An extragradient algorithm for quasiconvex equilibrium problems without monotonicity

TL;DR

This paper introduces an extragradient algorithm for quasiconvex equilibrium problems that does not require monotonicity or Lipschitz conditions, converging to approximate or exact solutions under certain quasiconvexity assumptions.

Contribution

It proposes a novel iterative extragradient algorithm for quasiconvex equilibrium problems that converges without monotonicity or Lipschitz conditions, broadening applicability.

Findings

Algorithm converges to a $ ho$-quasi-solution for semistrictly quasiconvex functions.

Algorithm converges to the exact solution for strongly quasiconvex functions.

No monotonicity or Lipschitz conditions are needed for convergence.

Abstract

We attempt to provide an algorithm for approximating a solution of the quasiconvex equilibrium problem that was proved to exist by K. Fan 1972. The proposed algorithm is an iterative procedure, where the search direction at each iteration is a normal-subgradient, while the step-size is updated avoiding Lipschitz-type conditions. The algorithm is convergent to a $\rho$- quasi-solution with any positive $\rho$ if the bifunction $f$ is semistrictly quasiconvex in its second variable, while it converges to the solution when $f$ is strongly quasiconvex. Neither monotoniciy nor Lipschitz property is required.