29 research outputs found
Approximation methods for solutions of some nonlinear problems in Banach spaces.
Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Durban 2016.Abstract available in PDF file
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
A study of optimization problems and fixed point iterations in Banach spaces.
Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
Nonlinear Analysis and Optimization with Applications
Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world
Nonlinear Analysis: Algorithm, Convergence, and Applications 2014
Department of Applied Mathematic
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
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
Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.
Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a shrinking projection method of an inertial type with
self-adaptive step size for finding a common element of the set of solutions of Split Gen-
eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable
family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step
size incorporated helps to overcome the difficulty of having to compute the operator norm
while the inertial term accelerates the rate of convergence of the propose algorithm. Under
standard and mild conditions, we prove a strong convergence theorem for the sequence
generated by the proposed algorithm and obtain some consequent results. We apply our
result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza-
tion Problem (SMP), and present numerical examples to illustrate the performance of
our algorithm in comparison with other existing algorithms. Moreover, we investigate the
problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion
Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo-
contractive mappings. We propose an iterative scheme which combines inertial technique
with viscosity method for approximating common solutions of these problems in Hilbert
spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and
apply our results to approximate the solutions of other optimization problems. Finally,
we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement
contemporary results in the literature in this direction