:if A LL C LEAR(l)then6:Output Solution v⊲Is l a solution? 7:for each out in Neighbors(v)do8:newlabel←U PDATE(v.jpc,l,out.jpc)9:A DD R EACHABLE(out,newlabel)reachable set.This label now permeates the graph recursively via the vertex’s outgoing edges.A label is calculated for each of the vertex’s neighbors,and if this label is added to the neighbors reachable set,then the process repeats itself.The process ends when no additional reachable labels are found. Note that if a vertex does not belong to the same connected component as the root vertex then its set of reachable labels is empty.Because of the recursive nature of Algorithm3, a vertex that serves as a bridge between the connected component containing the root vertex and another connected component will cause the reachable data to permeate through the SG-PEG.Algorithm4S OLVE(p,F,A)Input:a starting conﬁguration p,an environment F,and an abstract sampler A1:A DD V ERTEX(p,{0···0})2:while a solution has not been found do3:s←A.G ET S AMPLE()4:x←A DD V ERTEX(s)5:for each y in SG-PEG vertices do6:if(xy⊂F n)andlength(x,y)<maxlength andcycleLength(x,y)>mincycle then7:A DD E DGE(x,y)⊲Digraph edge 8:A DD E DGE(y,x)⊲Digraph edge 9:return E XTRACT S OLUTION(solution)VI.A LGORITHMIn this section we detail how our algorithm uses a SG-PEG to search for a pursuer solution strategy.Our algorithm (Algorithm4)begins by creating a SG-PEG vertex.This vertex’s joint pursuer conﬁguration is the initial joint pur-suer conﬁguration supplied to our algorithm and it’s set of reachable shadow labels contains only a single label whose shadows are all contaminated.This is the root vertex of our SG-PEG.We then proceed by obtaining samples in F n,checking these samples for potential connections with existing vertices in the SG-PEG graph,and update the SG-PEG where necessary when edges are created.A.Abstract SamplerOur main search algorithm uses an abstract sampler to return a joint pursuer conﬁguration(Algorithm4line3). Deﬁnition An abstract sampler is a joint probability density function whose continuous random variables are the pur-suers’positions in F.The only functionality that we require an abstract sampler to have is the ability to generate a point in F n.The beneﬁt of using an abstract sampler is that our algorithm is not dependent on a speciﬁc sampler to generate a solution strategy.This allows us to choose samplers that efﬁciently explore F n.Note that the goal of catching the evaders means that the best sampling strategies may differ from those used in traditional motion planning algorithms.However,forour algorithm to be probabilistically complete,the abstract sampler must have a support equal to F n(Section VI-D). We demonstrate the feasibility of using an abstract sample generator in our algorithm by providing simulation results that utilize various sample generators(Section VII).B.Edge CriteriaIn this section we discuss the constraints used in our main algorithm that determine whether an edge should connect two vertices(Algorithm4line6).The three constraints can be categorized as visibility,edge length,and cycle length constraints.1)Visibility Condition:The visibility condition states that for two vertices to share a pair of directed edges,the vertices corresponding joint pursuer conﬁgurations must be mutually visible to one another.This corresponds to the i th pursuer of one conﬁguration residing within the visibility region of the i th pursuer in a neighboring conﬁguration.Another way of interpreting this constraint is that only straight line motions are permitted between corresponding pursuers in neighboring vertices.This constraint prevents the generation of strategies in which the pursuers collide with obstacles.2)Edge Length:To limit the amount of time spent computing the reachable data when an edge is added in the SG-PEG we place a constraint on the length of the segment connecting the vertices joint pursuer conﬁgurations in F n. The idea is that given two joint conﬁgurations that are far apart,requiring multiple intermediary vertices as opposed to a single long connection is preferred.The intermediary vertices provide additional opportunities for any potential subsequent samples to become connected.3)Minimum Cycle Length:To avoid an oversaturation of edges we enforce a minimum cycle length in the SG-PEG.The intuition is that if a large number of samples in F n that are relatively close together,a large amount of resources could potentially be used computing all of the nearby transitions without necessarily revealing any new information.This optimization is aimed at minimizing the number of samples between which no shadow events occur.C.Search for a solution strategyThe intuition is that given an initial joint pursuer conﬁgu-ration,we assume that all the shadows in the shadow region are contaminated.We then build a SG-PEG using an abstract sampler to select new points in F n.Since we maintain the reachable shadow labels during the construction of the SG-PEG,we know that a solution strategy exists if we encounter a reachable shadow label that is completely cleared.At that point we use the reachable data stored in the vertices and the shadow label mappings stored in the edges to recover a solution by following those mappings back to the root.This solution should appear as a collection of vertices in the ing the joint pursuer conﬁgurations stored in the vertices as intermediary steps that the pursuers need to reach,we will have generated a joint motion strategy that is also a solution strategy.D.Probabilistic CompletenessFinally,we argue that under certain conditions,Algo-rithm4is probabilistically complete.Theorem1:If the abstract sampler has a support equal to F n,and there are no constraints on the edge length and cyclelength,then our algorithm is probabilistically complete.Thatis,the probability of our algorithmﬁnding a solution,if oneexists,tends to1as the number of samples goes to inﬁnity. Proof Sketch:The argument proceeds in the same fashion as the probabilistic completeness proof for PRM presentedby Kavraki,Kolountzakis,and Latombe[9].The only signif-icant difference is that,instead of considering the clearancebetween a solution strategy and the obstacle boundaries,wemust consider the clearance from the critical boundaries atwhich shadow events that are not part of theﬁnal solutionstrategy would occur.VII.S IMULATION R ESULTSWe implemented our algorithm in simulation and providesome results for three different environments,using threedifferent sample generators,and three different cycle con-straints.The environments(Figure4)all require at leasttwo pursuers to generate a solution strategy.As such wehave deployed two pursuers to test our algorithm.The threedifferent sample generators have the following behavior:•SG1-Returns a uniform sample in F n.This is a baseline sample generator that produces independentand identically distributed samples in F n.This samplegenerator satisﬁes the completeness constraint.•SG2-Chooses samples such that no two pursuers are mutually visible.By ensuring that the pursuers can not see one another,we attempt to maximize exploration by generating samples where the pursuers’visibility regions don’t overlap.Note that this sample generator does not satisfy the completeness constraint.•SG3-Selects an existing SG-PEG vertex,and for each pursuer selects a new target position from the pursuer’s current visibility region.This is a local randomized sam-pler.By sampling within an existing SG-PEG vertex’s ﬁeld-of-view,we are essentially causing the search to “bloom”from the root vertex.This sample generator does not satisfy the completeness constraint.For each combination of environment,sample generator,and cycle constraints we ran10trials,each with a uniquestarting position.The simulations were implemented in C++on a machine running Ubuntu12.0464-bit with an IntelCore2Duo E8400processor and4GB of RAM.Eachsimulation was given a maximum computation time limit of1200seconds.If the algorithm could not generate as solution strategy within the allotted time,we assumed that it failed. The cycle constraints represent the extremes and one intermediary constraint.By not allowing any cycles,the SG-PEG has a tree structure,and may encounter environments where this limitation prevents our algorithm from generating a solution strategy.The other extreme has no constraint on the cycles.This means that if the samples are close together,。

Adaptive differential evolution algorithm

Adaptive diﬀerential evolution algorithm for multiobjective optimization problemsWeiyi Qian *,Ajun liDepartment of Mathematics,Bohai University,Jinzhou,Liaoning 121000,PR ChinaAbstractIn this paper,a new adaptive diﬀerential evolution algorithm (ADEA)is proposed for multiobjective optimization problems.In ADEA,the variable parameter F based on the number of the current Pareto-front and the diversity of the current solutions is given for adjusting search size in every generation to ﬁnd Pareto solutions in mutation operator,and the select operator combines the advantages of DE with the mechanisms of Pareto-based ranking and crowding dis-tance sorting.ADEA is implemented on ﬁve classical multiobjective problems,the results illustrate that ADEA eﬃciently achieves two goals of multiobjective optimization problems:ﬁnd the solutions converge to the true Pareto-front and uni-form spread along the front.Ó2008Elsevier Inc.All rights reserved.Keywords:Multiobjective optimization problems;Diﬀerential evolution algorithm;Adaptive;Select operator1.IntroductionMultiobjective optimization problems (MOPs)which consist of several competing and incommensurable objective functions are frequently encountered in real-world problem such as scientiﬁc and engineering appli-cations.Consequently,there are two goals in multiobjective optimization:(i)to discover solutions as close to the Pareto-front as possible,(ii)to ﬁnd solutions as diverse as possible in the obtained nondominated front.In recent years,many optimization techniques have been proposed in some literatures to solve MOPs.Some of the most attractive algorithms are evolution algorithms (EAs)such as NSGAII [1],MODE [2],PAES [3].In contrast to traditional gradient-based techniques,EAs use a set of potential solutions to detect feasible region.So several solutions of a multiobjective problem can be obtained in a single run.The properties enable EAs converge fast to the true Pareto-front (the concept will be explained in the following section).In MOPs,a large number of optimal solutions exist,and each correspond to a diﬀerent trade-oﬀamong the objective functions.Diﬀerential evolution algorithm (DE)is designed for minimizing functions of real variable.It is extremely robust in locating the global minimum.DE is a simple yet powerful evolutionary optimization algorithm that0096-3003/$-see front matter Ó2008Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2007.12.052*Corresponding author.E-mail address:qianweiyi@ (W.Qian).Available online at Applied Mathematics and Computation 201(2008)431–440432W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440has been successfully used in solving single-objective problems by Price and Storn[4].After that,it was used to handle MOPs.Abbass was theﬁrst to apply DE to MOPs in the so-called Pareto diﬀerential evolution(PDE) algorithm[5,6].Madavan achieved good results with the Pareto diﬀerential evolution approach(PDEA)[7]. Xue introduced multiobjective diﬀerential evolution[2].Tea Robi^c proposed diﬀerential evolution for multi-objective optimization(DEMO)[8].In this paper,we propose a new way of extending DE to be suitable for solving MOPs.A novel approach, called adaptive diﬀerential evolution algorithm(ADEA)which incorporated a new select operator and an adaptive parameter,is introduced to search the global solutions.In each generation,the select operator emphasize Elitist to promote the research towards the Pareto-front.And the adaptive parameter F was used to adjust step size for the need of algorithm.From the simulate results onﬁve test problems,weﬁnd the speed of converging to the true Pareto-front of ADEA is more fast and the diversity is better than most of other optimization algorithms for multiobjective problems.The rest of the paper is organized as follows:Section2provides the concept of Pareto-front and Pareto-ranking,DE scheme which was used as a background for ADEA.In Section3,a method called ADEA is dis-cussed in detail.Section4outlines the applied test problems and performance measures.Experimental results showing the eﬀectiveness of our approach and the further comparison and discussion of the results are also provided in this section.Section5concludes the paper.2.Background2.1.Pareto-rankingTwo or more objectives in a multiobjective problem are usually conﬂict and compete,they cannot be min-imized concurrently.So one solution often cannot be said to be better or worse than another according their function values.There exists not a single optimal solution,but a set optimal solutions-called Pareto-front.Starting from a set of initial solutions,multiobjective evolutionary algorithms use an iteratively improving optimization techniques toﬁnd the optimal solutions.In every iterative progress,EAs favor population-based Pareto dominance as a measure ofﬁtness.That makes for exploring the potential promising areas of search space and obtain more theﬁnal approximation of the Pareto-optimal front.The Pareto-optimal front is deﬁned as follows:For any two decision vectors a and b,a1bða dominates bÞif fðaÞ6fðbÞK fðaÞ¼fðbÞ;a#bða weakly dominates bÞif fðaÞ6fðbÞ;a$bða is incomparable to bÞif fðaÞi fðbÞK fðbÞi fðaÞ;where relations 2f¼;<;6;i g are for every j2f1;...;n g if f jðaÞ f jðbÞ.A decision vector v is called Pareto-optimal solution when it is not dominate by any other decision vector ~v2X in all objectives.The set of all Pareto optimal solutions is called the Pareto-optimal front.We rest Par-eto-ranking–a concept that has been prevalent since Goldberg’s early work[2]and features in a host of tech-niques toﬁnd Pareto-optimal front.The computation of Pareto-ranking is very complex before Deb et al.[8] proposed a fast ranking approach which requires at most oðMN2Þcomputations in NSGAII.This approach can sort a population of size N according to the level of non-domination fast.2.2.Diﬀerential evolutionDE is an eﬃcient evolutionary optimization algorithm.It has been successfully applied on a plethora of applications.DE is a population set of solution vectors which are successively updated by addition,subtrac-tion and component swapping,until the populations converge,hopefully to optimum.An initial set S consists of N points with corresponding function values is generated randomly in X.Mutation and crossover are used to generate new points according S.A selection progress is then used to drive these points to the vicinity of the global minimizer.That is,theﬁnal minimizers will be obtained through irritating the initial populations.Now we will describe the progress of DE in detail.N points are sampled in X as initial set S¼f x1;x2;...;x N g.Take N)n,n being the dimension of the func-tion f(x).In each iteration,N competitions between target points and trial points which are generated through mutation and crossover operators are held to determine the members of S for the next generation.In mutation phrase,DE randomly selects three distinct points x(1),x(2),x(3)from the current set S.The weighted diﬀerent of any points is then added to the third points which can be mathematically described as: ^x i¼x pð1ÞþFðx pð2ÞÀx pð3ÞÞ;ð1Þwhere F61is a selecting factor.Then a trial point yi as a competition to target point x i will be found from itsparents x i and^x i using the following crossover rules:y j i¼^x jiif R j6C R or j¼I i;x j i if R j>C R and j¼I i;(ð2Þwhere I i is a randomly chosen integer in the set I,i.e.I i¼f1;2;...;n g;the superscript j represents the j th com-ponent of respective vectors:R j2ð0;1Þ;draw randomly for each j.The entity C R is a constant.Notice that for C R¼1,the trial vector y i is equal to the mutated vector,that is,only the mutation operation is used to repro-duction.The goal of the crossover operation is to get the trial vector yi .The selection mechanism decideswhich point(x i and yi )is accept.All the solutions in the population have the same chance of being selectedas parents without dependence on theirﬁtness value.The child produced after the mutation and crossover operations is evaluated.Then,the performance of the child vector and its parents is compared and the better one is selected.If the parent is still better,it is retained in the population.The diﬀerential evolution algorithm(DE):Step1.Determine the initial set S¼f x1;x2;...;x N gwhere the points x i;i¼1;...;N are sampled randomly in X.Evaluate f(x)at each x i2S.Take N)n, n being the dimension of the f(x).Set generation counter k=0.Step2.While stopping criterion not meet,doStep2.1.Generate points to replace points in S.For each x i2S,determine y i by the following two reproduction operations.Mutation:Randomly select three points from S except x i,ﬁnd the second parent^x i by the muta-tion rule(1).Crossover:Calculate the trial vector y i corresponding to x i from x i and^x i using the crossover rule(2).Step2.2.Replace points in S.Evaluate the trial vector yi ,if yiis better than its parent x i;yireplacesx i,otherwise yiis discarded.Set k=k+1.Go to step2.3.Adaptive diﬀerential evolution algorithmWhen applying DE to MOPs,we face many diﬃculties.How to replace the parent with trial solutions and preserve a uniform spread of nondominated solutions are two main challenge for DE algorithm.The selection operator of original DE is simple but it is notﬁt for the MOPs.The selection is easy in single-objective opti-mization,but the selection is not so straight forward in MOPs.It exists that the trial solutions and the parent solutions are incomparable.According to the original DE,if the trial solution is not dominance the parent,the trial is discarded.This method lead to miss the optimal solutions that have been found.Meanwhile,the sen-sitivity of the parameter F was not study,authors just chose their values randomly and mainly between0.5and 0.9.The eﬄuence of F to the new points was ignored.To overcome the above mentioned drawbacks,we proposed a adaptive parameter F and a new selection method.We do not compare trial solution with its parent when a trial point is generated,but keep it in the trial set.When all the members in the population set generate trial solutions,we get a new population size of two popsize with trial set and parent set.Then truncate it to the size of popsize for the next step of the algorithm.The truncation consists of sorting the individuals with nondominated sorting and evaluating the W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440433individuals of the same front with the crowding distance metric.The truncation procedure continue until only the best popsize individuals are get.The truncation is similar to what in NSGAII,but ADEA introduces a diﬀerent crowding distance metric.It is outlined in Fig.1.We use diagonal of the largest cuboid which enclos-ing the point x i without including any other point in the population as density estimation of the population.Fig.1outlines the crowing distance computation procedure of the solutions in an nondominated set I.In DE,the generation of new points is mainly dependent on mutation operator.In mutation,parameter F plays an important role.F has inﬂuence on the speed of converge and the diversity of the solutions.F decides the search ually,optimization algorithms favor global search at the early stage for exploring feasible domain and ﬁnding the global optimization solutions and local search at the latter stage for accelerating con-verge.Based on above strategy,we deﬁnes a parameter F as follows:F ¼max P k j ¼1P m j i ¼1j d ij À d j j þd f P j Q j Á d þd f ;1À2j P j j Q j ;l min !where d ij is the crowding distance of the i th solution in the j th Pareto level; dj is the average value of crowd-ing distance of the solutions in the j th Pareto level; dis the average value of crowding distance of the solu-tions in every iteration;j P j is the number of the nondominated solutions;j Q j is the number of the population set;the parameter d f represent the Euclidean distance between two boundary solutions in Q ;l min is a lower bound for F .ADEA is implemented as follows:Initially,a random parent population S is created.The truncation procedure is introduced for calculating F in the ﬁrst iteration and all the solutions in S are retained.In the other iteration,truncation procedure used to select the parent set (for next generation)and calculating F .F is decided by the number of the current Pareto-front and the diversity of the current solutions.It can adjust its size with the need of the algorithm.The gen-eration of new points is as the same as DE (according mutation rule and crossover rule).Then a combined population S ¼S [Q (including new points)is formed.The population S will be of size 2N .Since all previous and current population members are included in S ,the elitism is insured.The ADEA procedure is shown in Fig.2.4.Experimental results 4.1.Performance measuresTo validate our approach,we used the methodology normally adopted in the evolutionary multiobjective optimization literature.Because we wanted to compare ADEA to other MOEAs on their published results,we use three metrics that have been used in these studies.They represent both quantitative comparisons and qual-Fig.1.Crowding distance metric in ADEA.434W.Qian,A.li /Applied Mathematics and Computation 201(2008)431–440itative comparisons with MOEAs that are respective of the state-of-the-art:the Nondominated Sorting Genetic Algorithm II (NSGAII)the Strength Pareto Evolutionary Algorithm (SPEA),the Pareto Archived Evolution Strategy (PAES),the Pareto-frontier Diﬀerential Evolution Approach (PDEA),the Multiobjective Optimization Evolution Algorithm (MOEA).(1)Convergence metric Ç.This metric is deﬁned as:Ç¼P ni ¼1d in;where n is the number of nondominated vector found by the algorithm being analyzed and d i is the Euclidean distance (measured in the objective space)between the obtained nondominated front Q and the nearest member in the true Pareto front P .It measures the distance between the Q and P .(2)Generational distance (GD).This metric is similar to Ç.It is deﬁned as:GD ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃP ni ¼1d ip n ;where d i is the Euclidean distance (measured in the objective space)between the obtained nondominated front Q and the nearest member in the true Pareto-front P .It should be clear that a value of GD =0indicates that all the elements generated are in the true Pareto-front of the problem.(3)Diversity metric D .This metric measures the distant of spread achieved among the nondominated solutions.It is deﬁned as:D ¼d f þd l þP ni ¼1ðd i À dÞd f þd l þðn À1Þ dwhere d i is the Euclidean distance (measured in the objective space)between consecutive solutions in theobtained nondominated front Q and dis the average of these distances.The parameters d f and d l rep-resent the Euclidean distance between the extreme solutions of the true Pareto-front P and the boundary solutions of the obtained front Q .4.2.Test problems and resultsThe test problems for evaluating the performance of our methods are chosen based on signiﬁcant past stud-ies in multiobjective evolutionary algorithms.We chose ﬁve problems from benchmark domains commonly used in past multiobjective GA research (in the literature [1,7,2,10]).For every test problem,number of population points (NP)=100,number of iterations (NG)=200:Fig.2.Outline of ADEA.W.Qian,A.li /Applied Mathematics and Computation 201(2008)431–440435Test problem1(ZDT1):Convex Pareto-front:f1ðxÞ¼x1;f2ðxÞ¼gðxÞð1Àﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx1=gðxÞpÞ;gðxÞ¼1þ9nÀ1X ni¼2x iwith n=30and x i2½0;1 .Test problem2(ZDT2):Nonconvex Pareto-front f1ðxÞ¼x1;f2ðxÞ¼gðxÞð1Àðx1=gðxÞÞ2Þ;gðxÞ¼1þ9nÀ1X ni¼2x iwith n=30and x i2½0;1 .Test problem3(ZDT3):Discontinuous Pareto-front:f1ðxÞ¼x1;f2ðxÞ¼gðxÞð1Àﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx1=gðxÞpÀx1gðxÞsinð10p x1ÞÞ;gðxÞ¼1þ9nÀ1X ni¼2x iwith n=30and x i2½0;1 .Test problem4(ZDT4):Many local Pareto-fronts:f1ðxÞ¼x1;f2ðxÞ¼gðxÞð1Àﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃx1=gðxÞpÞ;gðxÞ¼1þ10ðnÀ1ÞþX ni¼2ðx2iÀ10cosð4p x iÞÞwith n=10and x12½0;1 x i2½À5;5 for i¼2; (9)Test problem5(ZDT6):Low density of solutions near Pareto-fronts: f1ðxÞ¼1ÀexpÀ4x1sinð6p x1Þ6;f2ðxÞ¼gðxÞð1Àðf1ðxÞ=gðxÞÞ2Þ;gðxÞ¼1þ9nÀ1X ni¼2x iwith n=10and x i2½0;1 .The results of ADEA is shown in Figs.3–7,the behavior of ADEA is compared to NSGAII on a number of test problems.To match the settings of the algorithms used for comparison,the population size was set100 and the algorithm was run for200generations.As can be seen,ADEA is able to generate uniform distribution of solutions and the results is better than NSGAII on both quality and quantity on zdt1,2,3,4,6.parison and discussionResults onﬁve test functions,in relation to the convergence metric C and diversity metric D,are presented in Table1–5,the mean and variance of the values are averaged over10runs.The results for ADEA and other ﬁve algorithms which are taken from the literature(see[9]for the results and parameter settings of both ver-sions of NSGAII,SPEA and PAES,[7]for PDEA,and[2]for MODE)are provided.436W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440437Results for PDEA in[6]were evaluated with generational distance instead of the convergence metric.As can be seen,ADEA is able to converge better in all problems except in ZDT1and ZDT2,where NSGA-II(binary-coded)and SPEA found better convergence.The challenge for MOEAs in theﬁrst three test prob-lems(ZDT1,ZDT2and ZDT3)lies in the high-dimensionality of these problems.Many MOEAs have achieved very good results on these problems in both goals of multiobjective optimization(convergence to the true Pareto-front and uniform spread of solutions along the front).The results for the problems ZDT1 and ZDT2(see Tables1and2)show that ADEA achieves good results,which are comparable to the results of the algorithms NSGA-II(real-coded),PAES and MODE.On ZDE3(see Table3),ADEA outperform all other algorithms used in comparison.438W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440Table1Statistics of results on test problems ZDT1Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.033482±0.0047500.390307±0.001876 NSGA-II(binary-coded)0.000894±0.0000000.463292±0.041622 SPEA0.001799±0.0000010.784525±0.004440 PAES0.082085±0.008679 1.229794±0.000742 PDEA N/A0.298567±0.000742 MODE0.005800±0.000000N/ASDE0.002741±0.0003850.382890±0.001435 Table2Statistics of results on test problems ZDT2Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.072391±0.0316890.430776±0.004721 NSGA-II(binary-coded)0.000824±0.0000000.435112±0.024607 SPEA0.001339±0.0000000.755184±0.004521 PAES0.126276±0.036877 1.165942±0.007682 PDEA N/A0.317958±0.001389 MODE0.005500±0.000000N/ASDE0.002203±0.0002970.345780±0.003900W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440439 Table3Statistics of results on test problems ZDT3Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.114500±0.0049400.738540±0.019706 NSGA-II(binary-coded)0.043411±0.0000420.575606±0.005078 SPEA0.047517±0.0000470.672938±0.003587 PAES0.023872±0.0000100.789920±0.001653 PDEA N/A0.623812±0.000225 MODE0.021560±0.000000N/ASDE0.002741±0.0001200.525770±0.043030 Table4Statistics of results on test problems ZDT4Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.513053±0.1184600.702612±0.064648 NSGA-II(binary-coded) 3.227636±7.3076300.479475±0.009841 SPEA7.340299±6.5725160.798463±0.014616 PAES0.854816±0.5272380.870458±0.101399 PDEA N/A0.840852±0.035741 MODE0.638950±0.500200N/ASDE0.100100±0.4462000.436300±0.110000 Table5Statistics of results on test problems ZDT6Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.296564±0.0131350.668025±0.009923 NSGA-II(binary-coded)7.806798±0.0016670.644477±0.035042 SPEA0.221138±0.0004490.849389±0.002713 PAES0.085469±0.006664 1.153052±0.003916 PDEA N/A0.473074±0.021721 MODE0.026230±0.000861N/ASDE0.000624±0.0000600.361100±0.036100 ZDT4is a hard optimization problem with manyð219Þlocal Pareto-fronts that tend to mislead the optimi-zation algorithm.In Table4,we can see that all algorithms have diﬃculties in converging to the true Pareto-front.But ADEA get very good result.With theﬁrst test problem ZDT6,there are two major diﬃculties.Theﬁrst one is thin density of solutions towards the Pareto-front and the second one nonuniform spread of solutions along the front.On this problem, ADEA outperform all other algorithms.In theﬁrst three and the last problems,crossover rate is0.7,ZDT4is0.3.For the diversity metric,ADEA outperform any other proposed algorithms.5.ConclusionADEA is a new DE implementation dealing with multiple objectives.The biggest diﬀerence between ADEA and other MOEAs is that ADEA introduced a new deﬁned self-parameter and a new select operator.We tested the approach onﬁve benchmark problems and it was found that our approach is competitive to most other approaches.We also experimented with diﬀerent crossover rate on these problems toﬁnd their best solu-tions.The crossover rate is found to be sensitive on problem4to the solutions.In the near future,we also plan to evaluate ADEA on additional test problems.440W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440References[1]K.Deb,A.Pratap,S.Agarwal,T.Meyarivan,A fast and elitist multiobjective genetic algorithm:NSGA-II,IEEE Transactions onEvolutionary Computation6(2002)182–197.[2]F.Xue,A.C.Sanderson,R.J.Graves,Pareto-based multi-objective diﬀerential evolution,Proceedings of the2003Congress onEvolutionary Computation(CEC’2003),vol.2,IEEE Press,Canberra,Australia,2003,pp.862–869.[3]J.Knowles,D.Corne,The Pareto archived evolution strategy:a new baseline algorithm for multiobjective optimization,in:Proceedings of the1999Congress on Evolutionary Computation,IEEE Service Center,Piscataway,NJ,1999,pp.98–105.[4]K.V.Price,R.Storn,Diﬀerential evolution-a simple evolution strategy for fast optimization,Dr.Dobb’s Journal22(1997)18–24.[5]H.A.Abbass,R.Sarker,C.Newton,PDE:A Pareto-frontier diﬀerential evolution approach for multi-objective optimizationproblems,Proceedings of the Congress on Evolutionary Computation2001(CEC’2001),vol.2,IEEE Service Center,Piscataway,NJ, 2001,pp.971–978.[6]H.A.Abbass,The self-adaptive Pareto diﬀerential evolution algorithm,Proceedings of the Congress on Evolutionary Computation(CEC’2002),vol.1,IEEE Service Center,Piscataway,NJ,2002,pp.831–836.[7]N.K.Madavan,Multiobjective optimization using a Pareto diﬀerential evolution approach,Proceeding of the Congress onEvolutionary Computation(CEC’2002),vol.2,IEEE Service Center,Piscataway,NJ,2002,pp.1145–1150.[8]Tea Rolicˇ,Bogdan Filipic ,in:C.A.Coello,et al.(Eds.),DEMO:Diﬀerential Evolution for Multiobjective Optimization,EMO2005,LNCS3410,2005,pp.520–533.[9]mpinen,A bibliography of diﬀerential evolution algorithm.<http://www2.lut.ﬁ/~jlampine/debiblio.htm>.[10]D.E.Goldberg,Genetic Algorithms in Search,Optimization,and Machine Learning,Addison-Wesley,Reading,MA,1989.。

《人工智能-一种现代方法》第四版习题答案

Chapter 2
2.1 Define in your own words the following terms: agent, agent function, agent program, rationality, reflex agent, model-based agent, goal-based agent, utility-based agent, learning agent. The following are just some of the many possible definitions that can be written:
2.2 Both the performance measure and the utility function measure how well an agent is doing. Explain the difference between the two. A performance measure（性能度量） is used by an outside observer to evaluate（评估） how successful an agent is. It is a function from histories to a real number. A utility function（效用函数） is used by an agent itself to evaluate how desirable（令人想要） states or histories are. In our framework, the utility function may not be the same as the performance measure; furthermore, an agent may have no explicit utility function at all, whereas there is always a performance measure.

An adaptive evolutionary algorithms for numerical optimization

1 Introduction
Many practical engineering applications can be formulated as numerical optimization problems. Methods for solving these problems can roughly be divided into local optimization techniques and global optimization techniques. Local optimization techniques such as gradient-based algorithms are limited to nding local optimum. They usually take the gradient information as the search guidance. However, not always such information is available because in the real world the objective functions are often very complex, multimodaland nondi erentiable. As a consequence, there is a continuing seek for new and more sophisticated techniques capable of handling such problems. In the past decade we have seen an increasing interest in simulated evolution approaches to optimization, including genetic algorithms (GAs), evolution strategies (ESs) and evolutionary programmings (EPs), which are all named as evolutionary algorithms (EAs). Unlike gradient-based algorithms, EAs use a multi-agent search strategy and exchange

基于自适应权重的多重稀疏表示分类算法_段刚龙_魏龙_李妮

网络出版时间：2012-08-16 10:45网络出版地址：/kcms/detail/11.2127.TP.20120816.1045.019.htmlComputer Engineering and Applications计算机工程与应用基于自适应权重的多重稀疏表示分类算法段刚龙, 魏龙, 李妮DUAN Ganglong, WEI Long, LI Ni西安理工大学信息管理系, 陕西西安 710048Department of Information Management, Xi’an University of Technology, Xi’an 710048, ChinaAdaptive weighted multiple sparse representation classification approach Abstract：An adaptive weighted multiple sparse representation classification method is proposed in this paper. To address the weak discriminative power of the conventional SRC (sparse representation classifier) method which uses a single feature representation, we propose using multiple features to represent each sample and construct multiple feature sub-dictionaries for classification. To reflect the different importance and discriminative power of each feature, we present an adaptive weighted method to linearly combine different feature representations for classification. Experimental results demonstrate the effectiveness of our proposed method and better classification accuracy can be obtained than the conventional SRC method.Key words：adaptive weight; multiple sparse representation; SRC摘要：提出了一种基于多特征字典的稀疏表示算法。

黏菌是什么生物

黏菌是什么生物黏菌是锥足变形虫门黏菌门的总称。

和变形虫是近亲。

在系统发育上，变形虫应该是动物和真菌共同的姐妹群。

关于走迷宫是酱紫的：nakagaki et al., (2000, 2001, 2007) 发现黏菌会伸展自己的细胞质并覆盖住整个迷宫平面直至发现食物，然后缩回多余的部分只剩下最短路径，如下图所示（nakagaki 2001):白线是最短路径，ag是起点和终点。

黏菌先布满整个迷宫，然后依最短路径缩回多余的部分。

reid et al.(2012) 进一步发现当黏菌无法事先布满整个空间的时候，它们会进行自由式的空间移动探索。

在探索的过程中会在介质表面留下粘液标记，并会避免回到有粘液标记的地方，这样就不会走回头路。

如图：d是已经被探索过并被粘液标记的地方，黏菌会避免再向那个方向生长并向新的空间探索。

关于机器人的故事其实没有那么夸张啦。

tsuda et al.(2007)做了个机器人如下图：机器人有六条腿，黏菌被保存在下面的六边形盒子里。

这个盒子将模拟机器人感受到的光的方向和强度，并给黏菌同样的刺激。

黏菌会跑到黑暗的地方，它的移动方向会反馈给机器人控制它的移动。

最后，机器人会跑到黑暗的地方。

这个故事告诉我们，路痴比黏菌更可恶。

路痴就是人间失格，嗷嗷嗷嗷。

做reference list 是因为今天做了泡菜坛子，感觉自己萌萌哒：nakagaki t. et al. 2000. maze-solving by an amoeboid organism. nature 407:470.nakagaki t. et al. 2001. path finding by tube morphogenesis in an amoeboid organism. biophys chem 92: 47-52.nakagaki t. et al. 2007. minimum-risk path finding by an adaptive amoebal network. phys rev lett 99: 068104.reid cr. et al. 2012. slime mold uses an externalized spatial "memory" to navigate in plex environments. pnas 109: 17490-17494.tsuda s. et al. 2001. robot control with biological cells. biosystems 87:215-223.。

面向模型未知的自由漂浮空间机械臂自适应神经鲁棒控制

WANGChao, JIN G Lijian， YEXiaoping，JIANGLihong，ZHANG Wenhui (School of Engineering,Lishui University,Lishui 323000,Zhejiang,China)
A bstract：In order to solve the problem that the precise mathematical model of free-floating space
确获得，利用神经网络控制器来补偿机械臂动力学模型, 设计网络权值的自适应学习律实现在线实时调整，避免
对数学模型的依赖 .设计自适应鲁棒控制器来抑制外界扰动和补偿逼近误差，提高系统鲁棒性和控制精度 .基于 Lyapunov理论，证明了闭环系统的稳定性.仿真试验验证了所提控制方法的有效性，对于自由漂浮空间机器人研究具有重要意义.
关键词：空间机器人 '神经网络 '鲁棒控制 '自适应；稳定性
中图分类号！T P 24
文献标志码： A
文章编号！1672- 5581(2019)02-0153 - 06
Adaptive neural robust control for free-floatin* space manipulator facin* unknown model
manipulators is difficult ot obtain and the parameters of the dynamic model will change due to the external

动态优化演化算法收敛性

Vector Prediction Approach to HandleDynamical Optimization ProblemsBojin Zheng1,2,Yuanxiang Li2, ,and Ting Hu31College of Computer Science,South-Central University For Nationalities,Wuhan,430074,China2State Key Lab.of Software Engineering,Wuhan University,Wuhan,430072,China 3Dept.of Computer Science,Memorial University of Newfoundland,NL,Canada zhengbojin@,yxli@,tingh@cs.mun.ca Abstract.The Dynamical Optimization Evolutionary Algorithms(DOEAs)have been applied to solve Dynamical Optimization Problemswhich are very common in real-world applications.But little work focusedon the convergent DOEAs.In this paper new deﬁnitions of convergenceare proposed and a new algorithm named Vector Prediction Approach isdesigned.This algorithmﬁrstly analyzes the genes of best individuals fromthe past,then predicts the next genes of best individual in every tick byGene Programming,such that the algorithm tracks the optima when timevarying.The numerical experiments on two test-bed functions show thatthis algorithm can track the optima when time varying.The convergenceof this algorithm under certain conditions is proved.1IntroductionReal-world optimization problems are often time-varying,so they are called Non-Stationary Optimization Problems or Dynamical Optimization Problems(DOPs). At present,applying Evolutionary Computation to solve Static Optimization Problems is under extensive investigation,but as to DOPs,many foundational questions are not answered yet.Since1960s,various Dynamical Optimization Evolutionary Algorithms (DOEAs)have been proposed.Kazuko Yamasaki et al.[1]pointed out that all the algorithms base on those typical characteristics of DOPs as follows:1.Reappearance.The representative DOEAs are memory-based DOEAs[2,3].2.Continuity.For example,some DOEAs employ neighborhood searchoperators.3.Rarity.The representative DOEAs are Diversity-based DOEAs.The Trig-gered Hyper-mutation Approach would be the representation of this kind of algorithm[4,5].4.Predictability.”Futurist Approach”[6]is the representative algorithm.Cur-rently,prediction-based DOEAs focus on how to rebuild the dynamical environment[7].T.-D.Wang et al.(Eds.):SEAL2006,LNCS4247,pp.353–360,2006.c Springer-Verlag Berlin Heidelberg2006354 B.Zheng,Y.Li,and T.HuFurthermore,some algorithms tried to combine the characteristics to improve the performance of DOEAs.For example,Multi-population Approach bases on the reappearance and rarity[8].So far,the convergence of DOEAs has not been well considered.Little work has focused on this topic.Actually,the DOEAs mentioned above do not con-verge except some memory-based DOEAs under some special conditions.Even if these memory-based DOEAs,they are just capable of dealing with some speciﬁc Dynamical Optimization Problems with small periods.In this paper,we focus on how to design a convergent DOEA.Firstly,we pro-pose the deﬁnitions of convergence.Secondly,we introduce a new approach named Vector Prediction Approach which analyzes the track of every gene of the chro-mosome of the best solutions from the past,and predict the next locations of the genes,such that the algorithm can track the optima of DOPs.Thirdly,we theoret-ically prove the convergence of the proposed algorithm under some certain condi-tions.Fourthly,we represent the experimental results which experimentally prove the convergence.At last,we discussed some important issues and future work. 2Introduction to the AlgorithmDeﬁnition1(Strong Convergence).For function f(−→x,t),if one algorithm can obtain an inﬁnite sequence of−→x∗t(−→x∗t1,−→x∗t2,···)to satisfy∀ε>0∧i>0|min f(−→x,t i)−f(−→x∗,t i)|<ε(1) then the algorithm is strongly convergent to f(−→x,t).Here−→x is independent variable vector,f(−→x,t)is a time-varying function,and the functional value is real number,t is time variable andεis an positive arbi-trary small constant,i is a positive integer.Deﬁnition2(Weak Convergence).For function f(−→x,t),if one algorithm can obtain an inﬁnite sequence of−→x∗t(−→x∗t1,−→x∗t2,···)to satisfy∀ε>0,∃N>0such that∀i>N|min f(−→x,t i)−f(−→x∗,t i)|<ε(2) then the algorithm is weakly convergent to f(−→x,t).The strong convergence is very hard to achieve.So we focus on the weak con-vergence to the predictable Dynamical Optimization Problems.It is a good choice to employ meta-learning approach to learn from the past and construct the−→x∗t’s expressions and predict the next value:−→x t+1.We suggest that two populations be used,one for evolving,and the other for predicting.The algorithm can be depicted as follows,Procedure VPAInitializing the parameters;Initializing the population;Vector Prediction Approach to Handle Dynamical Optimization Problems355 Initializing the predicting population;Evaluate and store the best;while(not termination-condition)dobeginif environment changed thenrecord the number of changesif prediction-condition thenﬁttingend ifpredict the current best solutioncopy the prediction solution into populationfor each solution in populationdo crossover and mutation operators to generate the new solutionif the new solution is better than current solutionreplace the current solution with the new solutionend ifend forstore the best solutionend ifendIn this algorithm,the population(also called evolutionary population)means the population to be used to solve problem in the current tick and the predicting population means the population to be used to predict.2.1Environmental DetectionEnvironmental detection would contribute to the convergence of DOEAs.Here we assume that the algorithm knows the ticks.That is,the algorithm knows when the environment would change.2.2Prediction ProcedureIn this algorithm,we use the improved GEP(Gene Expression Programming) [9,10]to predict the functions,e.g.,use the GEP to perform symbolic regression for obtaining some expressions,and then compute−→x t+1The procedure would be depicted as follows,Procedureﬁtting;BeginWhile(notﬁtting-termination-condition)dofor each individual in predicting populationevaluate the current individualdo crossover,mutation operator to generate a new individualevaluating the new individualif the new individual is better than current individual in predicting pop-ulation then356 B.Zheng,Y.Li,and T.Hureplace current individual in predicting population with the new one end ifend forend whilestore the best individual in the predicting populationend;In this algorithm,every individual’s chromosome is a vector,and the dimen-sion number is the same as the dimension number of−→x.Every element in this vector is a functional expression,and every expression is divided into two parts, theﬁrst is symbolic sequence,the second is a array to store the real constants.Therefore,the chromosome of every individual can be depicted as follows,⎧⎪⎪⎨⎪⎪⎩expression1expression2...expression n(3)And the evaluation function would beF itness=f(x1,x2,...,x n,t)x i=expression i(−→x best,t)(4) We store the symbolic sequence with the suﬃx expression.This make the encoding,decoding and evaluation easy,because it is unnecessary to construct the grammatical trees.2.3Prediction ConditionsWhen the individual which is generated by the predicting population is worse than the best individual which is generated by the evolutionary population,it is obvious that the prediction process should continue.But when the individual which is generated by the predicting population is better than the best individual which is generated by the evolutionary population and continue for several times, we can think the prediction results are good enough,and the prediction process would pause until the prediction is necessary again.3Convergence AnalysisHere we denote predicting population as P1and evolutionary population as P2. The basic deﬁnitions and theorems on Markov Chain are not introduced here, please refer to[11].Deﬁnition3(K−M Predictable).Given a sequence S and a universal ma-chine,if there exists a code C whose length is K bit,for any t,such that the machine should obtain S(t)in M step,then the sequence S is(K-M)predictable.Vector Prediction Approach to Handle Dynamical Optimization Problems 357Lemma 1.A homogeneous Markov chain with ﬁnite state space and irreducible transition matrix visits every state inﬁnitely often with probability one regardless of the initial distribution.Proposition 1.Assumed that the variable space is discrete and for any i ∈[1,n ],here n is the number of genes,the sequences of genes are S i (K i ,M i )predictable ,the populations sequence (P 1,P 2)t (t ≥0)is a homogeneous ﬁnite Markov chain with irreducible transition matrix.Theorem 1.If the variable space is discrete,the environmental changes are known and for any i ∈[1,n ],here n is the number of genes,the sequences of genes are S i (K i ,M i )predictable,then VPA should weakly converge with probability one.Proof.1.According to lemma 1,as the sequences of genes are S i (K i ,M i )pre-dictable,the maximum search space of predicting population would be 2ni =1K i.Assumed that the variable space is 2l ,the total size of state space would be 2n i =1K i +l.Because the transition matrix is irreducible,every state would be in-ﬁnitely often visited with probability one regardless of the initial distribution.2.Because the elitism is employed in this algorithm,once the best individual in evolutionary population and the best individual in predicting population emerge simultaneously,they would be stored and would not be replaced.3.Because the best predicting individual and the best evolutionary individual would emerge si-multaneously with probability one and would be stored forever,so this algorithm would weakly converge with probability one.4Numerical Experiments and Analysis 4.1Benchmark FunctionsCurrently,the moving parabola problem[12]is one of the foundations of the other problems.Here we design two test-bed functions based on it.Function Z1f (x 1,x 2,t )=100∗((x 1−sin(t )2+(x 2−cos(t ))2)(5)Here t=(current generation)-(current generation)mod 2,this means,the environment will change with a 2-generation interval.Function Z2f (x 1,x 2,x 3,t )= y 21+y 22+y 23y 1=x 1−sin(t 2)y 2=x 2−t y 3=x 3−x 1cos(t 2)(6)358 B.Zheng,Y.Li,and T.HuHere t is the same as in Z1.Function Z1has a small period,but Z2is not periodic.4.2Parameters SettingFor comparing with the proposed DOEA,we re-implemented a DOEA with Memory-Enhanced Strategy(MES).The size of memory is set to60.And the best individual in current tick would replace the eldest individual in the memory just after the environmental change.For both DOEAs,the size of population is set to300and the generation number is set to5000.For every function,the DOEAs are carried out30times.4.3Results and AnalysisFor direct representation,we plot the bestﬁtness sequences of MES and VPA on Z1and Z2(The data come from only one run).Fig.1.The comparison on Z1From theﬁgures we can see that this algorithm can obtain very nice results as the red lines have shown:after some generations(the very generation would be called as theﬁrst hitting generation),this algorithm obtained the optima at every tick,but MES did not,though it obtained very good results when dealing with Z1.Furthermore,we did notﬁnd the convergence trend of MES. Actually,theoretically,it would not converge when dealing with such problems.First Hitting Generation(FHG)and Oﬀ-Line Performance(OLP)are used to measure the performance of both the algorithms.In Table1,the average FHG(AVG FHG)and the standard error of FHG(SE FHG),average OLP (AVG OLP)and the standard error of OLP(SE OLP)are compared.The data in table1imply that the performance of VPA is much better than MES.Actually,VPA converged in all the runs,but MES did not.Vector Prediction Approach to Handle Dynamical Optimization Problems359Fig.2.The comparison on Z2parisons of Performance MeasuresAVG FHG SE FHG AVG OLP SE OLPZ1MES5000-0.03555430.0062121 VPA50.6666667 1.34740660.00003340.0000008Z2MES5000- 1.02054650.0005568 VPA651.466666722.55946040.00357940.00010915Discussions and Future WorkIn this paper,we propose a new approach:Vector Prediction Approach.The experimental results and theoretical analysis imply that the algorithm has a good performance.In contrast to the memory-based approach,the proposed approach could deal with non-periodic Dynamical Optimization Problems.We concentrate on only the dynamical optimization problems without random-ness,because the”domino eﬀects”of randomness are very diﬃcult to be catego-rized.Actually,the randomness probably make a mass of DOPs unsolvable or meaningless.Even if some of them are solvable,the”online”and”oﬀ-line”per-formance measures which are commonly used are somehow doubtable because the measures base on a hypothesis:the comparative algorithms should converge.Any-how,the randomness would be an interesting topic and we leave it as future work.Notwithstanding its limitation,this study does imply that Vector Prediction Approach should be feasible to deal with the DOPs which are predictable. AcknowledgementsThe authors gratefully acknowledge theﬁnancial support of the National Natural Science Foundation of China under Grant No.60473014,and the Specialized360 B.Zheng,Y.Li,and T.HuResearch Fund for the Doctoral Program of Higher Education of China under Grant No.20030486049.The authors gratefully thank to Ms.Lingling Wang, Feng Wang and Shujuan Peng for the modiﬁcation.References1.Yamasaki,K.,Kitakaze,K.,Sekiguchi,M.:Dynamic optimization by evolutionaryalgorithms applied toﬁnancial time series.In:Proceedings of the2002Congress on Evolutionary Computation,CEC’02.Volume2.(2002)2017–20222.Bendtsen,C.N.,Krink,T.:Dynamic memory model for non-stationary optimiza-tion.In:Proceedings of the2002Congress onEvolutionary Computation,CEC’02.Volume1.(2002)145–1503.Branke,J.:Memory enhanced evolutionary algorithms for changing optimizationproblems.In:Proceedings of the1999Congress on Evolutionary Computation, CEC99.Volume3.(1999)1875–18824.Cobb,H.G.:An investigation into the use of hypermutation as an adaptive oper-ator in genetic algorithms having continuous,time-dependent nonstationary envi-ronments.Technical Report Memorandum Report6760,Navy Research Labora-tory,Washington,D.C.,USA(1990)5.Morrison,R.:Design evolutionary algorithms for dynamic environments.PhDthesis,George Mason University,Fairfax,Virgina(2002)6.Hemert,J.,Hoyweghen,C.,Lukshandl,E.,Verbeeck,K.:A futurist approach todynamic environments.In Branke,J.,B¨a ck,T.,eds.:Evolutionary Algorithms for Dynamic Optimization Problems,San Francisco,California,USA(2001)35–38 7.Dorigo,M.,Gambardella,L.M.,Middendorf,M.,Stutzle,T.:Special section onant colony optimization.IEEE Transactions on Evolutionary Computation6(4) (2002)317–3208.Branke,J.:A multi-population approach to dynamic optimization problems.InParmee,I.,ed.:Fourth International Conference on Adaptive Computing in De-sign and Manufacture(ACDM2000),Plymouth,Grossbritannien,Springer Verlag (2000)299–3089.Ferreira,C.:Gene expression programming in problem solving.In:the6th OnlineWorld Conference on Soft Computing in Industrial Applications(WSC6).Volume1.(2001)1–2210.Ferreira,C.:Gene expression programming:A new adaptive algorithm for solvingplex Systems13(2)(2001)87–12911.Iosifescu,M.:Finite Markov Processes and Their Applications.Chichester:Wiley(1980)12.Branke,J.:Evolutionary Optimization in Dynamic Environments.Kluwer AcademicPublishers(2002)。

求解TSP问题的自适应邻域遗传算法

（5）在集合 M 中随机抽取选择城市 t，作为下一个城市，将 M 清空，从 V 中移除城市 t；
（6）如果 V 集合不空，令 i=t，跳到（3）；否则搜索结束。针对图 1 的城市图，假定从 C8 出发，采用邻域法将会在 C7、C3和 C4中作出选择，若选择 C7，其后会选择 C3，以此类推，当样本比较充分的情况下就存在很大概率得到图 3 所示的优化路径。
遗传算法（Genetic Algorithm）是模拟生物在自然环境中
的遗传和进化过程而形成的一种自适应全局优化概率搜索算法。Goldberg 等首次应用 GA 求解 TSP 问题[6]。文献[2]指出将 GA 应用求解 TSP 问题时的主要缺点是：对初始种群很敏感，初始种群的选择常常直接影响解的质量和算法效率；对于结构复杂的组合优化问题，搜索空间大，搜索时间比较长，往往会出现早熟收敛的情况。
2 邻域法 2.1 最近邻法
针对 TSP 问题最自然的启发寻优方式应该是最近邻法。在此方法中，从某个城市出发，其下一站是从未去过的最近城市。最近邻法可以采用如下方法描述：设推销员从城市 i 出发，则集合 V={1，2，…，i-1，i+1，…，n}表示未到过的城市，在该集合中寻找城市 k，满足 di，k=min{di，j：j Î V}。若集合 V 中还有元素，令 i＝k，从集合 V 中去除城市 k，则重复上述步骤，直到集合 V 为空，从而求得一个哈密尔顿（Hamilton）回路[1]。
摘要：提出结合自适应邻域法与遗传算法来求解 TSP 问题。在自适应邻域法中，从某个城市出发，下一城市不一定是其最近城市，而是在比其最近城市稍远的邻域范围进行动态随机选取。在求解 TSP 时，采用自适应邻域法对种群初始化，然后采用选择、交叉、变异进行迭代，在选择中仅保留父代 90％的样本，剩下的采用自适应邻域法产生新样本进行补充。仿真实验结果表明所提算法与其他算法相比具有竞争能力。关键词：遗传算法；旅行商问题；最近邻法；自适应邻域法 DOI：10.3778/j.issn.1002-8331.2010.27.005 文章编号：1002-8331（2010）27-0020-05 文献标识码：Ａ中图分类号：TP301

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An adaptive amoeba algorithm for constrained shortestpathsXiaoge Zhang a ,Yajuan Zhang a ,Yong Hu b ,Yong Deng a ,c ,⇑,Sankaran Mahadevan caSchool of Computer and Information Science,Southwest University,Chongqing 400715,ChinabInstitute of Business Intelligence and Knowledge Discovery,Guangdong University of Foreign Studies,Guangzhou 510006,China cSchool of Engineering,Vanderbilt University,Nashville,TN 37235,USAa r t i c l e i n f o Keywords:Constrained shortest path Adaptive amoeba algorithm Lagrangian relaxation Optimizationa b s t r a c tThe constrained shortest path problem (CSP)is one of the basic network optimization problems,which plays an important part in real applications.In this paper,an adaptive amoeba algorithm is combined with the Lagrangian relaxation algorithm to solve the CSP problem.The proposed method is divided into two steps:(1)the adaptive amoeba algorithm is modiﬁed to solve the shortest path problem (SPP)in a directed network;(2)the modiﬁed adaptive amoeba algorithm is combined with the Lagrangian relaxa-tion method to solve the CSP problem.In addition,the evolving processes of the adaptive amoeba model have been detailed in the paper.Two examples are used to illustrate the efﬁciency of the proposed method.The results show that the proposed method can deal with the CSP problem effectively.Ó2013Elsevier Ltd.All rights reserved.1.IntroductionThe shortest path problem (SPP)is one of the basic network optimization problems,which has been applied in many ﬁelds,such as road navigation in transportation (Deng,Chen,Zhang,&Mahadevan,2012;Liu,Zheng,&Cai,2013;Mahdavi,Nourifar,Hei-darzade,&Amiri,2009;Zhang,Zhang,Deng,&Mahadevan,2013a;Zhu,Zhang,Song,&Li,2011),trafﬁc routing in communication net-works (Cai,Zhang,Zhou,Cao,&Tang,2012;Cappanera &Scaparra,2011;Junior,Nedjah,&de Macedo Mourelle,2013),schedule plan-ning in robotic systems (Chen,Deng,&Wu,2013;Marzouk &Ali,2013)and others (Deng,Liu,Hu,&Deng,2013;Gao,Lan,Zhang,&Deng,2013;Kang,Deng,Sadiq,&Mahadevan,2012;Liu,Chan,Li,Zhang,&Deng,2012;Sura &Mahadevan,2011).The constrained shortest path problem (CSP)refers to ﬁnding the shortest path from the source node to the sink node under additional constraints for the paths in the network.Constrained shortest path applica-tions arise frequently in practice since in many contexts a company (e.g.a package delivery ﬁrm)wants to provide its services at the lowest possible cost and yet ensure a certain level of service to its customers (Hester,Adams,&Mahadevan,2010;Mahadevan &Overstreet,2012).The additional constraints to the SPP problem make CSP become NP-hard (Carlyle,Royset,&Kevin Wood,2008;Dumitrescu &Boland,2003).Many methods have been developed to solve CSP problems (Carlyle et al.,2008;Deng,Chan,Wu,&Wang,2011a;Deng,Jiang,&Sadiq,2011b;Royset,Carlyle,&Wood,2009).These methods can be divided into two categories:one is the k shortest paths algo-rithm (Lefebvre,Puget,&Vilím,2011),and the other is based on Lagrangian relaxation (Carlyle et al.,2008).In recent years,researchers have explored to bio-inspired algorithms due to their ﬂexibility and simplicity (Beheshti,Shamsuddin,&Hasan,2013;Das &Mishra,2013;Dudek,2013;Gómez-Gasquet,Andrés,&Lar-io,2012;Lee,Lai,Chen,&Yang,2013;Mousavi,Hajipour,Niaki,&Alikar,2013;Zhang,Hu,Chen,&Shen,2012;Zhang,Deng,Chan,&Zhang,2013c;Zhang et al.;Zhang &Wang,2013).For example,Mendes presented a genetic algorithm for Resource Constrained Project Scheduling (Mendes,Gonçalves,&Resende,2009)in 2009.Mohemmed investigated on the application of particle swarm optimization (PSO)to solve shortest path (SP)routing prob-lems (Mohemmed,Sahoo,&Geok,2008).Recently,a large amoeboid organism,the plasmodium of Physa-rum polycephalum ,has been shown to be capable of solving many graph theoretical problems (Baumgarten,Ueda,&Hauser,2010;Nakagaki &Yamada,2000;Tero,Kobayashi,&Nakagaki,2006;Tero et al.,2010;Zhang et al.,2013;Zhang,Zhang,Zhang,Wei,&Deng,2013),including ﬁnding the shortest path (Miyaji &Ohnishi,2008;Nakagaki,Yamada,&Tth,2001;Nakagaki et al.,2007;Nakagaki et al.,2007).Inspired by this intelligent organism,a path ﬁnding mathematical model has been established (Tero,Kobayashi,&Nak-agaki,2007).Bonifaci has proved that the mathematical model can convergence to the shortest path (Bonifaci,Mehlhorn,&Varma,2012).However,the model is only designed for undirected net-works while there are many directed networks in real applications,such as transportation networks.What’s more,it can only deal with SPP without constraints by now.In this paper,we improve the amoeba model to solve the CSP problem.The proposed method can be divided into 2parts:(1)0957-4174/$-see front matter Ó2013Elsevier Ltd.All rights reserved./10.1016/j.eswa.2013.07.054⇑Corresponding author at:School of Computer and Information Science,South-west University,Chongqing 400715,China.Tel./fax:+8602368254555.E-mail addresses:ydeng@ ,prof.deng@ (Y.Deng).the adaptive amoeba algorithm is modiﬁed to solve the shortest path problem (SPP)in directed network;(2)the modiﬁed adaptive amoeba algorithm is combined with Lagrangian relaxation method to solve the CSP problem.In addition,the evolving processes have been described in the paper.The remainder of this paper is organized as follows.Section 2introduces the basic theory including CSP and the amoeba model.In Section 3,an adaptive amoeba algorithm for CSP problem is pro-posed.Section 4gives two examples to illustrate the efﬁciency of the proposed method.Section 5ends the paper with conclusions.2.Basic theoriesIn this section,some basic theories including constrained short-est path problem,Lagrangian relaxation ,amoeba model are introduced.2.1.Constrained shortest path problemLet G ¼V ;E ðÞbe a network,where V ¼1;ÁÁÁ;n is the set of nodes and E ¼i ;j ðÞ:i ;j 2n ;i –j is the set of edges.Each edge has two non-negative weights c ij and t ij .c ij represents the generalized cost,and t ij represents the constrained variable,for instance travel time.The constrained shortest path problem can be stated as the following integer programming problem:minX i ;jc ij x ijð1Þsubject toX i ;j x ij ÀXj ;ix ji ¼1for j ¼1;0for j ¼2;...;n À1À1for j ¼n 8><>:ð2ÞXi ;jt ij x ij 6Tð3Þx ij 20;1f g ;8i ;jwhere x ij is binary variable,which is deﬁned as follows:x ij ¼1if x ij is in optimal path ;0others&The parameter T represents the maximum value sum of t ij .The shortest path problem is formulated by and (4).Constraint (3)make that CSP problem belongs to NP-hard grangian relaxationConsider the following optimization problem:min c T x s :t :Ax ¼b x 2XThe Lagrangian relaxation method uses the idea of explicit linear constraints by bringing them into the tion with associated vector k called the Lagrange lowing problem can be treated as Lagrangian original problem (6).min c T x þk T Ax Àb ðÞs :t :x 2XThen,the function L k ðÞ¼min c T x þk T Ax Àb ðÞ x 2ÈLagrangian function.2.3.Amoeba modelFrom the experiments on the amoeboid organism as described in Nakagaki,Yamada,and Toth (2001),the mechanism of tube for-mation can be obtained:tubes thicken in a given direction when shuttle streaming of the protoplasm persists in that direction for a certain time.It implies positive feedback between ﬂux and tube thickness,as the conductance of the sol is greater in a thicker channel.According to the mechanism,two rules describing the changes in the tubular structure of the amoeboid organism are:ﬁrst,open-ended tubes,which are not connected between the two food sources,are likely to disappear;second,when two or more tubes connect the same two food sources,the longer tube is likely to dis-appear (Tero et al.,2007).With these two rules,a mathematical model for maze solving problems has been constructed.Using the maze illustrated in Fig.1,the model can be described as follows.Each segment in the diagram represents a section of tube.Two special nodes,which are also called food source nodes,are named N 1and N 2,and the other nodes are denoted as N 3;N 4;N 5,and so on.The section of tube between N i and N j is de-noted as M ij .If several tubes connect the same pair of nodes,inter-mediate nodes will be placed in the center of the tubes to guarantee the uniqueness of the connecting segments.The variable Q ij is used to express the ﬂux through tube M ij from N i to N j .Assuming the ﬂow along the tube as an approximately Poiseuille ﬂow,the ﬂux Q ij can be expressed as:Q ij ¼D ijL ij ðp iÀp j Þð8Þwhere p i is the pressure at the node N i ;D ij is the conductivity of the edge M ij .Assume zero capacity at each node;by considering the conser-vation law of sol the following equation can be obtained:XQ ij ¼0ðj –1;2Þð9ÞFor the source node N 1and the sink node N 2the following two Graphical maze:the source node N 1and the sink node N 2are indicated and other nodes are shown by solid squares Tero et al.(2007)7608X.Zhang et al./Expert Systems with Applications 40(2013)7607–7616where I 0is the ﬂux ﬂowing from the source node.It can be seen thatI 0is a constant value in this model.In order to describe such an adaptation of tubular thickness we assume that the conductivity D ij changes over time according to the ﬂux Q ij .The following equation for the evolution of D ij ðt Þcan be usedddtD ij ¼f ðj Q ij jÞÀrD ij ð12Þwhere r is a decay rate of the tube.The equation implies that the conductivity ends to vanish if there is no ﬂux along the edge,while it is enhanced by the ﬂux.f is monotonically increasing continuous function satisfying f ð0Þ¼0.Then the network Poisson equation for the pressure can be ob-tained from Eqs.(8)–(10),as follows:X iD ijL ij ðp i Àp j Þ¼À1for j ¼1;þ1for j ¼2;0otherwise8><>:ð13ÞBy setting p 2=0as a basic pressure level,all p i can be determined by solving Eq.13and Q ij can also be obtained.Since f is monotonically increasing continuous function satisfy-ing f ð0Þ¼0in Eq.(12),f ðQ Þ¼j Q j is used in this paper and the rea-son lies that it has been certiﬁed in Tero et al.(2007)that f ðQ Þ¼j Q j is able to ﬁnd the shortest path with rapid speed when compared with other functions.With the ﬂux calculated,the con-ductivity can be derived,where Eq.(14)is used instead of Eq.(12),adopting the functional form f ðQ Þ¼j Q j .D n þ1ij ÀD n ij ¼j Q j ÀD n þ1ijð14ÞRemove the edges with conductivityequal to 0EndInitializationTermination criterion is met ornotCalculate the pressure of eachnodeGet the directed shortest pathCalculate the flux of each edgeWhether there is route fromnode i to node j?The pressure of node i is larger than node j?Change the conductivity of the edge from node i to node j to 0,the flux to be0Directed NetworkX.Zhang et al./Expert Systems with Applications 40(2013)7607–761676093.Proposed methodWith experiments of pathﬁnding process of the mathematical model described in Section2.3,it can be seen that the bio-inspired model can handle the shortest path problem.However,it is ob-served that the original amoeba algorithm can onlyﬁnd the short-est path in the undirected network.In this section,the proposed method is detailed to solve the CSP problem in a directed network. As a consequence,there are two issues that need to be solved.One is how to solve shortest path problem in the directed network,the other is how to solve CSP problem.As a result,the proposed meth-od is composed of two parts.3.1.Shortest path problem in directed networkLet G¼N;E;WðÞbe a directed network,where N denotes a set of n nodes,E denotes an edge set with m directed edges,and L de-notes a weight set for E.Assume that there is only one directed edge between two nodes in G.Given a source node s and a target node t,the directed shortest path problem can be deﬁned as how toﬁnd a path from s to t,which only consists of directed edges of E,with the minimum sum of weights on the edges.In the undirected networks,every edge is bidirectional.In origi-nal amoeba model,the arc starts from the node with higher pres-sure and ends in the node with lower pressure.There is a positive feedback mechanism:the higher the pressure,the more theﬂux.The more theﬂux,the higher the pressure.This positive feedback make the shortest path fade in when the iteration continues.How-ever,in a directed network,only considering the pressure of each node is not enough.The directivity of each arc should be taken into consideration.In the proposed method,we adopt a novel method to solve this issue.Theﬂow chart of the proposed method forﬁnding the directed shortest path from s to t in G is shown in Fig.2.Step1:Initialize tube length and conductivity for G.In G,each tube has two attributes,its length L i;j and conductivity D i;j.The length L i;j is initialized according to the weight of the edge L i;j in G,while the conductivity is assigned equally as0.5.pi states the pressure of node i.At the same time,the basic infor-mation of the directed network is constructed.Step2:Remove the edges with conductivity equal to0in order to reduce the computing time.Step3:Calculate the pressure of each node using its current conductivity and length according to Eq.(13).Step4:Calculate the pressure and conductivity of each node according to Eq.(8)and Eq.(14)during the next iteration.Step5:On the basis of the information of the directed network, it can be determined that whether there is a path from node i to node j.If it exists,R ij in the mark matrix will be1.Otherwise,R ij is0.During each iteration,the pressure of each node can be obtained.If there is a path from node i to node j(R ij¼1),check whether the pressure of node i is higher than that of node j in7610X.Zhang et al./Expert Systems with Applications40(2013)7607–7616the model of Physarum polycephalum.If pi is smaller than pj,thefollowing operations are processed:change the conductivity D ij value of edge from node i to node j to be0,theﬂux Q ij to be e!0þ.Otherwise,the original data such as D ij;Qijis kept.Step6:Check whether termination criterion is met or not.The termination criterion is set as the conductivity of each tube remaining unchanged.If it is satisﬁed,tubes with conductivity approximating1compose the directed shortest path.Other-wise,go to Step2and repeat until convergence.3.2.Constrained shortest pathsConsider the network shown in Fig.3.Each arc has two attri-butes:cost c ij and traversal time t ij.The numbers along each arc shows its cost and traversal time respectively.For example,the numbers(1,10)along the arc between node1and node2means the cost along the arc is1,and the traversal time is10.Assume that we wish toﬁnd the shortest path from source node1to sink node1,2,,N i1,2,,N i1,2,,NX.Zhang et al./Expert Systems with Applications40(2013)7607–761676116,but we wish to restrict our choice of paths to those that require no more than T ¼10time units for traversal.This is a typical CSP problem.In this paper,we adopt an indirect approach by combining time and cost into a single generalized cost instead of solving this problem directly.Instead of setting a limit on the total time,a ‘‘cost’’is set on each arc,which is proportional to the time that it takes to pass through that arc.Thus,we solve the CSP problem with the modiﬁed cost c ij þk t ij .For instance,if k ¼0,this problem will become the shortest path problem only consider-ing the original costs c ij and the shortest path is 1–2–4–6as shown in Fig.4(a).The total length of the path is 3while its cost time is 18.If we assume that k ¼2and solve the constrained shortest path problem,Fig.4(b)displays the shortest path with the modiﬁed costs c ij þ2Ãt ij .The shortest path is 1–2–5–6and 1–3–2–5–6.Their path length is 35with respect to the modiﬁed costs c ij þ2Ãt ij .However,the cost time of path 1–2–5–6is 15while the cost time of 1–3–2–5–6is 10.The total length,the cost timeTable 1Attribute value of each edge.Edge Length Toll Edge Length Toll 1?2120609?1020601?590509?1690402?31004010?1190502?6906010?1770403?7905011?1870604?8704012?1150505?41206013?18140405?9805014?13100606?10402015?20150506?12604015?1640407?12806016?2080607?13605017?2060508?15704018?1940408?161004019?2050407612X.Zhang et al./Expert Systems with Applications 40(2013)7607–7616will change with k.Fig.5illustrates how the constrained shortest path solution changes with k.As can be seen in Fig.5,the solution to the constrained shortest path problem can be easily obtained.Thisﬁgure displays how the different variables change with k.It can be seen that there is a solu-tion to the above CSP problem when k¼2.In this way,the con-strained shortest path can be obtained as1–3–2–5–6.3.3.The main steps of the proposed methodThe main steps of the proposed method can be summarized as follows.First of all,given the constrained shortest path problem, initialize every variable,such as the initialized conductivity matrix D,the length of each edge L ij,the originalﬂux matrix Q and so on. Secondly,taking the constraint factor into account,we convert the CSP problem into a shortest path problem by Lagrangian relaxation method.Next,the check procedure is processed after each iteration so as the amoeba model can be applied to directed network.Final-ly,the procedure is over when the termination criterion is met. Fig.6shows the procedures of the proposed method.4.ExamplesTwo examples are contained in order to illustrate the proposed method.Example4.1.Fig.7shows a directed transportation network with 20nodes.Each edge has2attributes:length(L),toll(T).Table1 displays the attribute value of every edge.Suppose that the shortest path between node1and node20needs to be found under the constraint that the total toll is less than200.Based on the proposed method,we solve this problem with the modiﬁed costs L ijþk T ij,where k P0.The initialized conductivity matrix is set0:5,the pressure of each node is0,the originalﬂux of each node Q is set with equal to0.The process of the Physarum algorithm is depicted in Fig.8when k¼1.Fig.8(a)represents the initial state of the network,before the implementation of the algorithm.The next twoﬁgures,Fig.8(b)and Fig.8(c)show the intermediate states of the network when some edges has been cut off by the algorithm.Theﬁnal path between two speciﬁed nodes is derived in Fig.8(d).The results are different for different values of k as shown in Fig.9.From Fig.9,the constrained shortest path can be obtained easily.The solution to this CSP problem is1–5–9–16–20.The tolls of the path is200and the length of the path is340.Table2The generated cost value of the random network.Edge Cost Edge Cost Edge Cost1?310.73?219.647?289.862?3 5.2411?2111.29?2812.31?410.414?2111.524?2811.34?515.417?2111.225?2814.43?68.892?22 2.682?2910.74?6 5.1121?227.757?2914.11?715.52?239.609?299.095?78.3215?238.1411?29 3.922?813.616?239.7922?29 5.583?813.44?24 6.2825?299.485?88.527?2411.35?30 6.057?8 6.228?24 6.119?3011.93?98.2510?248.8518?3015.05?910.912?2411.51?31 6.528?98.9313?247.743?3113.34?1012.115?2410.27?3110.31?1110.221?2412.38?3110.310?128.2410?2511.511?3111.111?1311.616?2513.213?318.7112?139.7419?25 6.1714?319.8010?149.4922?2512.619?31 6.3512?159.552?268.6529?3111.113?159.203?2616.53?3212.91?16 5.877?2611.94?3210.93?1613.18?2615.313?3211.87?179.1611?268.1915?328.7115?178.0914?26 6.9918?32 6.435?18 3.0415?2610.423?327.329?1810.017?2617.328?3211.310?18 4.5623?268.923?3312.32?198.342?279.136?337.943?199.046?27 5.999?337.017?198.628?2711.810?3313.12?208.169?27 5.3212?3312.93?209.4914?2714.319?33 5.896?207.0519?27 6.2020?337.058?20 4.4520?2713.823?3311.52?2110.96?287.51X.Zhang et al./Expert Systems with Applications40(2013)7607–76167613Example 4.2.The Delay Constrained Least Cost (DCLC)problem is to ﬁnd the least cost path in a graph while keeping the path delay below a speciﬁed value.This problem has been widely used in routing in computer networks (Jia &Varaiya,2006;Kun,2005;Leela,Thanulekshmi,&Selvakumar,2011;Rocha,Sousa,Cortez,&Rio,2011;Zhengying,Bingxin,&Erdun,2001).The DCLC prob-lem can be characterized as a directed,connected graph G ¼V ;E ðÞ,where V is the set of nodes and E &V ÂV .Correspond-ing to each link/edge ði ;j Þconnecting nodes i and j ,there are non-negative numbers c ij representing the cost of the link/edge and another nonnegative delays d ij representing transmission delay from node i to node j .Given a source node s 2V ,an ending node e 2V ,and a transmission delay constraint D delay ,the following equation can be obtained:min p 2P s ;e ðÞc p ðÞsuch thatd p ðÞ6D delay ð15Þwhere P s ;e ðÞis the set of all possible paths from node s to node e .As a consequence,DCLC is a typical CSP problem.In this paper,a random directed network is generated using the Waxman model (Waxman,1988),and the link probability between node i and node j is deﬁned as follows:P i ;j ðÞ¼c e Àd i ;j ðÞ=b L max ðÞ;0<c ;b 61ð16Þwhere d i ;j ðÞexpresses the Euclidean distance between node i and node j ,and L max is the maximum distance between any two nodes.c and b are two parameters controlling the link probability.In this example,c and b are 0.3and 0.9respectively.The edges’s costs and delays are randomly generated using normal distribution with mean value equal to 10and standard deviation equal to 3.As the test networks are randomly generated (not only topology but also the edge costs and delays),it is necessary to set an appropriate de-lay constraint (threshold)according to the delay values of the net-work.In order to prevent the DCLC problem from becoming trivial,the delay constraint is chosen as (Jia &Varaiya,2006):D delay ¼0:75Ãp ld þ0:25Ãp lcð17Þwhere p ld is the delay of the least delay path,and p lc is the cost of the least cost path.The path with the least cost and the path with the least delay can be found using the amoeba algorithm.The data of the generated network is shown in Tables 2and 3.Based on the data,it can be determined that the least cost path is 1!3!33while the least delay path is 1!4!10!33.According to Eq.(17),the constraint is D delay ¼0:75Ã21:8135þ0:25Ã23:13¼22:1438.As can be seen in Fig.10,the solution to this DCLC problem can be achieved.The answer is 1!4!10!33and the transmission delay is 21:8135,which is less than 22.1438.Table 3The generated delay value of the random network.Edge Delay Edge Delay Edge Delay 1?315.503?2111.47?287.602?312.5811?219.419?289.601?4 6.07614?2110.824?2814.04?511.0217?2114.725?288.233?618.302?2212.02?297.454?619.1021?229.267?2917.51?79.8102?23 6.509?2910.95?79.38515?2310.311?297.402?814.4616?2317.722?2912.33?814.254?2410.525?29 3.015?8 6.3777?24 4.205?3011.07?814.898?24 4.619?3011.33?913.1010?247.3318?3010.55?99.08912?248.361?3112.58?97.63813?248.193?3111.34?10 6.55815?2412.27?318.991?117.57121?249.418?3113.110?1214.3110?257.4811?3113.711?137.73516?25 6.7813?319.7912?13 4.86519?2510.314?319.3410?149.27522?25 4.1119?3110.012?1510.932?26 6.3729?3112.413?159.9093?2612.43?3211.41?1611.887?26 6.824?3211.83?1613.328?269.1813?32 6.917?1710.2311?269.1615?3210.915?17 6.65914?26 3.8418?3211.55?1814.5915?267.5223?327.179?1811.1117?2611.528?329.5610?1813.3523?2610.13?3315.02?1910.092?2713.36?338.543?1913.306?279.109?33 6.477?1910.258?279.2110?339.172?207.7739?279.1412?339.253?2017.0514?277.0619?3314.86?2012.2419?278.3920?339.318?2012.6620?2712.823?338.662?215.7936?289.937614X.Zhang et al./Expert Systems with Applications 40(2013)7607–76165.ConclusionThe constrained shortest path(CSP)problem plays an important role in many network problems.In this paper,an adaptive amoeba algorithm named Physarum polycephalum isﬁrstly modiﬁed to solve the shortest path problem in a directed network.Then the modiﬁed model is combined with Lagrangian relaxation method to solve the CSP problem.To the best of our knowledge,this is theﬁrst time that the adaptive amoeba algorithm is applied to this optimization problem.Examples in route selection in transporta-tion and computer networks are used to illustrate the proposed method.The results demonstrate that the proposed method is able to deal with CSP problem.AcknowledgementThe Ph.D candidates of the corresponding author in Shanghai Jiao Tong University,Xiaoyan Su and Peida Xu,provide valuable discussions on bio-inspired optimization.The work is partially supported by National Natural Science Foundation of China,Grant Nos.61174022and71271061,Chongqing Natural Science Founda-tion(for Distinguished Young Scholars),Grant No.CSCT, 2010BA2003,National High Technology Research and Develop-ment Program of China(863Program)(No.2012AA041101),Sci-ence and Technology Planning Project of Guangdong Province, China(2010B010600034,2012B091100192),Business Intelligence Key Team of Guangdong University of Foreign Studies(TD1202), Doctor Funding of Southwest University Grant No.SWU110021, China State Key Laboratory of Virtual Reality Technology and Sys-tems,the Fundamental Research Funds for the Central Universities Grant No.XDJK2013D010.ReferencesBaumgarten,W.,Ueda,T.,&Hauser,M.(2010).Plasmodial vein networks of the slime mold Physarum polycephalum form regular graphs.Physical Review E, 82(4),046113.Beheshti,Z.,Shamsuddin,S.M.H.,&Hasan,S.(2013).MPSO:Median-oriented 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