Accurate Continuous Approximation Theory And Alorithm Analsis For A Class Of Sparse Matrix Optimization Problems

Optimization problems are closely related to people's lives and learning.Optimization theory is the theoretical basis for solving the optimization problems.With the rapid development of computer technology in recent years,optimization theory is particularly important in contemporary important fields,such as image processing,artificial intelligence and compressed sensing.Among them,the sparse optimization problems are more commonly used.Sparse optimization is an important branch of opt imization theory and is increasingly used in economic,engineering,military and other fields.With the development of image restoration and reconstruction,artificial intelligence and compressed sensing,it is particularly important to solve the sparse optimization problems.The sparse optimization problems refer to finding the solution whose most of the elements are zero.The sparsity of a matrix can be defined by its cardinality(1-0 norm),so the optimization problem models with the cardinality term are the most direct and ideal models for this kind of problems.Both the theory of solving the sparse optimization problems and the analysis of algorithms have extremely important significance.For a class of non-continuous sparse optimization problems,we define a class of stationary points of the approximation problems,and give an exact continuous relaxation problem,and prove that the original problem and its approximation problem have the same global minimizers.Then we prove that the stationary point of the approximation problem is equivalent to the local optimal solution with the lower bound property of the original problem.Then we design a proximal gradient algorithm for the sparse matrix problems.We propose the related steps of the proximal gradient algorithm,and prove that the iterates of the approximation problem converges to the defined stationary point,which is a local minimizer of the original problem satisfing a desired lower bound property.Finally,we make three numerical experiments on the approximation problem with the proposed proximal gradient algorithm,verify the convergence of the iterates and find that the iterates always converge to a sparse local solution of the original problem.