Penalty Methods For A Class Of Sparse Constrained Optimization Problems

Sparse constraint optimization problem is a class of nonconvex and nonsmooth optimization problems due to the sparsity constraint.It has wide applications in many fields such as regression analysis,signal and image processing,machine learn-ing,and pattern recognition.Because of the nonconvexity and nonsmoothness of the problem,the classical optimization algorithms cannot be applied to solve it directly.It is of theoretical significance and application value to study numerical algorithms for solving sparse constrained optimization problems,which has attracted lots of attention from scholars.In this thesis,we consider the following sparse constrained optimization problem(?) Where f:R~n→R is continuously differentiable,(?) We propose two penalty methods(i.e.,classical penalty method and improved penal-ty method)to solve this problem.Under suitable conditions,we study the conver-gence of these two methods.In particular,we prove that each accumulation point of the iterate sequence generated by the improved penalty method is a L-stationary point of the original problem.Numerical results show the efficiency and the relia-bility of the improved penalty method.This thesis is divided into five chapters and the structure is arranged as follow.The first chapter describes the research background and research status of the relevant work in this paper.The development status of sparse constrained optimiza-tion problem and the penalty function are introduced.In the second chapter,the concept of L-stability point is proposed,a non-monotonic projection gradient method is obtained,and the convergence theorem is given.The third chapter puts forward a kind of sparse constraint optimization problem of classic penalty method,and the convergence theorem is given.In the fourth chapter,an improved penalty method is proposed,the correspond-ing penalty subalgorithm is proposed,and the convergence theorem is given.It is proved that any aggregation point of the iterative point series generated by the al-gorithm is the L-stable point of the sparse constrained optimization problem,and it is compared with the non-monotone projection gradient method and the classical penalty method.In the fifth chapter,the validity and reliability of the algorithm are verified by numerical experiments.