A linesearch projection algorithm for solving equilibrium problems without monotonicity in Hilbert spaces

TL;DR

This paper introduces a novel linesearch projection algorithm for equilibrium problems in Hilbert spaces that does not require monotonicity or Lipschitz conditions, demonstrating strong convergence and computational efficiency.

Contribution

It develops a new convergence proof avoiding Fejér monotonicity by projecting a fixed point, and employs an Armijo-linesearch without subgradients for faster computation.

Findings

Algorithm converges strongly to a solution.

Numerical experiments show improved efficiency.

No need for monotonicity or Lipschitz conditions.

Abstract

We propose a linesearch projection algorithm for solving non-monotone and non-Lipschitzian equilibrium problems in Hilbert spaces. It is proved that the sequence generated by the proposed algorithm converges strongly to a solution of the equilibrium problem under the assumption that the solution set of the associated Minty equilibrium problem is nonempty. Compared with existing methods, we do not employ Fej\'{e}r monotonicity in the strategy of proving the convergence. This comes from projecting a fixed point instead of the current point onto a subset of the feasible set at each iteration. Moreover, employing an Armijo-linesearch without subgradient has a great advantage in CPU-time. Some numerical experiments demonstrate the efficiency and strength of the presented algorithm.