Distributed Wireless Sensor Network Localization Via Sequential Greedy Optimization Algorithm

3328IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 6, JUNE 2010Distributed Wireless Sensor Network Localization Via Sequential Greedy Optimization AlgorithmQingjiang Shi, Student Member, IEEE, Chen He, Member, IEEE, Hongyang Chen, and Lingge Jiang, Member, IEEEAbstract—Node localization is essential to most applications of wireless sensor networks (WSNs). In this paper, we consider both range-based node localization and range-free node localization with uncertainties in range measurements, radio range, and anchor positions. First, a greedy optimization algorithm, named sequential greedy optimization (SGO) algorithm, is presented, which is more suitable for distributed optimization in networks than the classical nonlinear Gauss-Seidel algorithm. Then a unified optimization framework is proposed for both range-based localization and range-free localization, and two convex localization formulations are obtained based on semidefinite programming (SDP) relaxation techniques. By applying the SGO algorithm to the edge-based SDP relaxation formulation, we propose a second-order cone programming (SOCP)-based distributed node localization algorithm. Two distributed refinement algorithms are also proposed by using the SGO algorithm to nonconvex localization formulations. The proposed localization algorithms all can be implemented partially asynchronously in networks. Finally, extensive simulations are conducted to demonstrate the efficiency and accuracy of the proposed distributed localization algorithms. Index Terms—Distributed optimization, range-based node localization, range-free node localization, second-order cone programming (SOCP), semidefinite programming (SDP), sequential greedy optimization (SGO) algorithm, wireless sensor network (WSN).I. INTRODUCTIONWIRELESS SENSOR NETWORKS (WSNs) consist of a large number of tiny, low-power, and randomly deployed sensor nodes that have sensing, processing, and communication capabilities [1]. Most applications of WSNs, e.g., environmental monitoring, search and rescue, target tracking, etc., require the knowledge of positions of sensor nodes. In general, for economic consideration, only a small fraction of nodes’ positions are measured by global positioning system (GPS) or manual configuration (these nodes are commonlyManuscript received October 28, 2009; accepted February 04, 2010. Date of publication March 11, 2010; date of current version May 14, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mathini Sellathurai. This work was supported in part by the National Nature Science Foundation of China by Grants 60772100, 60832009, 60872017, by the Cooperative Research Project with Fujitsu R&D Bejing Center, and by the China Scholarship Council. This work was presented in part at the IEEE Global Communications Conference, HI, December 2009. Q. Shi, C. He, and L. Jiang are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China, 200240 (e-mail: shiqj@sjtu. ; chenhe@; lgjiang@). H. Chen is with the Institute of Industrial Science, The University of Tokyo, Tokyo, Japan (e-mail: hongyang@mcl.iis.u-tokyo.ac.jp). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 10.1109/TSP.2010.2045416called anchors), especially for large-scale sensor networks. Hence, developing efficient node self-localization algorithms is necessary for wireless sensor networks. The node localization problem in sensor networks includes range-based localization and range-free localization [2]. The former relies on range (or distance) measurements between nodes that can be estimated by the received signal strength (RSS) method or the time-of-arrival (TOA) method [3], while the latter uses only connectivity information (i.e., whether a node is in the transmission range of another node) [7], [10]. Correspondingly, localization algorithms can be grouped into range-based localization algorithms and range-free localization algorithms. The range-based localization algorithms can provide higher localization accuracy than the range-free localization algorithms but the latter is cheaper and simpler since they do not require special hardware for ranging. In terms of computational paradigm, the localization algorithms can also be divided into centralized algorithms and distributed algorithms [2], [8], [9]. The centralized algorithms require transmission of all range measurements or connectivity information between nodes to a fusion center (e.g., a sink node) for processing, resulting in large communication energy and bandwidth consumption and thereby shortening the lifetime of the whole network. The distributed algorithms are energy-efficient and scalable to the size of the networks, where the whole task of node localization is cooperatively carried out by all nodes with local information exchange between neighboring nodes. Hence, distributed localization algorithms are much more attractive for large-scale sensor networks. Many localization algorithms have been proposed for sensor network localization. Among the range-free localization algorithms, [4]–[6] are heuristic, simple, and allow distributed implementation, but they are less accurate unless many anchors are used. Based on the classical multidimensional scaling (MDS) technique, a set of localization algorithms are proposed in [7], which are applicable to both range-based localization and rangefree localization, and significantly outperform the heuristic algorithms. Among these algorithms, MDS-MAP (P, R) is the best for its ability to localize irregular networks. However, it is very complicated and costly because of its kernel strategy, i.e., first building a local map for each node and then merging these local maps together to form a global. Centralized computation is necessary when merging local maps. Hence, although the MDS-MAP (P, R) can be implemented in networks, it is not suitable for large-scale networks. A metric MDS-based algorithm is proposed for range-based localization in [8]. This algorithm is very suitable for distributed implementation. However, its local convergence property implies bad localization performance if1053-587X/$26.00 © 2010 IEEESHI et al.: DISTRIBUTED WSN LOCALIZATION3329a good initialization is not available. In [9], a robust multilateration-based iterative localization algorithm is proposed for range-based localization. It is quite lightweight and very suitable for distributed implementation. However, there is no theoretic guarantee for the convergence of the algorithm. Convex optimization-based localization algorithms are popular in sensor network localization due to their global convergence property. Most of them are for range-based localization [10]–[17], [19], while a few for range-free localization [10], [18]. The common philosophy of these algorithms is formulating the localization problem as a nonlinear optimization problem and then relaxing the resulting problem as a convex optimization problem solved by efficient interior-point algorithm [27]. Two convex relaxation techniques are involved in these algorithms, second-order cone programming (SOCP) relaxation in [10], [11], and [19] and semidefinite programming (SDP) relaxation in [12]–[18]. The SDP-based algorithms are very computationally expensive for large-scale networks (more than hundreds of nodes) and require centralized computation due to their complex structures. To reduce the computational load, [16] proposes a cluster-based SDP localization algorithm. Although this algorithm can be implemented in networks, it is not a good distributed algorithm since: 1) a cluster formation algorithm is necessary at the beginning of the algorithm, resulting in extra communication cost; 2) centralized computation is needed at each cluster; and 3) it may fail when there are not sufficient anchors. An alternative way to reduce the computational load is to further relax the range-based localization problem [17]. In [17], the node-based SDP relaxation and edge-based SDP relaxation are proposed. Although these two relaxations are weaker than the general SDP relaxation [13], they are both efficient and accurate in computation. Compared to the SDP relaxation, although weaker (and thus less accurate), the SOCP relaxation has simpler structure and allows efficient distributed implementation. In [19], a totally asynchronous distributed localization algorithm is proposed using SOCP for range-based localization. The idea can be easily extended to range-free localization. However, the authors of [19] show the convergence of their algorithm numerically, not theoretically. Unlike the convex optimization-based localization algorithms, considering all constraints imposed by distance measurements and/or connectivity information, new formulations of localization problems based on nondifferentiable optimization are proposed for both range-based localization [20] and range-free localization [21]. The resulting nondifferentiable problems are solved by the normalized incremental subgradient (NIS) algorithm [20]. The NIS-based localization algorithms have much better localization performance than most previous algorithms such as the robust SDP method [14] with refinement by the dwMDS method [8], the MDS-MAP (C) method [7], etc. However, distributed implementation of these algorithms requires constructing a path across all nodes beforehand. The localization algorithms mentioned above except [15] and [19] all assume accurate anchor positions. In fact, the anchor positions cannot be exactly known due to the limit accuracy of the civilian GPS and/or manual observation errors. In addition, there is uncertainty in the nodes’ maximum radio range due to radio irregularity [6]. In this paper, both the range-based lo-calization problem and the range-free localization problem are studied, considering the uncertainties in range measurements, radio range, and anchor positions. By using a greedy optimization algorithm, we propose two distributed localization algorithms for both range-based localization and range-free localization. The proposed localization algorithms can be implemented partially asynchronously in networks. Extensive simulations are conducted to demonstrate the efficiency and accuracy of the proposed distributed localization algorithms. Our main contributions are threefold: 1) the sequential greedy optimization (SGO) algorithm is proposed and its convergence property is proved and analyzed. The SGO algorithm is a natural extension of the nonlinear Gauss-Seidel algorithm [30] but it is more suitable for distributed optimization in networks. 2) a unified optimization framework is proposed for sensor network localization and two convex localization formulations are obtained based on the SDP relaxation techniques. By using the SGO algorithm, we show for the first time that the edge-based SDP (ESDP) relaxation formulation can be solved in a distributed way in networks through solving a sequence of second-order cone programming. 3) distributed refinement algorithms that can further improve localization accuracy are proposed for both range-based localization and range-free localization by applying the SGO algorithm to nonconvex localization formulations. The remainder of this paper is organized as follows. In the next section we state the SGO algorithm and demonstrate its convergence property. In Section III, we present the unified optimization framework for sensor network localization, derive two convex formulations, and develop distributed localization algorithms using ESDP formulation as well as refinement algorithms. Section IV provides numerical experiments to illustrate the performance of the proposed localization algorithms, while Section V concludes the paper. Throughout this paper, we use upper-case bold type for matrices, lower-case bold type for vectors, regular type for scalars. denotes the -dimensional Euclidean space. denotes denotes the cardinality of the Cartesian product over sets. a set . The supscript denotes the transpose. and , respectively, denote the zero vector and the unit column vector with its th element being one, whose dimension will be clear denotes the identity matrix. For from the context. a symmetrical matrix denotes the th entry of , denotes the principal submatrix extracted from and . For symmetthe rows and columns indexed by means that is positive rical matrices and semidefinite, and denotes the Frobenius inner product of denotes compomatrices and . For vectors and denotes the componentwise mulnentwise inequality, and tiplication of vectors and . II. SEQUENTIAL GREEDY OPTIMIZATION ALGORITHM Many optimization algorithms have been developed for multivariate problems [26], [27]. Most of them fall into the category of iterative algorithms which are generally based on search directions. In contrast with general iterative optimization algorithms, there exists a special type of iterative algorithms, in3330IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 6, JUNE 2010which decision variables are partitioned into independent blocks and, at each iteration the objective function is minimized over one of blocks while fixing the rest. This type of algorithms are sometimes called alternating minimization algorithm [22], coordinate descent algorithm [28], [29], or nonlinear GaussSeidel (NGS) algorithm [30]. Note that, the first two are special cases of the last, hence, we discuss only the NGS algorithm in the sequel. The NGS algorithm is applicable to unconstrained optimization problems or problems with constraints of Cartesian product set. It works greedily in the sense that it can guarantee the objective function nonincreases at each iteration. Particularly, using the NGS algorithm, a complex multivariate problem can be broken down into a sequence of smallscale subproblems and the implementation can be easily realized [23]–[25], even in a distributed way (if subproblems are assigned to different processors). In this section, we extend the NGS algorithm to a more general form. We refer to the resulting algorithm as SGO algorithm. In the SGO algorithm, the decision variables are not strictly splitted into independent blocks as in the NGS algorithm. In other words, the SGO algorithm allows some decision variables to be updated in different subproblems. This extension is natural but is important since it makes the SGO algorithm more suitable for distributed optimization in networks. The convergence property of the SGO algorithm is analyzed and proved by using first-order Karush-Kuhn-Tucker (KKT) condition, without the assumption of convexity of the objective function and/or the constraint set. In addition, we discuss the application of the SGO algorithm for general optimization problems where the constraints are coupled (i.e., not of Cartesian product type). A. Algorithm We say an optimization problem is sparse-constrained if it can be written in the form, is a differentiable function, where is the th iterate, and is a local minimizer of found by using a certain optimization method and such that . Note that, is uniquely found once a specific optimization method is selected (i.e., the algorithmic rule is determined). For example, we can use descent methods [27], from . In this sense, [30] to reach a local minimizer of (2) is well defined. In addition, we will use the following notations when describing the algorithm. Let , and . Let , be subsets of and such that , and be the th index set whose components are the indexes of which contain . For example, if , those . In the SGO algorithm, it means that the then is not only optimized in the subproblem 1, but also variable in the subproblems 2 and 5. The SGO algorithm is described as follows. Assume a . Given obtained after the random initialization th iteration, it then sequentially performs greedy steps or subiterations in the th iteration. . . (3)and finally gets , where denotes the new obtained after the th subiteration of the th iteration (note: ); (respectively, ) denotes the variable made up indexed by the components of (respectively, ); of . B. Convergence Analysis and Discussion(1) where, is a positive integer, In general, an iterative algorithm converges means that its iterates converge. However, it cannot be established that the iterates generated by the SGO algorithm must converge, unless certain strong assumptions are made on the objective function. In what follows, we assume the SGO algorithm (3) is well defined1 and give two propositions which indicate some properties of the SGO algorithm. be the sequence generated by the Proposition 1: Let SGO algorithm (3). Then the SGO algorithm converges in the converges. sense that the sequence Proof: Let , so we have . From (3), we conclude that, are differentiable functions; and , are nonnegative integers. We assume the exists and is finite. Note that, minimum of means there is no inequality (equality) constraint on . Particand are equal to zero, the problem (1) ularly, when all the is translated into a special sparse-constrained problem, i.e., an unconstrained optimization problem. In the NGS algorithm, subproblems are sequentially miniis a strictly convex function of each . mized, assuming However, given a general , it is hard to obtain a global optimum for subproblems. Here, we will describe the SGO algorithm with a new operator “dec” instead of “min.” The new operator used in an iterative algorithm is defined as(2)1The assumption here means that an exact local minimum can be found at each step of the algorithm. However, it should be pointed out that, we only need to find an approximate local minimum at each step of the SGO algorithm in practical applications if an exact local minimum is not available in reasonable time.SHI et al.: DISTRIBUTED WSN LOCALIZATION3331Thus we have for all . This implies the is nonincreasing, and must converge since sequence is bounded below by its minimum on . Proposition 2: Assume is a limit point generated by the SGO algorithm (3) and of the sequence . Then is a KKT point of the problem (1) under some regularity conditions [31]. is a limit point of the Proof: Because and , it must hold that sequenceRemark 2: When (3) reduces toand, the SGO algorithm(7) which coincides with the iterative form of the nonlinear GaussSeidel algorithm [30]. The SGO algorithm of (7) is more often used in practice although it is a special case of (3). Remark 3: Compared to (7), the algorithm (3) can be easily exploited to devise the distributed algorithms for in-network processing. Considering for an extreme example that, two nodes in a network, each with an unknown state variable and a coupled unknown parameter that is related to both, try to estimate their state variables in a decentralized . way by minimizing a cost function of the form Using (3), each node minimizes jointly over its state variable and the coupled variable once the update of the other node’s state variable is received, implying that the message-passing on the coupled variable is not needed between two nodes. However, it is readily known that using (7) will incur more computation and/or communication overhead. Remark 4: Optimization problems with coupled constraints often appear in distributed signal processing in networks. Directly using the SGO type algorithm (i.e., sequentially optimize the objective function over some variables while fixing others) to solve such problems is possible (due to its greedy nature) but not safe. For example, the following problem: (8) , however, the SGO has unique global solution . algorithm would converge to any point such that For problems (particularly for convex problems) with coupled constraints, we can use4 barrier method (e.g., log-barrier method [27]), to eliminate all coupled constraints (respectively, all constraints) and derive a sparse-constrained (respectively, unconstrained) approximated problem. It is interesting to see that, solving the resulting approximated problem by using the SGO algorithm is equivalent to solving the primal problem using a SGO type algorithm with each subproblem being approximately (not perfectly) solved by the barrier method or other equivalent methods (e.g., primal-dual interior point method [27]). In other words, it is safe to directly use a SGO type algorithm to the problems with coupled constraints while each subproblem is approximately solved. To clarify the argument above, revisit the example above. Directly solving (8) by using the SGO algorithm with perfectly solving two subproblems will lead to an unexpected convergence point. However, by the log-barrier method, the approximated problem is(4) implying that, for each subproblem, must be at least a local at by minimizer, otherwise we can further decrease finding a local minimizer for the subproblem that doesn’t hold, which contradicts the premise that is a limit point. Under the assumption of regularity for each subproblem, the first-order necessary condition, generally known as KKT condition, holds for each subproblem, i.e., for each and , there exist Lagrange multipliers and 2 , such that (5) where the Jacobi matrix of of is with respect to and is the Jacobi matrixw.r.t . Combining the KKT conditions of all subproblems, we have (6)where . This implies that for the point , there exist Lagrange multipliers and satisfying the KKT condition of the primal problem. Therefore, is a KKT point of the primal problem.3 Proposition 2 implies that if the problem (1) is a convex optimization problem, then the limit point is a global optimum. Some remarks are made as follows. Remark 1: At the limit point , the objective function value, , is equal to the limit of the function sequence i.e., since is continuous. Hence, although the convergence of the SGO algorithm cannot be established, an approximate limit can be easily found in terms of the point of the sequence reduction amount of the objective function. Specifically, is an approximate limit point (and, thus, an approximate KKT point) holds for a given sufficiently small if and positive scalar .2When , the inequality (equality) constraints-related terms in (5) should be canceled. For example, when and 6 , (5) can be 0 0g. In this paper, for simplified as fr simplicity, we write the first-order necessary condition in a unified framework. 3This KKT point is often also a local optimum since the objective function values keep nonincreasing.m = 0 (l = 0) m =0 l =0 f () + J ( ) = ;h ( ) = x  x h xwhere is a very small positive scalar which controls the approximation accuracy, and the SGO solution (which is the same4Lagrange dual method can also be used to transform a problem with coupled constraints to a sparse-constrained problem.3332IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 6, JUNE 2010as the solution achieved by directly using the SGO algorithm to (8) but each subproblem being approximately solved by the same log-barrier method) to this sparse-constrained problem is , which can arbitrarily approach the global solution of the problem by choosing an infinitely small positive . Remark 5: The order of the subproblems can be varied at different iterations. Moreover, it is not necessary to run all the subproblems at each iteration. These will make the algorithm much more suitable for asynchronous implementation5 in networks but not influence the convergence of the algorithm. III. DISTRIBUTED SENSOR NETWORK LOCALIZATION ALGORITHM In this section, we first propose a unified optimization framework for sensor network localization. Then we derive two centralized localization formulations, named FullSDP and ESDP, based on convex relaxation techniques. Next, by using the SGO algorithm, we show that the ESDP can be implemented in a distributed way with each subproblem being an SOCP. Finally, we propose distributed refinement algorithms for sensor network localization. A. Problem Formulation Assume a sensor network in ( 2 or 3) has sensor nodes in total, with anchor nodes whose locations are known sensor nodes whose locations (maybe inaccurate) and , and the maximum radio ranges of are unknown , denote the the nodes are all . Let and exact locations and the inaccurate locations of anchor nodes, re, denote the locaspectively, and let tions of sensor nodes. In range-free localization, only the connectivity information of any two nodes, i.e., whether they can communicate with each other directly (without relay), are obtained. We call such two nodes one-hop neighboring nodes, and with the set of all one-hop neighboring nodes’ index pairs is denoted by . In range-based localization, (noisy) range measurements are taken by one-hop neighboring nodes, for neighboring nodes and . In addition, we denoted by assume that the location errors of anchors are bounded by , . Then the range-based loi.e., calization problem with anchor position uncertainty is to find a such that realization ofNote that, there are constraints in total. To achieve distributed localization with high accuracy and reduce the computational complexity, we will neglect all ‘>’ constraints in range-based localization, while in range-free localization, we will neglect all ‘>’ constraints but the ones that are related to two-hop neighboring nodes. Here, two-hop neighboring nodes are the nodes that cannot directly communicate with each other but can communicate with just one relay. Hence, for node and node who are two-hop neighbors, their distance satisfies . Correspondingly, the set of their index pairs with are denoted by . Here we formulate sensor network localization problems in a unified framework as the following minimization problem:(11) and where is the one-hop neighboring nodes set is known in range-based localization; in range-free localizaand , and is unknown but tion, is the union of if and if . (11) can be interpreted as a least-squares localization formulation considering all measurement uncertainty including the uncertainty of the maximum radio range , range measurements, and anchor positions6. Moreover, it should be pointed out that, (11) will become ill-posed if the last term in the objective function is not included. B. Convex Relaxations of Sensor Network Localization The problem (11) is a very complex nonconvex problem that is difficult to solve. We below relax the problem as two convex optimization problems by using SDP relaxation techniques in [12] and [17], respectively, named FullSDP and ESDP as in [17]. For simplicity, we will use to denote the linear constraint on . Let . We first equivalently write (11) as(9) and the range-free localization problem with anchor position uncertainty is to find a realization such that By relaxing as which means (13)6The range-free localization problem is often formulated as a feasibility problem. However, the uncertainty in maximum radio range will lead to the problem infeasible, implying the method such as [18] or [10] that assume the problem has feasible solution may fail.(12)(10)5In implementing the SGO algorithm, it is not allowed that two subproblems, which contain coupled variables (i.e., when 6 ), run concurrently. Hence, “asynchronous” here doesn’t mean “totally asynchronous” [30]. To distinguish from the latter, we call it “partially asynchronous.”=0SHI et al.: DISTRIBUTED WSN LOCALIZATION3333according to Schur complement [12], [27], we then obtain the FullSDPrespectively. (16) and (17) can be further written in standard cone programming forms and solved by using some efficient convex cone programming solvers such as SeDuMi [32] or SDPT3 [33] in a centralized way. C. Distributed Localization via Edge-Based SOCP (ESOCP) Centralized solving the fullSDP or the ESDP formulation is quite computationally intensive when the problem has large size. It requires numerous memory resource, especially for the range-free localization formulation which involves two-hop constraints. In the following, we show that the ESDP formulation (15) can be solved in a distributed fashion by using the SGO algorithm. Particularly, each subproblem is an SOCP. We call such an SOCP-based localization method ESOCP. The edge-based formulation (15) can also be written as(14) whereAlso, using the edge-based SDP relaxation technique [17], we can relax (11) as the following convex optimization problem, i.e., ESDP, which decomposes the single constraint of large positive semidefinite matrix in (14) into a number of constraints of small positive semidefinite matrices.(18) For a positive semidefinite matrix, all of the leading principal minors are nonnegative. Hence, (18) is equivalent to(15) Notice that a constraint in the form of into a second-order cone constraint can be turnedHence, the FullSDP and ESDP can be equivalently written as(19) where . and are related to the node only while In (19), and are coupled variables related to both node and node . Hence, (19) is not sparse-constrained. However, it can be solved by using the SGO algorithm according to the Remarks 3 and 4. The subproblem at each anchor node is(16) and(17)(20)。