Iterative methods for solving monotone equilibrium problems via dual gap functions

Abstract

This paper proposes an iterative method for solving strongly monotone equilibrium problems by using gap functions combined with double projection-type mappings. Global convergence of the proposed algorithm is proved and its complexity is estimated. This algorithm is then coupled with the proximal point method to generate a new algorithm for solving monotone equilibrium problems. A class of linear equilibrium problems is investigated and numerical examples are implemented to verify our algorithms.

An algorithm for a class of split feasibility problems: application to a model in electricity production

“…The latter model has been investigated in some research papers (see e.g. Contreras et al 2004;Quoc and Muu 2012). In this equilibrium model, it is assumed that there are n companies, each company i may possess I i generating units.…”

Auxiliary problem principles for equilibria

“…for some mapping F : R n → R n , satisfy the assumption of Corollary 3.3 only for τ = 0, while the so-called linear EPs (see [27]), that is (EP) with…”

“…On the contrary, the class of problems (MEP α ) has been explicitly considered only for variational inequalities with α < 0 in [6,26], indirectly through gap functions with α > 0 in [27,28] and very recently in [29] to refine some existence results for EPs.…”

“…Hence, the following chain of inclusions holds: [17,20]). Case (b) with τ = 0, μ > 0 and α = 0 or α = μ was considered in [27]. Considering just variational inequalities, case (a) with μ = 0 and α ≤ 0 has been proved in [6,26], while case (b) with μ > 0 and α ≥ 0 in [28,Lemma 2.3].…”

“…A rather novel feature is the analysis of (EP α ) with α < 0 which, up to now, has been considered only for the so-called linear EP in [27]. Furthermore, a systematic analysis of (MEP α ) allows achieving new equivalence results also for α > 0.…”