M.Sc Thesis

M.Sc StudentBarshad Kay
SubjectIterative Coherent Methods for Solving Convex Optimization
Problems in Hilbert Space
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Simeon Reich


In this thesis we develop new iterative methods for solving convex feasibility and common fixed point problems in Hilbert space, based on the notion of coherence. Such methods are important in modern convex optimization, since many practical problems related to a wide range of fields can be reduced to one of the aforesaid problems. For example, solving a system of linear equations with real coefficients can be viewed as a convex feasibility problem of finding a point in the intersection of hyperplanes corresponding to the equations of the system, as well as a common fixed point problem of finding a common fixed point of operators that are orthogonal projections onto these hyperplanes. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.