Iterative methods for approximating solutions of certain optimization problems and fixed points problems.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Abstract available in PDF file

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

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

Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process

Biomorphs are graphic objects with very interesting shapes resembling unicellular and microbial organisms such as bacteria. They have applications in different fields like medical science, art, painting, engineering and the textile industry. In this paper, we present for the first time escape criterion results for general complex polynomials containing quadratic, cubic and higher order polynomials. We do so by using a more general iteration method also used for the first time in this field. This also generalizes some previous results. Then, biomorphs are generated using an algorithm whose pseudocode is included. A visualization of the biomorphs for certain polynomials is presented and their graphical behaviour with respect to variation of parameters is examined

Two Bregman Projection Methods for Solving Variational Inequality Problems in Hilbert Spaces with Applications to Signal Processing

Studying Bregman distance iterative methods for solving optimization problems has become an important and very interesting topic because of the numerous applications of the Bregman distance techniques. These applications are based on the type of convex functions associated with the Bregman distance. In this paper, two different extragraident methods were proposed for studying pseudomonotone variational inequality problems using Bregman distance in real Hilbert spaces. The first algorithm uses a fixed stepsize which depends on a prior estimate of the Lipschitz constant of the cost operator. The second algorithm uses a self-adaptive stepsize which does not require prior estimate of the Lipschitz constant of the cost operator. Some convergence results were proved for approximating the solutions of pseudomonotone variational inequality problem under standard assumptions. Moreso, some numerical experiments were also given to illustrate the performance of the proposed algorithms using different convex functions such as the Shannon entropy and the Burg entropy. In addition, an application of the result to a signal processing problem is also presented

An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method

An Inertial Generalized Viscosity Approximation Method for Solving Multiple-Sets Split Feasibility Problems and Common Fixed Point of Strictly Pseudo-Nonspreading Mappings

In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature

Halpern-Subgradient Extragradient Method for Solving Equilibrium and Common Fixed Point Problems in Reflexive Banach Spaces

In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior knowledge of the Lipschitz constants. We also prove a strong convergence result for the sequence that is generated by our algorithm under mild conditions. We apply our result to solving variational inequality problems, and finally, we give some numerical examples to illustrate the efficiency and accuracy of the algorithm

Common Fixed Point Results for a Pair of Multivalued Mappings in Complex-Valued b-Metric Spaces

Several fixed point results for the existence of common fixed points of multivalued contractive mappings have been established in complex-valued metric space. In this paper, we study the existence of common fixed points for a pair of multivalued contractive mappings satisfying some rational inequalities in the framework of complex-valued b-metric spaces. The contractive condition used in this paper generalizes many contractive conditions used by other authors in the literature. Employing our results, we check the existence solution to the Riemann-Liouville equation

Strong convergence of a Bregman projection method for the solution of pseudomonotone equilibrium problems in Banach spaces

In this paper, we introduce an inertial self-adaptive projection method using Bregman distance techniques for solving pseudomonotone equilibrium problems in reflexive Banach spaces. The algorithm requires only one projection onto the feasible set without any Lipschitz-like condition on the bifunction. Using this method, a strong convergence theorem is proved under some mild conditions. Furthermore, we include numerical experiments to illustrate the behaviour of the new algorithm with respect to the Bregman function and other algorithms in the literature.</p