Research On Primal-dual And Block-iterative Splitting Algorithm For Convex Minimization Problems

In this thesis, we mainly consider this class of convex minimization problems:(?) f(x) + g(x) + ?(Lx)and propose two different primal-dual splitting algorithms for solving convex minimiza-tion problems with the bounded linear operators. Both algorithms rely on the projective algorithm on half-space, but still differ clearly from each other.This paper mainly contains three parts.In the first part, we introduce the basic knowledge and present preliminary result about the convex minimization problems. In addition,we give the key points of the paper.In the second part, we use two different method to construct the half-space con-taining the Kuhn-Tucker set Z, and get the new iterative point by projecting the current iterate onto the recent constructed half-space. Moreover we discuss the convergence properties of the two approaches.The 21st century is an era of big data, and the complexity of the problem is increas-ing. In order to adapt this trend, we give the block-iterative version of the second method in the third part, which is to solve the complex form of above-mentioned problem. An important feature of the third approach is that only a subset of the operators needs to be calculate at each iteration, as opposed to all operators as in the second algorithm.By doing this, the computation load is reduced largely. In social production and the real life, block-iteration is of great practicability.