Methods For Solving Lp Regularized Nonlinear Programming Problems

Lp regularization problem withO<p<1 is a typical non-smooth,nonconvex and non-Lipschitz continuous optimization problem,and it is widely used in signal recovery,variable selection,image processing,compression sensing and computer vision.How to solve the problem effectively is one of the hotspots in the field of optimization,and the processing of |||x||pp in the objective function makes it a difficulty in the research.In this thesis,we focus on the numerical mehods for solving the unconstrained Lp regularization problem and the Lp regularization problem with simple box constraints.The main results of our research are as follows:1.Considering the unconstrained Lp regularization problem,in our first algorithm,we transform the unconstrained Lp problem into an equivalent smooth constrained optimization problem using |xi|p=ti?0,i= 1,2,…,n.We study the KKT condition of the equivalent problem and derive the analytic solution of its steepest descent direction.On the basis of the steepest descent direction,we propose a feasible descent direction algorithm with global convergence,and test its effectiveness with numerical experiments.2.Based on the idea of smoothing methods,we use a class of Chen-Mangasarian smoothing functions to transform the original problem into a series of smoothing approximation problems without constraints.By combining the nonmonotonic linear search technique,we propose a smoothing Barzilai-Borwein gradient algorithm,and we prove that it can converge to the first-order stationary point of the Lp problem for arbitary Chen-Mangasarian smoothing function.In the numerical experiment,we simply compare the convergence of the smoothing Barzilai-Borwein gradient algorithm under two different smoothing functions.3.For the Lp regularization problems with simple box constraints,we combine the projected gradient method to derive our new algorithm using the framework of the smoothing Barzilai-Borwein gradient algorithm mentioned above,which guarantees the feasibility of iteration points.We prove that under suitable assumptions,the iteration points of the algorithm can effectively converge to first-order stationary point of the problem.