Optimization of Mixed DiscreteContinuous Design Variable Systems Using Neural Networks
Self-adaptive differential evolution algorithm for numerical optimization
n
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
1 2 3,
(
“DE/best/1”: Vi ,G = Xbest ,G + F ⋅ Xr ,G − X r G 1 2,
Ant Colony Optimization3
ij (1 p) ij , (i, j) A
• where p (0,1] is a parameter. An iteration is a completer cycle involving ants’ movement, pheromone evaporation, and pheromone deposit.
– – – – – –
– – – – – – –
• Subset
Job Shop Open Shop Flow Shop Total tardiness (weighted/non-weighted) Project Scheduling Group Shop
Multi-Knapsack Max Independent Set Redundancy Allocation Set Covering Weight Constrained Graph Tree partition Arc-weighted L cardinality tree Maximum Clique
ij ij
k
• By using this rule, the probability increases that forthcoming ants will use this arc.
The Algorithm
• After each ant k has moved to the next node, the pheromones evaporate by the following equation to all the arcs:
• Ant reaches next node, selects next path • Continues until reaches starting node • Finished “tour” is a solution
optimal operation of multireservoir systems
Optimal Operation of Multireservoir Systems:State-of-the-Art ReviewJohn badie,M.ASCE 1Abstract:With construction of new large-scale water storage projects on the wane in the U.S.and other developed countries,attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects.Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational management and operational decisions.The purpose of this review is to assess the state-of-the-art in optimization of reservoir system management and operations and consider future directions for additional research and application.Optimization methods designed to prevail over the high-dimensional,dynamic,nonlinear,and stochastic characteristics of reservoir systems are scrutinized,as well as extensions into multiobjective optimization.Application of heuristic programming methods using evolutionary and genetic algorithms are described,along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules.DOI:10.1061/͑ASCE ͒0733-9496͑2004͒130:2͑93͒CE Database subject headings:Reservoir operation;State-of-the-art reviews;Optimization models;Stochastic models;Linear programming;Dynamic programming;Nonlinear programming.IntroductionAccording to the World Commission on Dams ͑WCD 2000͒,many large storage projects worldwide are failing to produce the level of benefits that provided the economic justification for their development.This may be due in some instances to an inordinate focus on project design and construction,with inadequate consid-eration of the more mundane operations and maintenance issues once the project is completed.Performance related to original project purposes may also be undermined when new unplanned uses arise that were not originally considered in the project au-thorization and development.These might include municipal/industrial water supply,minimum streamflow requirements for environmental and ecological concerns,recreational enhance-ment,and accommodating shoreline encroachment and develop-ment.Although there may exist some degree of commensurability among these diverse project purposes,there is more often conflict and competition,particularly during pervasive drought condi-tions.In addition,performance of publically owned reservoir sys-tems is often restricted by complex legal agreements,contracts,federal regulations,interstate compacts,and pressures from vari-ous special interests.With construction of new large-scale water storage projects at a virtual standstill in the U.S.and other developed countries,along with an increasing mobilization of opposition to large stor-age projects in developing countries,attention must focus on im-proving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects.In addition,many of the adverse impacts of large storage projects on aquatic ecosystems can be minimized through im-proved operations and added facilities,as demonstrated by the Tennessee Valley Authority ͑TV A ͒͑Higgins and Brock 1999͒.Construction of bottom outlets or selective withdrawal structures can pass sediments downstream and improve water quality con-ditions.Unfortunately,many existing reservoir operational poli-cies fail to consider a multifacility system in a fully integrated manner,but rather emphasize operations for individual projects.However,the need for integrated operational strategies confronts system managers with a difficult task.Expanding the scope of the working system for more integrated analysis greatly multiplies the potential number of alternative operational policies.This is further complicated by conflicting objectives and the uncertainties associated with future hydrologic conditions,including possible impacts of climate change.Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational operational puter simula-tion models have been applied for several decades to reservoir system management and operations within many river basins.Many models are customized for the particular system,but there is also substantial usage of public domain,general-purpose mod-els such as HEC 5͑Hydrologic Engineering Center 1989͒,which is being updated as HEC RESSIM to include a Windows-based graphical user interface ͑Klipsch et al.2002͒.Spreadsheets and generalized dynamic simulation models such as STELLA ͑High Performance Systems,Inc.͒are also popular ͑Stein et al.2001͒.Other similar system dynamics simulation models include POW-ERSIM ͑Powersim,Inc.͒applied by Varvel and Lansey ͑2002͒,and VENSIM ͑Ventana Systems,Inc.͒applied by Caballero et al.͑2001͒.These simulation or descriptive models help answer what if questions regarding the performance of alternative operational strategies.They can accurately represent system operations and1Professor,Dept.of Civil Engineering,Colorado State Univ.,Ft.Collins,CO 80523-1372.E-mail:labadie@Note.Discussion open until August 1,2004.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and pos-sible publication on August 22,2002;approved on November 27,2002.This paper is part of the Journal of Water Resources Planning and Management ,V ol.130,No.2,March 1,2004.©ASCE,ISSN 0733-9496/2004/2-93–111/$18.00.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y D A L I A N U N I VE R S I T Y OF o n 06/04/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .are useful for Monte Carlo analysis in examining long-term reli-ability of proposed operating strategies.They are ill-suited,how-ever,to prescribing the best or optimum strategies when flexibil-ity exists in coordinated system operations.Prescriptive optimization models offer an expanded capability to systemati-cally select optimal solutions,or families of solutions,under agreed upon objectives and constraints.The purpose of this paper is to assess the state-of-the-art in reservoir system optimization models and consider future direc-tions.This is an update of a review that appeared in Water Re-sources Update published by The Universities Council on Water Resources ͑UCOWR ͒͑Labadie 1997͒.The focus is primarily on optimization of systems of reservoirs,rather than a single reser-voir.This is not meant to imply that single reservoir optimization is unimportant,but rather the substantial technological challenges and rewards abide with integrated optimization of interconnected reservoir systems.Optimization methods designed to prevail over the high-dimensional,dynamic,nonlinear,and stochastic charac-teristics of reservoir systems are scrutinized,as well as extensions into multiobjective optimization.Heuristic programming methods using evolutionary and genetic algorithms are described,along with the application of artificial neural networks and fuzzy rule-based systems for inferring reservoir system operating policies.Overcoming Hindrances to Reservoir System OptimizationDespite several decades of intensive research on the application of optimization models to reservoir systems,authors such as Yeh ͑1985͒and Wurbs ͑1993͒have noted a continuing gap between theoretical developments and real-world implementations.Pos-sible reasons for this disparity include:͑1͒many reservoir system operators are skeptical about models purporting to replace their judgment and prescribe solution strategies and feel more comfort-able with use of existing simulation models;͑2͒computer hard-ware and software limitations in the past have required simplifi-cations and approximations that operators are unwilling to accept;͑3͒optimization models are generally more mathematically com-plex than simulation models,and therefore more difficult to com-prehend;͑4͒many optimization models are not conducive to in-corporating risk and uncertainty;͑5͒the enormous range and varieties of optimization methods create confusion as to which to select for a particular application;͑6͒some optimization methods,such as dynamic programming,often require customized program development;and ͑7͒many optimization methods can only pro-duce optimal period-of-record solutions rather than more useful conditional operating rules.Optimal period-of-record solutions are criticized in the Engineer Manual on Hydrologic Engineering Requirements for Reservoirs ͑U.S.Army Corps of Engineers 1997;pp.4–5͒,where it is stated that ‘‘...the basis for the system operation are not explicitly defined.The post processing of the results requires interpretation of the results in order to develop an operation plan that could be used in basic simulation and applied operation.’’Many of these hindrances to optimization in reservoir system management are being overcome through ascendancy of the con-cept of decision support systems and dramatic advances in the power and affordability of desktop computing hardware and soft-ware.Several private and public organizations actively incorpo-rate optimization models into reservoir system management through the use of decision support systems ͑Labadie et al.1989͒.Incorporation of optimization into decision support systems has reduced resistance to their use by placing emphasis on optimiza-tion as a tool controlled by reservoir system managers who bear responsibility for the success or failure of the system to achieve its prescribed goals.This places the focus on providing support for the decision makers,rather than overly empowering computer programmers and modelers.An example of an optimization model incorporated into a de-cision support system ͑DSS ͒is the MODSIM river basin network flow model ͑Labadie et al.2000͒,which is currently being used by the U.S.Bureau of Reclamation for operational planning in the Upper Snake River Basin,Idaho ͑Larson et al.1998͒.The Windows-based graphical user interface ͑GUI ͒in MODSIM al-lows the user to create any reservoir system topology by simply clicking on various icons and placing system objects in any de-sired configuration on the screen.Data structures embodied in each model object on the screen are controlled by a database management system,with formatted data files prepared interac-tively and a network flow optimization model automatically ex-ecuted from the interface.Results of the optimization are pre-sented in useful graphical plots,or even customized reports available through a scripting language included with MODSIM .Complex,non-network constraints on the optimization in MOD-SIM are incorporated through an iterative procedure using the embedded PERL scripting language.RiverWare ͑Zagona et al.1998͒affords similar DSS functionality with an imbedded pre-emptive goal programming model providing the optimization ca-pabilities.RiverWare has been successfully applied to the TV A system for operational planning ͑Biddle 2001͒.Although lacking a generalized Windows-based graphical user interface,CALSIM has been developed by the California Depart-ment of Water Resources to allow specification of objectives and constraints in strategic reservoir systems planning and operations without the need for reprogramming ͑Munevar and Chung 1999͒.Similar to the use of PERL script in MODSIM,CALSIM employs an English-like modeling language called WRESL ͑Water Re-sources Engineering Simulation Language ͒that allows planners and operators to specify targets,objectives,guidelines,con-straints,and associated priorities,in ways familiar to them.Simple text file output,along with time series and other data read from relational data bases,are passed to a mixed integer linear programming solver for period by period solution.CALSIM II replaces the DWRSIM and PROSIM models that required con-tinual reprogramming as new objectives and constraints were specified,for coordinated operation of the Federal Central Valley and California State Water Projects.OASIS ͑HydroLogics,Inc.͒is a similar modeling package to CALSIM that uses an Operations Control Language ͑OCL ͒for developing linear programming models for multiobjective analysis of water resource systems.The explosion of readily available information through the In-ternet has increased the availability of advanced optimization methods and provided freely accessible software and data re-sources for successful implementation.Many powerful optimiza-tion software packages are available through the Internet,such as from the Optimization Technology Center ͑Northwestern Univer-sity and Argonne National Laboratory,Argonne,Illinois ͒at ͗/otc/otc.html ͘.In addition,several spreadsheet software packages available on desktop computers include linear and nonlinear programming solvers in their numeri-cal toolkits.The generalized dynamic programming package CSUDP ͑Labadie 1999͒facilitates the use of dynamic program-ming models,avoiding the need to develop new code for each application.CSUDP software is freeware and can be downloaded at ͗ftp:///distrib/͘.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y D A L I A N U N I VE R S I T Y OF o n 06/04/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .The power and speed of the modern desktop computer have reduced the degree of simplifications and approximations in res-ervoir system optimization models required in the past,and opened the door to greater realism in optimization modeling.The primacy of the system manager over the model is also empha-sized in the incorporation of knowledge-based expert systems into reservoir system modeling which recognize the value of the in-sights and experience of reservoir system operators.Despite these advances,optimization of the operation of an integrated system of reservoirs still remains a daunting task,particularly with attempts to realistically incorporate hydrologic uncertainties.Reservoir System Optimization Problem Objective FunctionAccording to the ASCE Task Committee on Sustainability Crite-ria ͑1998͒,‘‘Sustainable water resource systems are those de-signed and managed to fully contribute to the objectives of soci-ety,now and in the future,while maintaining their ecological,environmental and hydrological integrity.’’Objective functions used in reservoir system optimization models should incorporate measures such as efficiency ͑i.e.,maximizing current and future discounted welfare ͒,survivability ͑i.e.,assuring future welfare ex-ceeds minimum subsistence levels ͒,and sustainability ͑i.e.,maxi-mizing cumulative improvement over time ͒.Loucks ͑2000͒states that ‘‘sustainability measures provide ways by which we can quantify relative levels of sustainability...One way is to express relative levels of sustainability as separate weighted combinations of reliability,resilience and vulnerability measures of various cri-teria that contribute to human welfare and that vary over time and space.These criteria can be economic,environmental,ecological,and social.’’The strategy of shared vision modeling ͑Palmer 2000͒is useful for enhancing communication among impacted stakeholders and attaining consensus on planning and operational goals.A generalized objective function for deterministic reservoir system optimization can be expressed asmax ͑or min ͒r͚t ϭ1T␣t f t ͑s t ,r t ͒ϩ␣T ϩ1T ϩ1͑s T ϩ1͒(1)where r t ϭn -dimensional set of control or decision variables ͑i.e.,releases from n interconnected reservoirs ͒during period t ;T ϭlength of the operational time horizon;s t ϭn -dimensional state vector of storage in each reservoir at the beginning of period t ;f t (s t ,r t )ϭobjective to be maximized ͑or minimized ͒;T ϩ1(s T ϩ1)ϭfinal term representing future estimated benefits ͑or costs ͒be-yond time horizon T ;and ␣t ϭdiscount factors for determining present values of future benefits ͑or costs ͒.The dynamic nature of this problem reflects the need to repre-sent an uncertain future for sustainable water management;i.e.,‘‘...a future we cannot know,but which we can surely influence’’͑Loucks 2000͒.The time step t used in this formulation may be hourly,daily,weekly,monthly,or even seasonal,depending on the nature and scope of the reservoir system optimization prob-lem.Hierarchical strategies may also be pursued whereby results from long-term monthly or seasonal studies provide input to more detailed short-term operations over hourly or daily time periods ͑Becker and Yeh 1974;Divi and Ruiu 1989͒.The objective function may be highly nonlinear,such as for maximizing hydropower generationf t ͑s t ,r t ͒ϭ͚i ϭ1nK •e i ͑s it ,s i ,t ϩ1,r it ͒•h ¯it ͑s it ,s i ,t ϩ1͒•r it •⌬t it (2)where e i ϭoverall powerplant efficiency at reservoir i as a func-tion of average head and discharge during period t ;h ¯it ϭaveragehead as a function of beginning and ending period storage levels ͑calculated from the reservoir mass balance or system dynamics equation ͒,as well as possibly the discharge if tailwater effects are included;K ϭunit conversion factor;and ⌬t it ϭnumber of on-peak hours related to the load factor for powerplant i .This is a highly nonconvex function characterized by many local maxima ͑Tauxe et al.1980͒,and may be discontinuous and nondifferentiable if loading of individual turbines in the powerplant is considered.Other objective functions related to vulnerability criteria may at-tempt to minimize deviations from ideal target storage levels,water supply deliveries,discharges,or power capacities.If eco-nomic benefit and cost estimates are available for these uses,then the objective may be to maximize total expected net benefits from operation of the system,but with consideration of long-term sus-tainability.ConstraintsThe system dynamics or state-space equations are written as fol-lows,based on preservation of conservation of mass throughout the system:s t ϩ1ϭs t ϩCr t ϩq t Ϫl t ͑s t ,s t ϩ1͒Ϫd t͑for t ϭ1,...,T ͒(3)where s t ϭstorage vector at the beginning of time t ;q t ϭinflow vector during time t ;C ϭsystem connectivity matrix mapping flow routing within the system;l t ϭvector combining spills,evaporation,and other losses during time t ;and d t ϭrequired de-mands,diversions,or depletions from the system.In some formu-lations,diversions are treated as decision variables and included in the objective function as related to benefits of supplying water.Accurate calculation of evaporation and other water losses in the term l t (s t ,s t ϩ1)creates a set of nonlinear implicit equations in s t ϩ1which can be difficult to evaluate and constitute a nonconvex feasible set.Initial storage levels s 1are assumed known and all flow units in Eq.͑3͒are expressed in storage units per unit time.Spatial connectivity of the reservoir network is fully described by the routing or connectivity matrix C .For the example reservoir system of Fig.1,the connectivity matrixisFig.1.Example reservoir system configurationD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y D A L I A N U N I VE R S I T Y OF o n 06/04/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .C ϭͫϪ10000Ϫ1000ϩ1Ϫ10ϩ1ϩ1Ϫ1ͬAdditional state variable nodes with zero storage capacity may represent nonstorage locations where inflows and diversions occur.For more complex system configurations that are nonden-dritic,such as bifurcating systems and off-stream reservoirs,a more complex link-node connectivity matrix is gged routing of flows can be considered by replacing the term Cr t inEq.͑3͒with ͚ϭ0kC r t Ϫ,where elements of the routing matrices C may be fractions representing lagging and attenuation of downstream releases.Explicit lower and upper bounds on storage must be assigned for recreation,providing flood control space,and assuring mini-mum levels for dead storage and powerplant operation.s t ϩ1,min рs t ϩ1рs t ϩ1,max͑for t ϭ1,...,T ͒(4)Limits on reservoir releases are specified asr t ,min рr t рr t ,max͑for t ϭ1,...,T ͒(5)These limits maintain minimum desired downstream flows for water quality control and fish and wildlife maintenance,as well as protection from downstream flooding.In some cases,it may be necessary to specify these limits as functions of head where al-lowable discharges depend on reservoir storage levels.Additional constraints may be imposed on the change in release from one period to the next to provide protection from scouring of down-stream channels.When evaluating long term historical or syn-thetic hydrologic sequences,or multiple short-term sequences,difficulties may arise in finding feasible solutions that satisfy these constraints.In these cases,it may be necessary to relax these as explicit constraints and indirectly consider them through use of weighted penalty terms on violation of these constraints in the objective function.Other constraints may represent alternative objectives that must be maintained at desired target levels :f ¯͑s ,r ͒у(6)Example targets might include annual water supply requirements or power capacity maintenance.These targets may be adjusted parametrically to compute tradeoff relations between the primary objective of Eq.͑1͒and secondary objectives as a means of pro-viding multiple objective solutions ͑Cohon 1978͒.The optimization model defined in Eqs.͑1͒–͑6͒is challenging to solve since it is dynamic,potentially nonlinear,and nonconvex;and large-scale.In addition,unregulated inflows,net evaporation rates,hydrologic parameters,system demands,and economic pa-rameters should often be treated as random variables,giving rise to a complex large-scale,nonlinear,stochastic optimization prob-lem.In this formulation,it is assumed that calibration and verifi-cation studies have been carried out to assure the model is capable of reasonably reproducing historical energy production,storage levels,and flows throughout the system.This review explores several solution strategies,including implicit stochastic optimiza-tion,explicit stochastic optimization,real-time optimal control with forecasting,and heuristic programming methods.For more detailed treatment of these topics,the reader is referred to a num-ber of important books written over the years dealing with opti-mization of water resource systems in general,and optimal opera-tion of reservoirs in particular.These include:Maass et al.͑1962͒;Hall and Dracup ͑1970͒;Buras ͑1972͒;Loucks et al.͑1981͒;Mays and Tung ͑1992͒;Wurbs ͑1996͒;and ReVelle ͑1999͒.Implicit Stochastic OptimizationThe solution of Eqs.͑1͒–͑6͒may be accomplished by implicit stochastic optimization ͑ISO ͒methods,also referred to as Monte Carlo optimization,which optimize over a long continuous series of historical or synthetically generated unregulated inflow time series,or many shorter equally likely sequences ͑Fig.2͒.In this way,most stochastic aspects of the problem,including spatial and temporal correlations of unregulated inflows,are implicitly in-cluded and deterministic optimization methods can be directly applied.The primary disadvantage of this approach is that optimal operational policies are unique to the assumed hydrologic time series.Attempts can be made to apply multiple regression analy-sis and other methods to the optimization results for developing seasonal operating rules conditioned on observable information such as current storage levels,previous period inflows,and/or forecasted inflows.Unfortunately,regression analysis may result in poor correlations that invalidate the operating rules,and at-tempting to infer rules from other methods may require extensive trial and error processes with little general applicability.Linear Programming ModelsSince ISO models can be extremely large-scale,covering a lengthy time horizon,it is critical that only the most efficient optimization methods are applied.One of the most favored opti-mization techniques for reservoir system models is thesimplexFig.2.Implicit stochastic optimization ͑ISO ͒procedureD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y D A L I A N U N I VE R S I T Y OF o n 06/04/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .method of linear programming and its variants ͑Nash and Sofer 1996͒.These models require all relations associated with Eqs.͑1͒–͑6͒to be linear or linearizable.The advantages of linear pro-gramming ͑LP ͒include:͑1͒ability to efficiently solve large-scale problems;͑2͒convergence to global optimal solutions;͑3͒initial solutions not required from the user;͑4͒well-developed duality theory for sensitivity analysis;and ͑5͒ease of problem setup and solution using readily available,low-cost LP solvers.Recent al-ternatives to the simplex method,such as the affine scaling and interior projection methods ͑Terlaky 1996͒,are particularly attrac-tive for solving extremely large-scale problems.Hiew et al.͑1989͒applied ISO using LP to the eight-reservoir Colorado-Big Thompson ͑C-BT ͒project in northern e of a 30year historical hydrologic record of monthly unregu-lated inflows to the system resulted in a linear programming prob-lem with 12,613variables and 5,040constraints.Multiple regres-sion analysis was applied to the LP model results to produce optimal lag-one storage guide curves:s t ϩ1ϭA ¯s t*ϩB ¯q t Ϫ1ϩc ¯(7)where s t *ϭoptimal storage levels obtained from the linear pro-gramming solution;q t ϭobserved hydrologic inflows;and corre-lation matrices A ¯,B ¯and vector c ¯are calculated from multiple regression analysis performed on the LP results.Coefficients of determination obtained from this analysis ranged from 0.795to 0.996for the larger reservoirs,with the remaining reservoirs ei-ther small or with water levels only allowed to vary a few feet per year.Simulation of the system operations using the optimal stor-age guide curves of Eq.͑7͒confirmed their validity.This study was successful because of the ability of linear models to accu-rately represent the system behavior,along with the fortunate cal-culation of high correlation coefficients obtained from the mul-tiple regression analysis.For other systems,these advantages may not be in evidence.Other extensions of linear programming into binary,integer,and mixed-integer programming may be valuable for representing highly nonlinear,nonconvex terms in the objective function and constraints ͑e.g.,Trezos 1991͒,but these methods are consider-ably less efficient computationally and would likely be intractable for use in ISO.Needham et al.͑2000͒applied mixed integer lin-ear programming to deterministic flood control operations in the Iowa and Des Moines Rivers,but noted the potential for exces-sive computer times when extended to stochastic evaluation.This study came to the rather counterintuitive conclusion that coordi-nated operation of reservoir systems does not necessarily improve performance,which stands in stark contrast with other studies that have shown just the opposite ͑e.g.,Shim et al.2002͒.Piecewise linear approximations of nonlinear functions are often used in separable programming applications and solved with various extensions of the simplex method,although problem size can become excessive in some cases.Functions of more than one variable can be approximated using multilinear interpolation methods over a multidimensional grid.For minimization prob-lems,these functions must be convex;otherwise,more time con-suming restricted basis entry simplex algorithms must be applied which fail to guarantee convergence to global optima.Crawley and Dandy ͑1993͒applied separable programming to the multi-reservoir Metropolitan Adelaide water supply system in Australia.Network Flow Optimization ModelsIt is evident from Fig.1that an interconnected reservoir system can be represented as a network of nodes and links ͑or arcs ͒.Nodes are storage or nonstorage points of confluence or diver-sion,and links represent reservoir releases,channel or pipe flows,carryover storage,and evaporation and other losses.If all rela-tions in Eqs.͑1͒–͑5͒are linear,then the following dynamic,mini-mum cost network flow problem results:minimize͚t ϭ1T͚ᐉAc ᐉt x ᐉt(8)subject to͚j O ix jt Ϫ͚k I ix kt ϭ0͑for all i N ;for all t ϭ1,...,T ͒(9)l ᐉt рx ᐉt рu ᐉt ͑for all ᐉA ;for all t ϭ1,...,T ͒(10)where A ϭset of all arcs or links in the network;N ϭset of nodes;O i ϭset of all links originating at node i ͑i.e.,outflow links ͒;I i ϭset of all links terminating at node i ͑i.e.,inflow links ͒;x ᐉt ϭflow rate in link ᐉduring period t ;c ᐉt ϭcosts,weighting factors,or priorities per unit of flow rate in link ᐉduring period t ;and l ᐉt and u ᐉt ϭlower and upper bounds,respectively,on flow in link ᐉ.Fig.3illustrates a fully dynamic network where the horizontal arcs represent carryover storage ͑i.e.,s t )in the same physical reservoir from one period to the next,whereas the vertical arcs are flows,releases,and diversions ͑i.e.,r t )during the current period.Eqs.͑8͒–͑10͒define a pure network formulation where all network data can be represented by a set of arc parameters ͓l ᐉt ,u ᐉt ,c ᐉt ͔.For fully circulating networks,additional artificial nodes and links must be added for satisfying overall mass balance throughout the entire parative studies by Kuczera ͑1993͒and Ardekaaniaan and Moin ͑1995͒have shown the dual coordinate ascent RELAX algorithm ͑Bertsekas and Tseng 1994͒to be the most efficient network solver,as compared to primal-based algorithms and variations on the out-of-kilter method ͑Ford and Fulkerson 1962͒.Several network algorithms allow designation of node supply and demand ͓i.e.,entry of values other than zero on the right-hand side of Eq.͑9͔͒without requiring specification of artificial nodes and links,although this is only possible when no demand shortages occur.For so-called networks-with-gains ,Eq.͑9͒must be adjusted with coefficients not equal to Ϫ1,0,or ϩ1to allow for channel losses,evaporation losses,and return flows.Further extensions into generalized networks allow inclusion of side con-straints ͓i.e.,Eq.͑6͔͒that violate the pure network structure.All of these deviations from the pure network format exact acompu-Fig.3.Illustration of dynamic network showing carryover storagearcsD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y D A L I A N U N I VE R S I T Y OF o n 06/04/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。
组合优化问题及算法
启发式算法
邻域概念 对于组合优化问题(D,F,f),D上的一个映射： N:SD N(S)2D
称为一个邻域映射，其中2D表示D的所有子集构成的集合，N(S)称为S的邻域。 邻域的构造依赖于问题决策变量的表示，邻域的结构在现代化优化算法中起重要作用。
5. 二维装箱问题（平面上的套裁问题） 原料的尺寸大于需求的尺寸，需求的品种尺寸可以不同，最终的目标是在满足需求的前提下，使边角余
料最小。 6. 车间作业调度问题(job shop scheduling) n个工件，J1,…,Jn在m台机器M1,M2,…,Mm上加工。每个工件Ji有ni个工序，Oi1,…,Oini,第Oij工序的加工
类似可定义k-opt(k2)
启发式算法
局部最优与全局最优
若s*满足 f(s*)()f(s),其中sN(s*)F,
则称s*为f在F上的局部(local)最小（最大）解。 若s*满足 f(s*)()f(s),其中sF,
则称s*为f在F上的全局(global)最小（最大）解。
启发式算法
启发式算法定义 一个基于直观或经验构造的算法，在可接受的花费（计算时间、占用空间等）下给出待解决问题
s.t. xij 1, i 1,, n
j1
n
xij 1, j 1,, n
i1
xij | S | 1, 2 | S | n 2,
i, jS
xij {0,1}, i, j 1,, n, i j.
S {1,2,, n}
一些例子
3.有约束的机器调度问题(capacitated machine scheduling) n个加工量为{di|i=1,2,…,n}的产品在一台机器上加工，机器在第t个时段的工作能力为ct,求完成所有
two-stage stochastic programming
two-stage stochastic programmingTwo-stage stochastic programming is a mathematical optimization approach used to solve decision-making problems under uncertainty. It is commonly applied in various fields such as operations research, finance, energy planning, and supply chain management. In this approach, decisions are made in two stages: the first stage involves decisions made before uncertainty is realized, and the second stage involves decisions made after observing the uncertain events.In two-stage stochastic programming, the decision-maker aims to optimize their decisions by considering both the expected value and the risk associated with different outcomes. The problem is typically formulated as a mathematical program with constraints and objective functions that capture the decision variables, uncertain parameters, and their probabilistic distributions.The first stage decisions are typically made with theknowledge of the uncertain parameters, but without knowing their actual realization. These decisions are usually strategic and long-term in nature, such as investment decisions, capacity planning, or resource allocation. The objective in the first stage is to minimize the expected cost or maximize the expected profit.The second stage decisions are made after observing the actual realization of the uncertain events. These decisions are typically tactical or operational in nature, such as production planning, inventory management, or scheduling. The objective in the second stage is to minimize the cost or maximize the profit given the realized values of the uncertain parameters.To solve two-stage stochastic programming problems, various solution methods can be employed. One common approach is to use scenario-based methods, where a set of scenarios representing different realizations of the uncertain events is generated. Each scenario is associated with a probability weight, and the problem is then transformed into a deterministic equivalent problem byreplacing the uncertain parameters with their corresponding scenario values. The deterministic problem can be solved using traditional optimization techniques such as linear programming or mixed-integer programming.Another approach is to use sample average approximation, where the expected value in the objective function is approximated by averaging the objective function valuesover a large number of randomly generated scenarios. This method can be computationally efficient but may introduce some approximation errors.Furthermore, there are also robust optimization techniques that aim to find solutions that are robust against the uncertainty, regardless of the actualrealization of the uncertain events. These methods focus on minimizing the worst-case cost or maximizing the worst-case profit.In summary, two-stage stochastic programming is a powerful approach for decision-making under uncertainty. It allows decision-makers to consider both the expected valueand the risk associated with uncertain events. By formulating the problem as a mathematical program and employing various solution methods, optimal or near-optimal solutions can be obtained to guide decision-making in a wide range of applications.。
基于弹性网和直方图相交的非负局部稀疏编码
DOI： 10． 11772 / j． issn． 1001-9081． 2018071483
基于弹性网和直方图相交的非负局部稀疏编码
*பைடு நூலகம்
万 源，张景会 ，陈治平，孟晓静
( 武汉理工大学 理学院，武汉 430070) ( * 通信作者电子邮箱 Jingzhang@ whut． edu． cn)
摘 要： 针对稀疏编码模型在字典基的选择时忽略了群效应，且欧氏距离不能有效度量特征与字典基之间距离 的问题，提出基于弹性网和直方图相交的非负局部稀疏编码方法（ EH-NLSC） 。首先，在优化函数中引入弹性网模型， 消除字典基选择数目的限制，能够选择多组相关特征而排除冗余特征，提高了编码的判别性和有效性。然后，在局部 性约束中引入直方图相交，重新定义特征与字典基之间的距离，确保相似的特征可以共享其局部的基。最后采用多 类线性支持向量机进行分类。在 4 个公共数据集上的实验结果表明，与局部线性约束的编码算法（ LLC） 和基于非负 弹性网的稀疏编码算法（ NENSC） 相比，EH-NLSC 的分类准确率分别平均提升了 10 个百分点和 9 个百分点，充分体现 了其在图像表示和分类中的有效性。
Key words: sparse coding; elastic net model; locality; histogram intersection; image classification
0 引言
图像分类是计算机视觉领域的一个重要研究方向，广泛 应用于生物特征识别、网络图像检索和机器人视觉等领域，其 关键在于如何提取特征对图像有效表示。稀疏编码是图像特 征表示 的 有 效 方 法。考 虑 到 词 袋 （ Bag of Words，BoW） 模 型［1］和空 间 金 字 塔 匹 配 （ Spatial Pyramid Matching，SPM） 模 型［2］容易造成量化误差，Yang 等［3］ 结合 SPM 模型提出利用 稀疏编 码 的 空 间 金 字 塔 的 图 像 分 类 算 法 （ Spatial Pyramid Matching using Sparse Coding，ScSPM） ，在图像的不同尺度上 进行稀疏编码，取得了较好的分类效果。在稀疏编码模型中， 由于 1 范数在字典基选择时只考虑稀疏性而忽略了群体效 应，Zou 等［4］提出一种新的正则化方法，将弹性网作为正则项 和变量选择方法。Zhang 等［5］提出判别式弹性网正则化线性
混沌搜索和模式搜索的混合优化方法
华东理工大学学报(自然科学版)Journal of East China University of Science and Technology (Natural Science Edition )Vol.34No.12008202收稿日期:2007203213作者简介:柳 贺(19792),女,安徽凤阳人,博士生,研究方向为智能控制理论与应用、间歇过程建模与控制研究。
通讯联系人:黄 道,E 2mail :dhuang @ 文章编号:100623080(2008)0120126205基于混沌搜索和模式搜索的混合优化方法柳 贺1, 黄 猛1, 柳桂国1,2, 黄 道1(1.华东理工大学信息科学与工程学院,上海200037;2.浙江工商职业技术学院,浙江宁波315020) 摘要:混沌搜索能够有效跳出局部极小,然而其细搜索能力不足;模式搜索具有很强的细搜索能力,但是其搜索结果的好坏在很大程度上依赖于初始点的选择。
为了提高基于混沌搜索的优化方法的搜索精度,基于混沌搜索和模式搜索,本文提出了一种混合混沌模式搜索方法。
该方法在混沌搜索的基础上再进行模式搜索得到最终的搜索结果。
混沌搜索结果的精度不需要很高,却可以为模式搜索提供有效的初始点,避免搜索陷入局部极小,只需要简单搜索即可得到理想的最优解。
仿真结果表明混合混沌模式搜索方法简单、高效。
关键词:最优化;混沌搜索;模式搜索法中图分类号:TP301文献标识码:AA H ybrid Optimization MethodB ased onChaotic Search and P attern SearchL I U He 1, H UA N G Meng 1, L I U Gui 2g uo1,2, H UA N G D ao1(1.S chool of I nf orm ation S cience and Engi neeri ng ,East Chi na U ni versit y of S cience andTechnolog y ,S hang hai 20037,Chi na;2.Zhej i ang B usi ness Technolog y I nstit ute ,N i ngbo 315020,Zhej i ang ,Chi na )Abstract :Chaotic search can effectively jump out of local minima but it has poor fine search ability.Pattern search met hod has excellent local search ability ;however it s search result mostly depends on t he initial point.To enhance t he precision of optimization result s ,a hybrid chaotic pattern search met hod (HCPSM )is presented in t his paper based on t he chaotic search and pattern search met hod.The final result is found by pattern search based on t he result of chaotic search.The result of chaotic search is not required to be high p recise ,but it can p rovide an effective initial point for pattern search met hod to avoid t he local minima.The ideal optimum can be found by simple pattern search based on good initial point.Simulation result s show t hat t he HCPSM is simple and high effective.K ey w ords :optimization ;chaotic search ;pattern search met hod 混沌是非线性系统中一种较为普遍的现象,具有随机性、遍历性和规律性。
随机优化算法求解混合整数优化问题
随机优化算法求解混合整数优化问题引言混合整数优化问题（Mixed Integer Optimization Problem）是指在一组约束条件下，求解同时包含连续变量和离散变量的最优解的问题。
这类问题广泛应用于实际工程、管理和经济等领域，例如生产优化、资源分配、路径规划等。
由于离散变量的引入，混合整数优化问题具有较高的计算复杂度，传统的优化算法难以求解。
而随机优化算法由于其随机性和全局搜索能力，成为求解混合整数优化问题的有效方法。
本文将介绍常见的随机优化算法，并探讨如何应用这些算法求解混合整数优化问题。
随机优化算法概述随机优化算法是一类基于概率和随机性的优化方法，其主要特点是通过随机性来搜索解空间，并逐步逼近最优解。
相较于确定性优化算法，随机优化算法无需求解函数的解析表达式，只需能够评估目标函数值即可。
常见的随机优化算法包括模拟退火算法、遗传算法、粒子群算法等。
模拟退火算法模拟退火算法（Simulated Annealing, SA）源自固体物理学中的退火过程，在优化问题中被广泛应用。
该算法通过模拟物质的退火过程，将系统的能量降至较低的状态，从而找到全局最优解。
模拟退火算法的基本思想是通过接受劣解的概率来避免陷入局部最优解，以全局搜索为目标。
算法从一个初始解开始，通过改变解的状态（即变量的取值），计算目标函数的变化，并依概率接受新解。
随着迭代的进行，算法逐渐降低接受劣解的概率，以达到更优解。
模拟退火算法的关键参数包括初始温度、降温速度和终止温度。
初始温度越高，接受劣解的概率越大，有助于跳出局部最优解；降温速度则决定了搜索过程的速度，过快的降温速度可能导致搜索停滞；终止温度通常设定为一个较小的值，当温度低于终止温度时算法停止。
遗传算法遗传算法（Genetic Algorithm, GA）模拟了自然界中生物进化的过程，通过模拟遗传、变异、选择等操作来搜索最优解。
该算法常被用于解决搜索空间巨大而复杂的问题，可以在多个解之间进行搜索，并通过自然选择和遗传操作来进化优秀的个体。
Modified PSO algorithm for solving planar graph coloring problem
4.3.2. Instance 2
4.3.3. Instance 3
5. Inter-cluster load balancing through self-organizing cluster approach
5.1. Performance evaluation
935
On Efficient Sparse Integer Matrix Smith Normal Form Computations Original Research Article
Journal of Symbolic Computation, Volume 32, Issues 1-2, July 2001, Pages 71-99
6.3.2. Liveliness property
6.3.3. Deadlock
7. Conclusion
References
Vitae Purchase
Research highlights
? A hybrid load balancing (HLB) approach in trusted clusters is proposed for HPC. ? HLB reduces network traffic by 80%–90% and increases CPU utilization by 40%–50%. ? The AWT and MRT of remote processes are reduced by 13%–26% using ReJAM. ? The stability analysis of JMM using PA ensures the finite sequences of transitions. ? On the basis of these properties, JM model has been proved safe and reliable.
Mixed Integer-Discrete-Continuous Optimization by Differential Evolution, Part 1 the optimi
5th International Mendel Conference on Soft Computing MENDEL’99, Brno, Czech Republic, 9–12 June 1999
Jouni Lampinen – Ivan Zelinka Mixed Integer-Discrete-Continuous Optimization by Differential Evolution
trial vector cost value parameter 1 parameter 2 parameter 3 parameter 4 parameter 5 SELECTION: Select target vector or trial vector, the fittest one survive 2.56 -0.09 0.83 0.14 0.77 0.91
Байду номын сангаас
5th International Mendel Conference on Soft Computing MENDEL’99, Brno, Czech Republic, 9–12 June 1999
Jouni Lampinen 1999
Jouni Lampinen – Ivan Zelinka Mixed Integer-Discrete-Continuous Optimization by Differential Evolution
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 29 20 21 do 1 icount=1,GenMax do 2 i=1,NP A=IDINT(Rndgen()*NP+1) if(A.EQ.i)goto 3 B=IDINT(Rndgen()*NP+1) if((B.EQ.i).OR.(B.EQ.A))goto 4 C=IDINT(Rndgen()*NP+1) if((C.EQ.i).OR.(C.EQ.A).OR.(C.EQ.B))goto 5 do 6 j=1,iD trial(j)=P(i,j)+K*(P(C,j)-P(i,j))+F*(P(A,j)-P(B,j)) continue call CostFunction(score,iD,trial) if(score.LE.cost(i))then do 7 j=1,iD P(i,j)=trial(j) continue cost(i)=score iBest=i endif continue continue
具有截断控制参数学习效应及退化效应加工时间依赖于资源的单机排序问题
具有截断控制参数学习效应及退化效应加工时间依赖于资源的单机排序问题翟雯瑾;罗成新【摘要】讨论具有截断控制参数学习效应和退化效应且工件的加工时间依赖于资源分配的单机排序问题.在凸资源消费函数条件下研究问题,每个任务有一个松弛工期窗口,任务的实际加工时间依赖于截断控制参数、工件的开始加工时间.分别考虑了在工件的提前惩罚、延误惩罚等费用受限的前提下,最小化资源费用;资源消耗总费受限的前提下,使带有提前、延误、交货期开始时间、交货期大小、最大完工时间及总完工时间加权和最小的单机排序问题.将问题转化为指派问题,证明了该问题是在多项式时间内可解的,并分别给出了两个多项式时间的最优算法,并给出了一个算例.【期刊名称】《沈阳航空航天大学学报》【年(卷),期】2017(034)005【总页数】6页(P86-91)【关键词】排序;资源分配;截断控制参数;退化效应;指派问题【作者】翟雯瑾;罗成新【作者单位】沈阳师范大学数学与系统科学学院,沈阳110034;沈阳师范大学数学与系统科学学院,沈阳110034【正文语种】中文【中图分类】Q221.7经典排序中，任务的加工时间是固定的常数，但在实际生产中，任务等待或机器等原因都会引起任务加工时间的增长，即任务的实际加工时间与该任务的开始加工时间有关。
然而考虑到学习效应、退化效应、资源分配等情况，任务的加工时间不再是固定不变的。
通常情况下，给任务分配一定额度资源，任务的加工时间变小。
文献[2]讨论了带有学习和退化效应的单机排序问题，目标函数包括提前、延误和工期的总费用。
由于实际生产活动中的需要，带有资源分配的问题逐渐引起关注，文献[3-5]研究了关于资源分配的单机排序问题。
文献[6]在工件的加工时间与学习指数相关的凸函数下，研究了工件提前、延迟的工期指派问题。
文献[8-13]讨论了带有提前、延误的工期指派问题。
文献[14]研究了具有截断控制参数学习效应和退化效应且工件的加工时间依赖于资源分配的单机排序问题，并求得最大加工时间与总完工时间最小值时的最优算法，本文与文献[14]的差别在于目标函数不同。
Convex Matroid Optimization
a r X i v :m a t h /0207136v 1 [m a t h .C O ] 16 J u l 2002Convex Matroid Optimization Shmuel Onn ∗Abstract We consider a problem of optimizing convex functionals over matroid bases.It is richly ex-pressive and captures certain quadratic assignment and clustering problems.While generally NP-hard,we show it is polynomial time solvable when a suitable parameter is restricted.1Introduction Let M =(N,B )be a matroid over N :={1,...,n }with collection of bases B ⊆2N .Let w :N −→R d be a weighting of matroid elements by vectors in d -space.For any subset J ⊆N let w (J ):= j ∈J w (j )with w (φ):=0.Finally,let c :R d −→R be a convex functional on R d .We consider the following algorithmic problem.Convex matroid optimization.Given data as above,find a basis B ∈B maximizing c (w (B )).We begin with some examples of specializations of this problem.Example 1.1Linear matroid optimization.This is the special case of our problem with d =1,w :N −→R a weighting of elements by scalars,and c :R −→R :x →x the identity.The problem is to find a basis of maximum weight,and is quickly solvable by the greedy algorithm.Example 1.2Positive semidefinite quadratic assignment.This is the NP-hard problem[6]of finding a vector x ∈{0,1}n maximizing ||W x ||2=x T W T W x with W a given d ×n matrix.For fixed d it is solvable in polynomial time [3].The variant of this problem in which one asksfor x with restricted support |supp(x )|=r is the special case of our problem with M :=U r n the uniform matroid of rank r over N ,with w (j ):=W j the j th column of W for all j ∈N ,andwith c :R d −→R :x →||x ||2the l 2-norm (squared or not).The positive semidefinite quadratic assignment problem can be solved by solving the variant for r =0,...,n and picking the best x .2Convex Matroid Optimization Example1.3Balanced clustering.This is the problem of partitioning a given set{w1,...,w n} of points in R d into two clusters C1,C2of equal size m:=nmw j∈C1||w j−(1mw j∈C2||w j−(12over N,with w(j):=w j for all j∈N,and with c:R d−→R:x→||x||2+||w(N)−x||2with w(N)= n j=1w j the sum of all points.While the linear matroid optimization problem(Example1.1)is greedily solvable(cf.[4]),the general convex matroid optimization problem is NP-hard as indicated by Example1.2.Neverthe-less,in this article we show that,so long as d isfixed,the problem can be solved in polynomial time for an arbitrary matroid M and an arbitrary convex functional c.We assume that c is presented by an evaluation oracle that given x∈R d returns c(x),and that M is presented by an independence oracle that given J⊆N asserts whether or not J is an independent set of M.In this article we establish the following theorem.Theorem1.4For anyfixed d,the convex matroid optimization problem with oracle presented matroid M over N:={1,...,n},weighting w:N−→R d,and oracle presented convex functional c:R d−→R,can be solved in polynomial oracle time using O(n2d−1log n)operations and queries. The computational complexity is measured in terms of the number of real arithmetic operations and oracle queries.For rational input the algorithm is(strongly)polynomial time in the Turing computation model where the input includes the binary encoding of the weighting w:N−→Q d and the binary encoding of an upper bound U:=max J⊆N c(w(J))on the relevant values of the convex functional,but we do not dwell on the details here.The special case of the convex matroid optimization problem for uniform matroids coincides with the special case of the so-called shaped partition problem[7]for two parts.Therefore,the specializations to two-parts of the lower bounds of[1,2]imply a lower bound ofΩ(n d−1)on the complexity of the convex matroid optimization problem.It would be very interesting to further study a plausible common generalization of the convex matroid optimization problem for arbitrary matroids and the shaped partition problem for arbitrary number of parts.2Proof of the theoremFor a matroid M=(N,B)and a weighting w:N−→R d,consider the following convex polytopeP M w:=conv{w(B):B∈B}⊂R d.The convex matroid problem can be reduced to maximizing the convex functional c over P M w: there will always be an optimal basis B∈B for which w(B)is a vertex of P M w and so theShmuel Onn3 problem can be solved by picking the best such vertex.However,as the number of matroid bases is typically exponential in n it is not possible to construct P M w directly in polynomial time.To overcome this we consider the following zonotope:P w:=[−1,1]·(w(i)−w(j))⊂R d.1≤i<j≤nProposition2.1Fix any d.Then the number of vertices of the zonotope P w is O(n2(d−1)). Further,in polynomial time using that many arithmetic operations,all its vertices can be listed, each vertex v along with a linear functional a(v)∈R d uniquely maximized over P w at v. Proof.The zonotope P w is the Minkowski sum of m:= n2 line segments in R d and therefore (cf.[5,8])has O(m d−1)=O(n2(d−1))vertices which can all be enumerated,each v along with a vector a(v)uniquely maximized at v,using that many arithmetic operations.Let P M:=conv{1B:B∈B}⊂R n be the basis polytope of the matroid M=(N,B),where 1B:= j∈B e j is the incidence vector of B∈B with e j the j th standard unit vector in R n.We include the short proof of the following statement.Proposition2.2Every edge of the basis polytope is equal to e i−e j for some pair i,j∈N.Proof.Consider any pair A,B∈B of bases such that[1A,1B]is an edge(that is,a1-face) of P M,and let a∈R n be a linear functional uniquely maximized over P M at that edge.If A\B={i}is a singleton then B\A={j}is a singleton as well in which case1A−1B=e i−e j and we are done.Suppose then,indirectly,that it is not,and pick an element i in the sym-metric difference A∆B:=(A\B)∪(B\A)of A and B of minimum value a i.Without loss of generality assume i∈A\B.Then there is a j∈B\A such that C:=A\{i}∪{j}is a basis of M.Since|A∆B|>2,C is neither A nor B.By the choice of i,this basis satisfies a·1C=a·1A−a i+a j≥a·1A,and hence1C is also a maximizer of a over P M so lies in the 1-face[1A,1B].But no{0,1}-vector is a convex combination of others,yielding a contradiction.The normal cone of a face at a polyhedron P in R d is the relatively open cone of those linear functionals a∈R d uniquely maximized over P at that face.The collection of normal cones of all faces of P is called the normal fan of P.A polyhedron P is a refinement of a polyhedron Q if the normal fan of P is a refinement of that of Q,that is,the closure of each normal cone of Q is the union of closures of normal cones of P.We have the following lemma.Lemma2.3The zonotope P w is a refinement of the polytope P M w.Proof.Letπ:R n−→R d:e j→w(j)be the natural projection sending the unit vector e j corresponding to the matroid element j∈N to the vector w(j)∈R d.Then for each B∈B we haveπ(1B)=w(B)and henceP M w=conv{w(B):B∈B}=conv{π(1B):B∈B}=π(P M)4Convex Matroid Optimization so P M w is a projection of P M.Thus,each edge of P M w is the projection of some edge of P M and hence,by Proposition2.2,is equal toπ(e i−e j)=w(i)−w(j)for some pair i,j∈N.Thus,the zonotope P w= 1≤i<j≤n[−1,1]·(w(i)−w(j))is the Minkowski sum of a set of segments contain-ing all edge directions of P M w and hence its normal fan is a refinement of the normal fan of P M w.We are now in position to prove our theorem.Proof of Theorem1.4.Given data M,w,c,the algorithm proceeds with the following steps:first,compute via Proposition2.1the list of O(n2(d−1))vertices v of P w,each v along with a linear functional a(v)∈R d uniquely maximized over P w at v.Second,for each v do the following: let a:=a(v)and define the following weighting of matroid elements by scalars:b:M−→R:j→a·w(j)=di=1a i w(j)i;now apply a greedy algorithm to obtain a basis B(v)∈B of maximum weight b(B),that is, sort N by decreasing b-value(using O(n log n)operations)andfind,using at most n calls to the independence oracle presenting M,the lexicographicallyfirst basis B(v).Third,for each v compute the value c(w(B(v)))using the evaluation oracle presenting c;an optimal basis for the convex matroid optimization problem is any B(v)achieving maximal such value among the bases B(v)of vertices v of P w.The complexity is dominated by the second step which takes O(n log n) operations and queries and is repeated O(n2(d−1))times,giving the claimed bound.We now justify the algorithm.First,we claim that each vertex u of P M w satisfies u=w(B(v)) for some B(v)produced in the second step of the algorithm.Consider any such vertex u.Since P w refines P M w by Lemma2.3,the normal cone of u at P M w contains the normal cone of some (possibly more than one)vertex v of P w.Then a:=a(v)is uniquely maximized over P M w at u. Now,consider the second step of the algorithm applied to v and let b be the corresponding scalar weighting of matroid elements.Then the b-weight of any basis B satisfiesb(B)= j∈B a·w(j)=a· j∈B w(j)=a·w(B)≤a·uwith equality if and only if w(B)=u.Thus,the maximum b-weight basis B(v)produced by the greedy algorithm will satisfy u=w(B(v)).Thus,as claimed,each vertex u of P M w is obtained as u=w(B(v))for some B(v).Now,since c is convex,the maximum value c(w(B))of any basis B∈B will occur at some vertex u=w(B(v))of P M w=conv{w(B):B∈B}.Therefore,any basis B(v)with maximum value c(w(B(v)))is an optimal solution to the convex matroid optimization problem.The third step of the algorithm produces such a basis and so the algorithm is justified.ReferencesShmuel Onn5[1]N.Alon and S.Onn,Separable partitions,Discrete Applied Mathematics,91:39–51,1999.[2]S.Aviran and S.Onn,Momentopes and the vertex complexity of partition polytopes,Dis-crete and Computational Geometry,27:409–417,2002.[3]K.Allemand,K.Fukuda,T.M.Liebling and E.Steiner,A polynomial case of unconstrainedzer-one quadratic optimization,Mathematical Programming Series A,91:49–52,2001. [4]W.J.Cook,W.H.Cunningham,W.R.Pulleyblank and A.Schrijver,Combinatorial Opti-mization,John Wiley&Sons,1997.[5]P.Gritzmann and B.Sturmfels,Minkowski addition of polytopes:complexity and applica-tions to Gr¨o bner bases,SIAM Journal on Discrete Mathematics,6:246–269,1993.[6]L.P.Hammer,P.Hansen,P.M.Pardalos and D.J.Rader,Maximizing the product of twolinear functions in0−1variables,RUTCOR Research Report,Rutgers University,1997. [7]F.K.Hwang,S.Onn and U.G.Rothblum,A polynomial time algorithm for shaped partitionproblems,SIAM Journal on Optimization,10:70–81,1999.[8]S.Onn and L.J.Schulman,The vector partition problem for convex objective functions,Mathematics of Operations Research,26:583–590,2001.Shmuel OnnTechnion-Israel Institute of Technology,32000Haifa,Israel,andUniversity of California at Davis,Davis,CA95616,USA.email:onn@ie.technion.ac.il,onn@,http://ie.technion.ac.il/∼onn。
Finding community structure in networks using the eigenvectors of matrices
M. E. J. Newman
Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109–1040
We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity” over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in neteasure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.
低开销三进制域Eta双线性对硬件加速器
低开销三进制域 Eta 双线性对硬件加速器*
李翔宇
清华大学 微电子学研究所 北京信息科学与技术国家研究中心, 北京 100084 通信作者: 李翔宇, E-mail: xiangyuli@
摘 要: 基于身份标识的加密 (Identity-Based Encryption, IBE) 在物联网领域有很高的潜在应用价值， 双线性对运算是其中的关键运算. 本文针对物联网需求设计了一种低开销的双线性对硬件加速器. 它选择 了水线低执开行销,的增超加奇了异电椭路圆的曲吞线吐上率的. 三M进ille制r 域算法et硬a 件对实. 该现设中计通将过将MiMlleilrle算r 循法环与中幂的运算GF分(为36两m)个上硬的件稀部疏分乘, 流法 与立方结合并重新调度和优化, 减少中间值相关的开销. 优化后的方案, 具有更简单的运算单元, 减少了寄 存制器较的复使杂,用采和用存了储微器码的控读制写的. 方核式心进模行乘实运现算.采本用文了选最取高定位义优在先的GF字(串39行7)结上构的. 椭考圆虑曲到线电上路的的e规ta模对较进大行, 了控 ASIC 实现, 在 90 nm 工艺下, 版图面积 650 × 650 µm2, 计算时间为 16.7 µs, 面积延时积比现有 eta 对 ASIC 实现减小了 38.8% . 关键词: 基于身份标识的加密; 双线性对; 专用集成电路; eta 对 中图分类号: TN495 文献标识码: A DOI: 10.13868/ki.jcr.000445
2015 年, Chung 等人 [10] eta 对 ASIC 硬件加速器.
3 Eta 对和 Miller 算法
Eta 对是定义在三进制域 GF (3m) 超奇异椭圆曲线 y2 = x3 − x + b 上的对称双线性对, 其中参数 b ∈ {1, −1}. 目前普遍采用的三进制 eta 对的计算方法是无立方根的逆向 eta 对算法, 如算法 1 所示:
SIAMJ.Optimization DISCRETIZEDCONTINUOUSMINIMAXPROBLEMSANDOTHER Vol.x,No.x,pp.x{xx,xxx19xx
H = fH = H T : 1 kdk2 hd; Hdi
2 kdk2
8d 2 IRn g:
let d(x; H; ^ ) be the unique KKT point of QP (x; H; ^ ). Then Lemma 3.2 implies that d(x ; H; ^ ) = 0 for all H 2 H. Since H is compact, in view of Assumptions 4 and 5, it follows from a classical result of Robinson's 1, Theorem 2.1] that, given > 0, there exists ^ > 0 such that, for all x with kx ? x k < ^ and all H 2 H, all ! 2 max (x ) are binding for QP (x; H; ^ ) and kd(x; H; ^ )k < . That ^ > 0 can be chosen independent of ^ follows from niteness of . b. Lemma 3.14'. For k large enough, max(x ) k Proof. Let 1 ; 2 > 0 be as given by Assumption 3, and let > 0 be as given by Lemma 3.13' (for an arbitrary > 0). Since xk ! x as k ! 1 (Proposition 3.11) ^ ^ it follows from Lemma 3.12 that there exists k such that kxk ? x k < for all k k
基于Hadoop求解大规模VRPTW问题的并行遗传算法
摘要车辆路径问题广泛应用于各个领域，不论是机器人自主无碰运动、服务网络规划等学术研究领域，还是数字地图导航、仓库AGV无导引小车运作等工业生产环境，甚至是与人们生活息息相关的快递配送业，都要用到车辆路径问题的优化理论。
车辆路径问题的研究，不仅具有重要的学术研究意义，而且有重要的生产实用价值。
带有时间窗的车辆路径问题在车辆路径问题的基础上考虑了时间成本的影响，更加符合实际需求。
针对带有时间窗的车辆路径问题的研究已经比较成熟，包括精确算法、启发式算法、元启发式算法等，但这些算法基本都是串行的集中式算法，大都只能求解中小规模的车辆路径问题，然而现在的车辆路径问题动辄就是上千个节点的规模，加上时间窗的约束，传统串行算法求解效率比较低，短时间内很难求解出可接受解。
当今大数据、云计算等计算机技术的蓬勃发展，为并行计算提供了技术支持，也为并行化解决大规模带有时间窗的车辆路径问题提供了新的思路。
针对集群式并行计算具有高容错性、高扩展性、高可用性和廉价性等方面的优势，本研究选用了经典的集群分布式并行计算平台----Hadoop作为并行计算的基础架构，基于此使用MapReduce并行框架进行分布式并行算法的设计与优化，用以解决大规模带有时间窗的车辆路径问题。
本研究在基础算法的设计上，选取了具有天然并行特性的遗传算法。
为了最大限度地降低遗传算法的巨大计算开销，本文选择并改进了比较优秀的选择、交叉、变异算子。
在MapReduce框架中，map和reduce阶段的设计上，充分考虑了大规模车辆路径问题的遗传基因的长度带来的影响，同时考虑了如何降低集群间信息传输的压力，最终采用粗粒度并行模型----遗传算法岛屿模型嵌入MapReduce框架。
在键值对的处理上，利用键值对中“键”的不变性保持遗传算法解个体和适应度值的一致性，并将迁徙操作与shuffle阶段结合起来，保证迁徙过程顺利执行。
本文使用带有时间窗的车辆路径问题的大规模标准算例----Gehring & Homberger (1999)算例进行了算法验证，分别从并行算法的有效性、串行和并行算法的对比、集群处理器数量对算法的影响和处理器配置对算法的影响等四个方面进行了数值实验与精确的分析，并论述了本文研究的有效性和重要价值。
数学建模优化问题
•使用fminunc和 fminsearch可能会得到局部最优解.
无约束最优化问题 求解无约束最优化问题的的基本思想 *无约束最优化问题的基本算法 返回
求解无约束最优化问题的基本思想
标准形式：
m f X in
X E n
其 中 f : E n E 1
m f X = m [ f X ] a i
X E n X E n
求解的基本思想 ( 以二元函数为例 )
其中（3）、（4）、（5）的等式右边可选用（1）或（2）的等式右边。 函数fminbnd的算法基于黄金分割法和二次插值法，它要求目标函数必须是连续函数，并可能只给出局部 最优解。
例 1 求 f = 2 e x s x 在 0 < x < 8 中 的 i 最 小 值 与 最 大 n 值
主程序为wliti1.m:
数学建模优化问题
一般优化问题概述
离散优化discrete optimization 或组合优化combinatorial optimization
整数规划(IP) 决策变量(全部或部分)为整数 Integer programming
✓ 整数线性规划(ILP)，整数非线性规划(INLP) ✓ 纯整数规划(PIP), 混合整数规划(MIP) Pure (mixed) Integer programming 一般整数规划，0-1（整数）规划 Zero-one programming
融合禁忌搜索的混合果蝇优化算法
融合禁忌搜索的混合果蝇优化算法张彩宏;潘广贞【摘要】基本果蝇优化算法(FOA)种群初始位置分布不均匀,搜索后期常跳入局部最优,导致寻优速度慢、寻优精度低,为此融合禁忌搜索的“禁忌”与“特赦”思想进行搜索更新,提出融合禁忌搜索算法(TS)的果蝇优化算法(TS-FOA).将Kent混沌映射的序列作为果蝇种群初始位置,保证果蝇群体在搜索空间中的均匀性、多样性;利用果蝇优化算法进行前期寻优,定义群体适应度方差判断其局部收敛状态;达到局部收敛状态时,引入禁忌搜索,继续深度寻优,提高寻优精度和寻优速度.设计仿真实验测试5个经典标准函数的寻优性能,实验结果表明,TSFOA在寻优精度、寻优速度上均优于基本FOA算法.【期刊名称】《计算机工程与设计》【年(卷),期】2016(037)004【总页数】7页(P907-913)【关键词】果蝇优化算法;禁忌搜索算法;Kent混沌映射;适应度方差【作者】张彩宏;潘广贞【作者单位】中北大学计算机与控制工程学院,山西太原030051;中北大学计算机与控制工程学院,山西太原030051【正文语种】中文【中图分类】TP18;TP301.6相比其它群智能算法，适合全局智能搜索的果蝇优化算法[1-6](fruit fly optimization algorithm，FOA)有许多优势：①思路清晰、简明易懂，寻优判别式是一阶微分方程，易于计算，而粒子群算法(particle swarm optimization，PSO)的寻优判别式是二阶微分方程[7]；②程序设计简单易实现，运行效率高，收敛精度和速度相对较好；③仅需初始化3个参数，而其它类似的智能优化算法需要初始化更多的参数，如PSO[7]、人工鱼群算法[8](artificial fish swarm algorithm，AFSA)、遗传算法[10](genetic algorithm，GA)需5个初始参数，蚁群算法(ant colony optimization，ACO)需设置7个初始参数[9]，细菌觅食优化算法设置的初始参数高达11个[10]。
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Graduate earch Assistant, Member AIAA
inadequate or at least untested when faced with the task of automating or optimizing the design of mixed discrete/continuous systems. A second and sometimes equally important issue is the vast amount of information required for the design of complex, multidisciplinary engineering systems. The computational capabilities provided by current computers can allow the designer the opportunity to produce vast amounts of data and consider many potential design variations. Providing a framework for these design studies and a method of storage of the information generated so that it can be useful in practical design decision making is critical to the success of automated design efforts. These issues result in questions as to how one can generate optimal or at least improved designs given real constraints on time and computational resources. This is particularly important in combinatorial design problems involving discrete design variables which often require an extremely large number of objective function evaluations. For these problems the reduction of the number or cost of these objective function evaluations is a high priority. The purpose of this study was to address issues pertaining to design space representation and evaluation for structural design problems which contain both discrete and continuous design variables. The method by which the design space de nition (response surface mapping) for the mixed discrete/continuous design variable problem was addressed in this study was through the use of arti cial neural networks. These networks have been shown to provide a useful tool for storage and manipulation of design data obtained through conventional analysis techniques for systems of either continuous 1, 2, 3] or discrete 1, 4, 5, 6] design variables, but problems containing both continuous and discrete variables have only recently been considered in some detail 7, 8]. This paper employs feedforward, back propagation neural networks to provide an approximation to the mixed discrete/continuous design space and to replace conventional numerical analysis methods at the system level in the optimization process. The problem considered in this paper is an extension of a preliminary study initially presented in Reference 9]. This earlier work indicated the potential for using neural networks in the combined continuous/discrete problem but the design space considered did not provide a particularly demanding application and the subspace design problem was rather limited. The methods considered in the present study were evaluated by application to a relatively straightforward structural design problem. This problem was selected since it contained the basic characteristics of both continuous and discrete design variables. The design space 1
AIAA Copyright c 1994 by Stephen M. Batill. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.
y Professor, Associate Fellow AIAA z Clark Equipment Assistant Professor, Member
Optimization of Mixed Discrete/Continuous Design Variable Systems Using Neural Networks R.S. Sellar S.M. Batilly J.E. Renaud z Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana ABSTRACT Emerging multidisciplinary design techniques must be capable of dealing with a wide variety of disciplinespeci c design methodologies. Of particular concern is the development of methods which can e ectively integrate continuous and discrete design variables. This paper documents a study in which material selection, structural arrangement and component sizing are considered for a single, simple structural system. A hierarchical design problem is formulated in which component sizing based upon nite element analysis is performed at the subsystem level. Arti cial neural networks are used to provide response surface mapping of the subsystem for use at the system level. The system level problem which includes the discrete material selection design variables and continuous structural arrangement design variables is formulated as a discrete problem and optimum designs were identi ed using simulated annealing algorithms for two di erent merit functions. This study identi es requirements for the e ective use of neural network mapping of the subspace and how this representation of the subspace in uences the system level optimization. INTRODUCTION Optimization methods have a long history and these methods have played important roles in engineering analysis and design. The foundation for many of the current optimization methods used in engineering design are analytic or numerical techniques which are well-suited for speci c classes of problems. As the engineering community attempts to expand the in uence of optimization methods into the realm of \multidisciplinary design optimization" it is faced with the problem of adapting optimization methods to problems which are more complex and represent a variety of problem classes. The current study was concerned with just one case in which both continuous and discrete design variables were required for a single design problem. There are a variety of optimization techniques that can be used to nd the solution to a design problem when the system in question is composed solely of either continuous (spar cap cross sectional area, wing sweep angle,: : : ) or discrete (material choice from a nite set, number of spars,: : : ) design variables. These techniques, however, are in many cases