An Approximate Dual Subgradient Algorithm for Multi-Agent Non-Convex Optimization

Transcription

1 1534 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE 2013 [12] M Fliess, C Join, and M Mboup, Algebraic change-point detection, Applicable Algebra Eng, Commun, Comp, vol 21, no 2, pp , 2010 [13] E Fridman, F Gouaisbaut, M Dambrine, and J-P Richard, A descriptor approach to sliding mode control of systems with time-varying delays, Int J Syst Sci, vol 34, no 8 9, pp , 2003 [14] E Fridman, Stability of systems with uncertain delays: A new complete Lyapunov-Krasovskii functional, IEEE Trans Autom Control, vol 51, no 5, pp , May 2006 [15] J-E Fonde, Delay Differential Equation Models in Mathematical Biology, PhD dissertation, Univ Michigan, Dearborn, 2005 [16] J P Gauthier, H Hammouri, and S Othman, A simple observer for nonlinear systems Applications to bioreactors, IEEE Trans Autom Control, vol 37, no 6, pp , Jun 1992 [17]MGhanes,JBBarbot,JDeLeon,andAGlumineau, Arobust output feedback controller of the induction motor drives: New design and experimental validation, Int J 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Observability and observers for nonlinear systems with time delays, Kybernetika, vol 38, no 4, pp , 2002 [26] H Mounier and J Rudolph, Flatness based control of nonlinear delay systems: A chemical reactor example, Int J Control, vol 71, no 5, pp , 1998 [27] K Natori and K Ohnishi, A design method of communication disturbance observer for time-delay compensation, taking the dynamic property of network disturbance into account, IEEE T Indust Electron, vol 55, no 5, pp , 2008 [28] S-I Niculescu, C-E de Souza, L Dugard, and J-M Dion, Robust exponential Stability of uncertain systems with time-varying delays, IEEE Trans Autom Control, vol 43, no 5, pp , May 1998 [29] S-I Niculescu, Delay Effects on Stability: A Robust Control Approach New York: Springer LNCIS, 2001 [30] P Picard, O Sename, and J F Lafay, Observers and observability indices for linear systems with delays, in Proc IEEE Conf Computat Eng Syst Appl (CESA 96), 1996, vol 1, pp [31] J-P Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, vol 39, no 10, pp , 2003 [32] J-P Richard, F Gouaisbaut, and W Perruquetti, Sliding mode control in the presence of delay, Kybernetica, vol 37, no 4, pp , 2001 [33] O Sename, New trends in design of observers for time-delay systems, Kybernetica, vol 37, no 4, pp , 2001 [34] O Sename and C Briat, H1 observer design for uncertain time-delay systems, in Proc IEEE ECC 07, 2007, pp [35] A Seuret, T Floquet, J-P Richard, and S K Spurgeon, A sliding mode observer for linear systems with unknown time-varying delay, in Proc IEEE ACC 07, 2007, pp [36]ASeuret,TFloquet,J-PRichard,andSKSpurgeon, Topics in Time-Delay Systems: Analysis, Algorithms and Control Berlin, Germany: Springer Verlag, 2008 [37] E Shustin, L Fridman, E Fridman, and F Castaos, Robust semiglobal stabilization of the second order system by relay feedback with an uncertain variable time delay, SIAM J Control Optim, vol 47, no 1, pp , 2008 [38] R Villafuerte, S Mondie, and Z Poznyak, Practical stability of time delay systems: LMI s approach, in Proc IEEE CDC, 2008, pp [39] Z Wang, B Huang, and H Unbehausen, Robust H1 observer design for uncertain time-delay systems: (I) the continuous case, in Proc IFAC 14th World Congress, Beijing, China, 1999, pp [40] J Zhang, X Xia, and C H Moog, Parameter identifiability of nonlinear systems with time-delay, IEEE Trans Autom Control, vol47, no 2, pp , Feb 2006 [41] G Zheng, J P Barbot, D Boutat, T Floquet, and J P Richard, On obserability of nonlinear time-delay systems with unknown inputs, IEEE Trans Autom Control, vol 56, no 8, pp , Aug 2011 An Approximate Dual Subgradient Algorithm for Multi-Agent Non-Convex Optimization Minghui Zhu and Sonia Martínez Abstract We consider a multi-agent optimization problem where agents subject to local, intermittent interactions aim to minimize a sum of local objective functions subject to a global inequality constraint and a global state constraint set In contrast to previous work, we do not require that the objective, constraint functions, and state constraint sets are convex In order to deal with time-varying network topologies satisfying a standard connectivity assumption, we resort to consensus algorithm techniques and the Lagrangian duality method We slightly relax the requirement of exact consensus, and propose a distributed approximate dual subgradient algorithm to enable agents to asymptotically converge to a pair of primal-dual solutions to an approximate problem To guarantee convergence, we assume that the Slater s condition is satisfied and the optimal solution set of the dual limit is singleton We implement our algorithm over a source localization problem and compare the performance with existing algorithms Index Terms Dual subgradient algorithm, Lagrangian duality I INTRODUCTION Recent advances in computation, communication, sensing and actuation have stimulated an intensive research in networked multi-agent systems In the systems and controls community, this has translated into how to solve global control problems, expressed by global objective functions, by means of local agent actions Problems considered include multi-agent consensus or agreement [11], [17], coverage control [4], formation control [7], [21] and sensor fusion [24] The seminal work [2] provides a framework to tackle optimizing a global objective function among different processors where each processor knows the global objective function In multi-agent environments, a problem of focus is to minimize a sum of local objective functions by a group of agents, where each function depends on a common global decision vector and is only known to a specific agent This problem is motivated by others in distributed estimation [16], [23], distributed source localization [20], and network utility maximization [12] More recently, consensus techniques have been proposed to address the issues of switching topologies, asynchronous computation and coupling in objective functions; see for instance [14], [15], [27] More specifically, the paper [14] presents the first Manuscript received October 14, 2010; revised January 25, 2012; accepted October 08, 2012 Date of publication November 16, 2012; date of current version May 20, 2013 This work was supported by the NSF CAREER Award CMMI Recommended by Associate Editor E K P Chong M Zhu is with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge MA USA ( mhzhu@mitedu) S Martínez is with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA USA ( soniamd@ucsdedu) Digital Object Identifier /TAC /$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE analysis of an algorithm that combines average consensus schemes with subgradient methods Using projection in the algorithm of [14], the authors in [15] further address a more general scenario that takes local state constraint sets into account Further, in [27] we develop two distributed primal-dual subgradient algorithms, which are based on saddle-point theorems, to analyze a more general situation that incorporates global inequality and equality constraints The aforementioned algorithms are extensions of classic (primal or primal-dual) subgradient methods which generalize gradient-based methods to minimize non-smooth functions This requires the optimization problems of interest to be convex in order to determine a global optimum The focus of the current technical note is to relax the convexity assumption in [27] In order to deal with all aspects of our multi-agent setting, our method integrates Lagrangian dualization, subgradient schemes, and average consensus algorithms Distributed function computation by a group of anonymous agents interacting intermittently can be done via agreement algorithms [4] However, agreement algorithms are essentially convex, and so we are led to the investigation of nonconvex optimization solutions via dualization The techniques of dualization and subgradient schemes have been popular and efficient approaches to solve both convex programs (eg, in [3]) and nonconvex programs (eg, in [5], [6]) Statement of Contributions: Here, we investigate a multi-agent optimization problem where agents desire to agree upon a global decision vector minimizing the sum of local objective functions in the presence of a global inequality constraint and a global state constraint set Agent interactions are changing with time The objective, constraint functions, as well as the state-constraint set, can be nonconvex To deal with both nonconvexity and time-varying interactions, we first define an approximate problem where the exact consensus is slightly relaxed We then propose a distributed dual subgradient algorithm to solve it, where the update rule for local dual estimates combines a dual subgradient scheme with average consensus algorithms, and local primal estimates are generated from local dual optimal solution sets This algorithm is shown to asymptotically converge to a pair of primal-dual solutions to the approximate problem under the following assumptions: firstly, the Slater s condition is satisfied; secondly, the optimal solution set of the dual limit is singleton; thirdly, dynamically changing network topologies satisfy some standard connectivity condition A conference version of this manuscript was published in [26], and an enlarged archived version of this paper is [25] Main differences are the following: i) by assuming that the optimal solution set of the dual limit is a singleton, and changing the update rule in the dual estimates, we are able to determine a global solution in contrast to an approximate solutionin[26];ii)wepresentasimplecriteriontocheckthenewsufficient condition for nonconvex quadratic programs; iii) new simulation results of our algorithm on a source localization example and a comparison of its performance with existing algorithms are performed Due to space limitations, details of the technical proofs and simulations can be found in [25] II PROBLEM FORMULATION AND PRELIMINARIES Consider a networked multi-agent system where agents are labeled by The multi-agent system operates in a synchronous way at time instants, and its topology will be represented by a directed weighted graph, for Here, is the adjacency matrix, where the scalar is the weight assigned to the edge pointing from agent to agent,and is the set of edges with non-zero weights The set of in-neighbors of agent at time is denoted by Similarly, we define the set of out-neighbors of agent at time as Weheremakethe following assumptions on communication graphs: Assumption 21 (Non-Degeneracy): There exists a constant such that,and,for,satisfies,forall Assumption 22 (Balanced Communication): It holds that for all and,and for all and Assumption 23 (Periodical Strong Connectivity): There is a positive integer such that, for all, the directed graph is strongly connected The above network model is standard to characterize a networked multi-agent system, and has been widely used in the analysis of average consensus algorithms; eg see [17], and distributed optimization in [15], [27] Recently, an algorithm is given in [9] which allows agents to construct a balanced graph out of a non-balanced one under certain assumptions The objective of the agents is to cooperatively solve the following primal problem : where is the global decision vector The function is only known to agent, continuous, and referred to as the objective function of agent Theset, the state constraint set, is compact The function are continuous, and the inequality is understood component-wise; ie,,for all, and represents a global inequality constraint We will denote and We will assume that the set of feasible points is non-empty; ie, Since is compact and is closed, then we can deduce that is compact The continuity of follows from that of In this way, the optimal value of the problem is finite and,thesetofprimal optimal solutionss, is non-empty We will also assume the following Slater s condition holds: Assumption 24 (Slater s Condition): There exists a vector such that Such is referred to as a Slater vector of the problem Remark 21: All the agents can agree upon a common Slater vector through a maximum-consensus scheme This can be easily implemented as part of an initialization step, and thus the assumption that the Slater vector is known to all agents does not limit the applicability of our algorithm; see [25] for an algorithm solving this problem In[27],inordertosolvetheconvexcaseoftheproblem (ie; and areconvexfunctionsand isaconvexset),weproposetwodistributed primal-dual subgradient algorithms where primal (resp dual) estimates move along subgradients (resp supergradients) and are projected onto convex sets The absence of convexity impedes the use of the algorithms in [27] since, on the one hand, (primal) gradient-based algorithms are easily trapped in local minima; on the other hand, projection maps may not be well-defined when (primal) state constraint sets are nonconvex In thesequel,wewillemploylagrangiandualization,subgradientmethods and average consensus schemes to design a distributed algorithm which can find an approximate solution to the problem Towards this end, we construct a directed cyclic graph where We assume that each agent has a unique in-neighbor (and out-neighbor) The out-neighbor (resp in-neighbor) of agent is denoted by (resp ) With the graph,wewill study the following approximate problem of problem : (1) (2)

3 1536 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE 2013 where, with a small positive scalar, and is the column vector of ones Problem (2) provides an approximation to,and will be referred to as problem In particular, the approximate problem (2) reduces to the problem when Its optimal value and the set of optimal solutions will be denoted by and,respectively Similarly to the problem, is finite and Remark 22: Thecyclicgraph can be replaced by any strongly connected graph Given, each agent is endowed with two inequality constraints: and,for each out-neighbor This set of inequalities implies that any feasible solution of problem satisfies the approximate consensus; ie, For simplicity, we will use the cyclic graph, with a minimum number of constraints, as the initial graph A Dual Problems Before introducing dual problems, let us denote by,,, and The dual problem associated with is given by where, and Here, the dual function is given as,where is the Lagrangian function (3) Note that is not separable since depends on neighbor s multipliers, B Dual Solution Sets The Slater s condition ensures the boundedness of dual solution sets for convex optimization; eg, [10], [13] We will shortly see that the Slater s condition plays the same role in nonconvex optimization To achieve this, we define the function as follows: Let be a Slater vector for problem Then with is a Slater vector of the problem Similarly to (3) and (4) in [27], which employ Lemma 32 of [27], we have that for any, it holds that where Let be zero in (5) This leads to the upper bound on can be computed locally We de- where note (5) (6) (7) We denote the dual optimal value of the problem by and the set of dual optimal solutions by We endow each agent with the local Lagrangian function and the local dual function defined by In the approximate problem, the introduction of, renders the and separable As a result, the global dual function can be decomposed into a simple sum of the local dual functions More precisely, the following holds: Since and are continuous and is compact, then is continuous; see Theorem 1416 in [1] Similarly, is continuous Since is bounded, then Remark 23: The requirement of exact agreement on in the problem is slightly relaxed in the problem by introducing a small In this way, the global dual function is a sum of the local dual functions,asin(4); is non-empty and uniformly bounded These two properties play important roles in the devise of our subsequent algorithm C Other Notation Define the set-valued map as ; ie, given, the set are the solutions to the following local optimization problem: (8) Notice that in the sum of, each for any appears in two terms: one is,andthe other is With this observation, we regroup the termsinthesummationintermsof, and have the following: (4) Here, is referredtoas themarginal map ofagent Since is compact and are continuous, then in (8) for any In the algorithm we will develop in next section, each agent is required to obtain one (globally) optimal solution and the optimal value the local optimization problem (8) at each iterate We assume that this can be easily solved, and this is the case for problems of,or and being smooth (the extremum candidates are the critical points of the objective function and isolated corners of the boundaries of the constraint regions) or having some specific structure which allows the use of global optimization methods such as branch and bound algorithms In the space, we define the distance between a point to a set as, and the Hausdorff distance between two sets as

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE We denote by III DADS ALGORITHM where In this section, we devise a distributed approximate dual subgradient algorithm which aims to find a pair of primal-dual solutions to the approximate problem For each agent,let be the estimate of the primal solution to the approximate problem at time be the estimate of the multiplier on the inequality constraint (resp ) 1 be the estimate of the multiplier associated with the collection of the local inequality constraints (resp ), for all Welet, for to be the collection of dual estimates of agent And denote where and are convex combinations of dual estimates of agent and its neighbors at time At time, we associate each agent a supergradient vector defined as,where, has components,,and for, while the components of are given by:,,and,for For each agent,wedefine the set for some where Let to be the projection onto the set Itiseasy to check that is closed and convex, and thus the projection map is well-defined The Distributed Approximate Dual Subgradient (DADS) Algorithm is described in Table I Algorithm 1 The DADS Algorithm and Initialization: Initially, all the agents agree upon some in the approximate problem Each agent chooses a common Slater vector, computes and obtains through a max-consensus algorithm where is given in (7) After that, each agent chooses initial states and Iteration: Every, each agent executes the following steps: 1) For each,given, solve the local optimization problem (8), obtain a solution and the dual optimal value 2) For each, generate the dual estimate according to the following rule: where the scalar 3) Repeat for is a step-size Remark 31: The DADS algorithm is an extension of the classical dual algorithm, eg, in [19] and [3] to the multi-agent setting and nonconvex case In the initialization of the DADS algorithm, the value serves as an upper bound on InStep1,one solution in is needed, and it is unnecessary to compute the whole set To assure primal convergence, we assume that dual estimates converge to the set where each has a single optimal solution 1 We will use the superscript to indicate that and are estimates of some global variables (9) Definition 31 (Singleton Optimal Dual Solution Set): The set of is the set of such that the set is a singleton, where for each The primal and dual estimates in the DADS algorithm converge to a pair of primal-dual solutions to the approximate problem We formally state this in the following theorem: Theorem 31: (Convergence Properties of the DADS Algorithm): Consider the problem and the corresponding approximate problem with some We let the non-degeneracy assumption 21, the balanced communication assumption 22 and the periodic strong connectivity assumption 23 hold In addition, suppose the Slater s condition 24 holds for the problem Consider the dual sequences of,, and the primal sequence of of the distributed approximate dual subgradient algorithm with satisfying,, 1) (Dual estimate convergence) There exists a dual solution where and such that the following holds for all 2) (Primal estimate convergence) If the dual solution satisfies,ie is a singleton for all, then there is such that, for all IV DISCUSSION Before outlining the technical proofs for Theorem 31, we would like to make the following observations First, our methodology is motivated by the need of solving a nonconvex problem in a distributed way by a group of agents whose interactions change with time This places a number of restrictions on the solutions that one can find Timevarying interactions of anonymous agents can be currently solved via agreement algorithms; however these are inherently convex operations, which does not work well in nonconvex settings To overcome this, one can resort to dualization Admittedly, zero duality gap does not hold in general for nonconvex problems A possibility would be to resort to nonlinear augmented Lagrangians, for which strong duality holds in a broad class of programs [5], [6], [22] However, we find here another problem, as a distributed solution using agreement requires separability, as the one ensured by the linear Lagrangians we use here Thus, we have looked for alternative assumptions that can be easier to check and allow the dualization approach to work More precisely, Theorem 31 shows that dual estimates always converge to a dual optimal solution The convergence of primal estimates requires an additional assumption that the dual limit has a single optimal solution We refer to this assumption as the singleton dual optimal solution set (SD for short) The assumption may not be easy to check apriori, however it is of similar nature as existing algorithms for nonconvex optimization In [5] and [6], subgradient methods are defined in terms of (nonlinear) augmented Lagrangians, and it is shown that every accumulation point of the primal sequence is a primal solution provided that the dual function is required to be differentiable at the dual limit An open question is how to resolve the above issues imposed by the multi-agent setting with less stringent conditions on the nature of the nonconvex optimization problem In the following, we study a class of nonconvex quadratic programs for which a sufficient condition guarantees that the SD assumption holds Nonconvex quadratic programs hold great importance from both

5 1538 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE 2013 theoretic and practical aspects In general, nonconvex quadratic programs are NP-hard, and please refer to [18] for detailed discussion The aforementioned sufficient condition only requires checking the positive definiteness of a matrix Consider the following nonconvex quadratic program: Lemma 52 (Lipschitz Continuity of ): There is a constant such that for any, it holds that In the DADS algorithm, the error induced by the projection map is given by (10) where and are real and symmetric matrices The approximate problem of is given by A basic iterate relation of dual estimates in the DADS algorithm is the following Lemma 53 (Basic Iterate Relation): Under the assumptions in Theorem 31, for any with for all, we have for all (11) We introduce the dual multipliers as before The local Lagrangian function can be written as follows: where the term independent of is dropped and is a linear function of The dual function and dual problem can be defined as before Consider any dual optimal solution Ifforall : (P1) is positive definite; (P2) ; then the SD assumption holds The properties (P1) and (P2) are easy to verify in a distributed way once a dual solution is obtained We would like to remark that (P1) is used in [8] to determine the unique global optimal solution via canonical duality when is absent V CONVERGENCE ANALYSIS This section outlines the analysis steps to prove Theorem 31 Please refer to [25] for more details Recall that is continuous and is compact Then there are such that and for all We start our analysis from the computation of supergradients of Lemma 51 (Supergradient Computation): If,then is a supergradient of at That is, for any A direct result of Lemma 51 is that the vector is a supergradient of ; ie, the following supergradient inequality holds for any : (12) at (14) Asymptotic convergence of dual estimates is shown next Lemma 54 (Dual Estimate Convergence): Under the assumptions in Theorem 31, there exists such that,,and The remainder of the section is dedicated to characterizing the convergence properties of primal estimates Lemma 55 (Properties of Marginal Maps): The set-valued marginal map is closed In addition, it is upper semicontinuous at ; ie, for any,thereis such that for any, it holds that Lemma 56 (Primal Estimate Convergence): Under the assumptions in Theorem 31, for each, it holds that where The main result of this technical note, Theorem 31, can be shown next In particular, we will show complementary slackness, primal feasibility of, and its primal optimality, respectively Proof for Theorem 31: Claim 1:, and Proof: Rearranging the terms related to in (14) leads to the following inequality holding for any with for all : (13) It can be seen that (9) of dual estimates in the DADS algorithm is a combination of a dual subgradient scheme and average consensus algorithms The following establishes that is Lipschitz continuous with some Lipschitz constant (15)

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 58, NO 6, JUNE Summing (15) over [0, ], and dividing by (16) We now proceed to show for each Let, for and, in (16) Recall that is not summable but square summable, and is uniformly bounded Take, and then it follows from Lemma 51 in [27] that: (17) On the other hand, since,wehave given the fact that is an upper bound of Let where Then we could choose a sufficiently small and in (16) such that where is given in the definition of and is given by:, for Following the same lines toward (17), it gives that Hence, it holds that The rest of the proof is analogous and thus omitted The proofs of the following two claims can be found in [25] Claim 2: is a primal feasible solution to the approximate problem Claim 3: is a primal solution to the problem VI CONCLUSION We have studied a distributed dual algorithm for a class of multiagent nonconvex optimization problems The convergence of the algorithm has been proven under the assumptions that i) the Slater s condition holds; 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