Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets

📝 Abstract

In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.

💡 Analysis

In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.

📄 Content

Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets∗ Guodong Shi, Karl Henrik Johansson†and Yiguang Hong‡ Abstract In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis. Keywords: Multi-agent systems, Optimal consensus, Connectivity Conditions, Distributed optimization, Intersection computation 1 Introduction In recent years, multi-agent dynamics has been intensively investigated in various areas including engineering, natural science, and social science. Cooperative control of multi-agent systems is an ∗This work has been supported in part by the Knut and Alice Wallenberg Foundation, the Swedish Research Council, KTH SRA TNG, and the NNSF of China under Grant 61174071. †G. Shi and K. H. Johansson are with ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden. Email: guodongs@kth.se, kallej@ee.kth.se ‡Y. Hong is with Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email: yghong@iss.ac.cn 1 arXiv:1112.1333v2 [cs.MA] 18 Mar 2012 active research topic, and rapid developments of distributed control protocols via interconnected communication have been made to achieve the collective tasks, e.g., [16, 15, 12, 25, 10, 9, 20, 22, 17, 18]. However, fundamental challenges still lie in finding suitable tools to describe and design the dynamical behavior of these systems and thus providing insights in their functioning principles. Different from the classical control design, the multi-agent studies aim at fully exploiting, rather than avoiding, interconnection between agents in analysis and synthesis in order to deal with distributed design and large-scale information process. Consensus is a basic problem of the study of multi-agent coordination, which usually requires that all the agents achieve the same state, e.g., a certain relative position or velocity. To achieve collective behavior, connectivity plays a key role in the coordination of multi-agent network, and various connectivity conditions have been used to describe frequently switching topologies in different cases. The “joint connection” or similar concepts are important in the analysis of stability and convergence to guarantee a suitable convergence. Uniform joint-connection, i.e., the joint graph is connected during all intervals which are longer than a constant, has been employed for different consensus problems [16, 15, 24, 19, 7]. On the other hand, [t, ∞)-joint connectedness, i.e., the joint graph is connected in the time intervals [t, ∞), is necessary [22, 25], and therefore the most general form to secure the global coordination. Moreover, distributed optimization of a sum of convex objective functions, PN i=1 fi(z), where each component fi is known only to node i, has attracted much attention in recent years, due to its wide application in multi-agent systems and wireless networks [29, 30, 32, 31, 33]. A class of subgradient-based incremental when some estimate of the optimal solution can be passed over the network via deterministic or randomized iteration were studied in [29, 30, 34]. Then a non-gradient-based algorithm was proposed in [33], where each node starts at its own optimal solution and updates using a pairwise equalizing protocol. In view of multi-agent systems, the local information transmitted over the neighborhood is usually limited to a convex combina- tion of its neighbors [16, 15, 25]. Combining the ideas of consensus algorithms and subgradient methods, a number of significant results were obtained. A subgradient method in combination with consensus steps was given for solving coupled optimization problems with fixed undirected topology in [31]. Then, an important work on multi-agent optimization was [27], where a decentralized algorithm was proposed as a simple sum of an averaging (consensu

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