Iterative Methods for Large Scale Convex Optimization

Abstract

This thesis presents a detailed description and analysis of Bregman’s iterative method for convex programming with linear constraints. Row and block action methods for large scale problems are adopted for convex feasibility problems. This motivates Bregman type methods for optimization. A new simultaneous version of the Bregman’s method for the optimization of Bregman function subject to linear constraints is presented and an extension of the method and its application to solving convex optimization problems is also made. Closed-form formulae are known for Bregman’s method for the particular cases of entropy maximization like Shannon and Burg’s entropies. The algorithms such as the Multiplicative Algebraic Reconstruction Technique (MART) and the related methods use closed-form formulae in their iterations. We present a generalization of these closed-form formulae of Bregman’s method when the objective function variables are separated and analyze its convergence. We also analyze the algorithm MART when the problem is inconsistent and give some convergence results.