Distributed ergodic algorithms for mixed equilibrium problems: Absent of cut property

Abstract

In this paper, the problem of distributively solving mixed equilibrium problems is studied by employing a multi-agent network. The goal of agents is to cooperatively find a point in a convex set such that the sum of multiple bifunctions with a free variable is non-negative. Different from the existing works, here we consider the case where bifunctions do not possess the cut property. To solve this problem, we propose a distributed ergodic algorithm based on the ergodic algorithm and consensus algorithm. When running the proposed algorithm, each agent makes decisions only using its own bifunction information and the local state information received from its immediate neighbors. Under mild conditions on bifunctions and the graph, we prove that outputs of all agents weakly converge to a common solution of the mixed equilibrium problem. Finally, a numerical simulation example is worked out to demonstrate the effectiveness of theoretical results.