Algorithmic Approach to a Minimization Problem

Abstract

We first construct an implicit algorithm for solving the minimization problem ${\text{min}}_{x\in \Omega }\parallel x\parallel$ , where $\mathrm{\Omega }$ is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. We prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.