Distributed Subgradient Algorithm for Multi-Agent Convex Optimization with Global Inequality and Equality Constraints

Abstract

Abstract:In this paper, we present an improved subgradient algorithm for solving a general multi-agent convex optimization problem in a distributed way, where the agents are to jointly minimize a global objective function subject to a global inequality constraint, a global equality constraint and a global constraint set. The global objective function is a combination of local agent objective functions and the global constraint set is the intersection of each agent local constraint set. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. Our main focus is on constrained problems where the local constraint sets are identical. Thus, we propose a distributed primal-dual subgradient algorithm, which is based on the description of the primal-dual optimal solutions as the saddle points of the penalty functions. We show that, the algorithm can be implemented over networks with changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically converge to optimal solution with optimal value of the optimization problem under the Slater's condition.

Gradient-free distributed optimization and nash equilibrium seeking

“…In terms of the theoretical analysis, whether the set constraints are the same or not does not have a significant influence, as long as the optimal solution set still lies in the intersection of these set constraints (i.e., x ∈ X = ∩ N i X i ). Hence, similar to the work in [34,35,42…”

“…and the distributed penalty primal-dual subgradient method in [41][42][43]. The work in [44] proposed an accelerated distributed Nesterov gradient descent for convex and smooth functions, while Yuan et al in [45] developed an optimal distributed stochastic mirror descent for strongly convex functions.…”