Fixed point approach for solving optimization problems in Hilbert, Banach and convex metric spaces.
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Date
2023
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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.
Description
Doctoral Degree. University of KwaZulu-Natal, Durban.