Distributed strategies for mixed equilibrium problems: Continuous-time theoretical approaches

Abstract

In this paper, the mixed equilibrium problem is studied by employing a continuous-time multi-agent system, where the objective of agents is to cooperatively find a point from the feasible set such that the sum of bifunctions with a free-variable is nonnegative. Different from existing works on mixed equilibrium problems with only convex set constraints, we consider a more general case where the feasible set is constrained by a set of convex inequalities. Moreover, each agent only has access to the information of its own bifunction and inequality constraint function, as well as the information of a local convex set. To handle this problem, the Karush–Kuhn–Tucker condition that is suitable for continuous-time theoretical approaches is provided, and a continuous-time distributed primal–dual algorithm is proposed. Under mild assumptions on the graph and bifunctions, we prove that the states of all agents asymptotically converge to the common solution of the mixed equilibrium problem. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.