Abstract
This paper considers a distributed constrained optimization problem, where the objective function is the sum of local objective functions of distributed nodes in a network. The estimate of each agent is restricted to different convex sets. To solve this optimization problem which is not necessarily smooth, we study a novel distributed projected subgradient algorithm for multi-agent optimization with nonidentical constraint sets and switching topologies. The algorithm shows that each agent minimizes its own objective function while communicating information locally with other agents over a network with time-varying topologies but satisfying a standard connectivity property. Under the assumption that the network topology is weight-balanced, the novel distributed subgradient algorithm we proposed is proven to be convergent. Particularly, we suppose the step-size is various, which is different from previous work on multi-agent optimization that makes worst-case assumption with constant step-size.
Highlights
In many networked systems, multi-agent are enforced to solve a distributed convex optimization problem, where the global objective function is the sum of local objective functions, each of them can not be known or shared by other agents
To figure out distributed optimization problems with asynchronous stepsizes or inequality–equality constraints, Distributed Lagrangian primal-dual subgradient algorithm and penalty primal-dual subgradient algorithm were shown in Zhu et al [6], Towfic and Sayed [12], both of them were designed for function constrained problems
Zhu et al [22] proposed a distributed Lagrangian primal–dual subgradient method which is based on the Applied and Computational Mathematics 2016; 5(3): 150-159 characterization of the primal–dual optimal solutions as the saddle points of the Lagrangian function associated with the problem
Summary
Multi-agent systems and distributed algorithms have received considerable research attentions due to its wide applications in many engineering systems and large-scale networks [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], including resource allocation in computer network [16,17,18], distributed estimation in sensor networks [19], distributed finite-time optimal rendezvous problem [21], and distributed demand response control problem in smart grid [22]. Nedić and Ozdaglar [1] presented an analysis of the consensus-based subgradient method for solving the distributed convex optimization problem. Inspired by the works of [1, 3, 33], a multi-agent unconstrained convex optimization problem through a novel combination of average consensus algorithms with subgradient methods was solved in Nedić and Ozdaglar [1]. Contributions: Inspired by the previous studies, this paper proposes a novel distributed subgradient algorithm for multiagent convex optimization with local constraint sets. Based on the conditions that each agent is restricted to different convex sets and the digraph is weight-balanced, we introduce a novel distributed projected subgradient algorithm under the case of various step-sizes.
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