Iterative algorithms for approximating solutions of some optimization problems in Hadamard spaces.

Masters Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF.Some text in red

A viscosity of Cesàro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems

In this paper, we introduce and study a iterative viscosity approximation method by modify Cesàro mean approximation for finding a common solution of split generalized equilibrium, variational inequality and fixed point problems. Under suitable conditions, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. The results presented in this paper generalize, extend and improve the corresponding results of Shimizu an

Self-adaptive inertial algorithms for approximating solutions of split feasilbility, monotone inclusion, variational inequality and fixed point problems.

Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a self-adaptive hybrid inertial algorithm for approximating a solution of split feasibility problem which also solves a monotone inclusion problem and a fixed point problem in p-uniformly convex and uniformly smooth Banach spaces. We prove a strong convergence theorem for the sequence generated by our algorithm which does not require a prior knowledge of the norm of the bounded linear operator. Numerical examples are given to compare the computational performance of our algorithm with other existing algorithms. Moreover, we present a new iterative algorithm of inertial form for solving Monotone Inclusion Problem (MIP) and common Fixed Point Problem (FPP) of a finite family of demimetric mappings in a real Hilbert space. Motivated by the Armijo line search technique, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumptions of monotonicity and Lipschitz continuity of the MIP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with the non-inertial version and some related methods in the literature. Furthermore, we propose a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problems and common fixed points of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm

Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others

Inertial Krasnosel’skiĭ-Mann iterative algorithm with step-size parameters involving nonexpansive mappings with applications to solve image restoration problems

In this work, we propose and study an inertial Krasnosel’ski˘ ı-Mann iterative algorithm with step-size parameters involving nonexpansive mapping to find a solution of a fixed point problem of a nonexpansive mapping in the frame work of Hilbert spaces. Strong convergence of the new proposed algorithm is proved under some useful properties of a nonexpansive mapping and inequalities on real Hilbert spaces together with the appropriate conditions of scalar controls without relying on the concept of viscosity methods. For the applications, we employ the obtained results to find a zero point of some monotone inclusion problems and then apply to solve image restoration problems. For representing the advantage of the new algorithm, the signal to noise ratio (SNR) with various types of blurring operators and some numerical experiments are presented to compare and illustrate the behavior of the new algorithm with numerical results of some previous existing algorithms

A study of optimization problems and fixed point iterations in Banach spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF