Simple proximal-type algorithms for equilibrium problems

Abstract

This paper proposes two simple and elegant proximal-type algorithms to solve equilibrium problems with pseudo-monotone bifunctions in the setting of Hilbert spaces. The proposed algorithms use one proximal point evaluation of the bifunction at each iteration. Consequently, prove that the sequences of iterates generated by the first algorithm converge weakly to a solution of the equilibrium problem (assuming existence) and obtain a linear convergence rate under standard assumptions. We also design a viscosity version of the first algorithm and obtain its corresponding strong convergence result. Some popular existing algorithms in the literature are recovered. We finally give some numerical tests and compare our algorithms with some related ones to show the performance and efficiency of our proposed algorithms.