Fixed point approach for solving optimization problems in Hilbert, Banach and convex metric spaces.

Abstract

In this thesis, we study the fixed point approach for solving optimization problems in real Hilbert, Banach and Hadamard spaces. These optimization problems include the variational inequality problem, split variational inequality problem, generalized variational inequality problem, split equality problem, monotone inclusion problem, split monotone inclusion problem, minimization problem, split equilibrium problem, among others. We consider some interesting classes of mappings such as the nonexpansive semigroup in real Hilbert spaces, strict pseudo-contractive mapping in real Hilbert spaces and 2-uniformly convex real Banach spaces, nonexpansive mapping between a Hilbert space and a Banach space, and quasi-pseudocontractive mapping in Hilbert spaces and Hadamard spaces. We introduce several iterative schemes for approximating the solutions of the various aforementioned optimization problems and fixed point problems and prove their convergence results. We adopt and implement several inertial methods such as the inertial-viscosity-type algorithm, relaxed inertial subgradient extragradient, modified inertial forward-backward splitting algorithm viscosity method, among others. Furthermore, we present several novel and practical applications of our results to solve other optimization problems, image restoration problem, among others. Finally we present several numerical examples in comparison with some results in the literature to illustrate the applicability of our proposed methods.