Splitting gradient algorithms for solving monotone equilibrium problems

TL;DR

This paper introduces a modified splitting gradient algorithm for monotone equilibrium problems that guarantees convergence without Lipschitz or H"older conditions, demonstrated through solving a Cournot-Nash model.

Contribution

The paper develops a new splitting algorithm that converges for paramonotone bifunctions without requiring Lipschitz or H"older continuity, extending previous methods.

Findings

Algorithm converges for paramonotone bifunctions without Lipschitz conditions.

Ergodic sequence converges even without paramonotonicity.

Efficient computational results on a Cournot-Nash model.

Abstract

It is well known that the projection method is not convergent for monotone equilibrium problems. Recently Sosa \textit{et al.} in \cite{SS2011} proposed a projection algorithm ensuring convergence for paramonotone equilibrium problems. In this paper we modify this algorithm to obtain a splitting convergent one for the case when the bifunction is the sum of the two ones. At each iteration, two strongly convex subprograms are required to solve separately, one for each component bifunction. We show that the algorithm is convergent for paramonotone bifunction without any Lipschitz type condition as well as H\"older continuity of the involved bifunctions. Furthermore, we show that the ergodic sequence defined by the algorithm's iterates converges to a solution without paramonotonicity property. We use the proposed algorithm to solve a jointly constrained Cournot-Nash model. The computational results show that this algorithm is efficient for the model with a restart strategy.