Projection Algorithm For Sparse Constrained Optimization Problems

Sparse constrained optimization problem is a very important field in applied mathematics.It has wide applications in signal and image processing,machine learning,compressive sensing and pattern recognition and so on.Recently,it has been successfully applied to face recognition,target detection,computer vision and other problems.Designing effective algorithm for solving the spare constrained optimization problem has important theoretical significance and application value,which is currently one of the most popular research topics.At present,there are few studies for the sparse constrained optimization problem whose objective function is a general nonlinear function.Therefore,to design algorithms for solving sparse constrained optimization problem is a more meaningful research topic.This paper is divided into four chapters.The main structure is as follows:The first chapter is an introduction.We mainly describe the specific definition,application backgrounds and the research situation of the sparse constrained optimization problem and the main results obtained in this article.In the second chapter,we present a new projection algorithm for solving the constrained optimization problem.Under this new step size rule,the algorithm does not require the gradient of the objective function to be Lipschitz continuous.It is shown that arbitrary accumulation point of the sequence of the iterates generated by the algorithm is an ?-stationary point of the problem.If the objective function is convex,it converges to the optimal solution.Finally,a numerical example is given to illustrate the feasibility and effectiveness of the proposed algorithm.The third chapter considers the optimization problem with sparsity constraints and closed convex set constraints.We design a gradient projection algorithm with Armijo step size rule,and prove that the sequence of the iterates generated by this algorithm can converge to an ?-stationary point of the problem.Finally,a numerical example is given to demonstrate the effectiveness of the algorithm.The fourth chapter considers the problem of finding the solution of sparsity split feasibility problem.By transforming the sparsity split feasibility problem into a sparsity constraints optimization problem whose objective function is convex,we present a new algorithm for solving the problem.It is proved that the sequence of iterates generated by this algorithm can converge to a solution of the problem.The numerical example is given to prove the effectiveness of the algorithm.