An extragradient algorithm for quasiconvex equilibrium problems without monotonicity

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 ?-quasi-solution with any positive ? 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. The main subprogram needed to solve at each iteration is a proximal regularized minimization problem whose objective function is the sum of a quasiconvex function and the one ||.||(2). We also discuss several cases where this global optimization problem can be solved efficiently.