Algorithms for approximating minimization problems in Hilbert spaces

Abstract

In this paper, we study the following minimization problem

$$\underset{x\in Fix\left(S\right)\cap \Omega }{min}\frac{\mu }{2}〈Bx,x〉+\frac{1}{2}{\Vert x\Vert }^2-h\left(x\right)\text{,}$$

where $B$ is a bounded linear operator, $\mu \ge 0$ is some constant, $h$ is a potential function for $\bar{\gamma }f$, $Fix\left(T\right)$ is the set of fixed points of nonexpansive mapping $S$ and $\Omega$ is the solution set of an equilibrium problem. This paper introduces two new algorithms (one implicit and one explicit) that can be used to find the solution of the above minimization problem.