Multi-step algorithms for solving EPs

Abstract

The paper introduces and analysizes the convergence of two multi-step proximal-like algorithms for pseudomonotone and Lipschitz-type continuous equilibrium problems in a real Hilbert space. The algorithms are combinations between the multi-step proximal-like method and Mann or Halpern iterations. The weakly and strongly convergent theorems are established with the prior knowledge of two Lipschitz-type continuous constants. Moreover, by choosing two sequences of suitable stepsizes, we also show that the multi-step proximal-like algorithm for strongly pseudomonotone and Lipschitz-type continuous equilibrium problems where the construction of solution approximations and the establishing of its convergence do not require the prior knowledge of strongly pseudomonotone and Lipschitz-type continuous constants of bifunctions. Finally, several numerical examples are reported to illustrate the convergence and the performance of the proposed algorithms over classical extragradient-like algorithms.

Keywords

• proximal-like method
• extragradient method
• equilibrium problem
• multi-step method
• Lipschitz-type continuous