Pure Mathematics
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Browsing Pure Mathematics by Subject "Algorithms."
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Item Fixed point approach for solving optimization problems in Hilbert, Banach and convex metric spaces.(2023) Ogwo, Grace Nnennaya.; Mewomo, Oluwatosin Temitope.; Alakoya, Timilehin Opeyemi.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.Item Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.(2021) Olona, Musa Adewale.; Mewomo, Oluwatosin Temitope.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.Item Self-adaptive inertial algorithms for approximating solutions of split feasilbility, monotone inclusion, variational inequality and fixed point problems.(2020) Owolabi, Abd-semii Oluwatosin-Enitan.; Mewomo, Oluwatosin Temitope.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.Item A study of optimization problems and fixed point iterations in Banach spaces.(2019) Jolaoso, Lateef Olakunle.; Mewomo, Oluwatosin Temitope.Abstract available in PDF.