On Theory And Algorithm For A Class Of Non-Lipschitz Constrained Optimization

Recent years,the theory and applications of compressed sensing have been one of the hottest research fields.The theory of compressed sensing includes three main aspects such as sparse representation,coding measurement,signal reconstruction.The key idea of compressed sensing is to compress and sample the original sparse signals simultaneously,then to reconstruct the original signal from the observation via the sparse optimization models.Sparse optimization aims to obtain sparse solutions by solving optimization problems.The theory of compressed sensing has brought significant change to the information sampling theory,and so it has wide value in applications.Since it is a key step to reconstruct the original signal by solving the sparse optimization models,the models and algorithms for sparse optimization have been researched intensively.This paper studies a class of box constrained optimization model.It is a typical sparse optimization problem,which is widely used in the fields of image reconstruction,signal processing and variable selection.The following problems exist in the process of solving such problems:(1)the optimality condition of the target model cannot be obtained directly by using the existing method since the objective function is nonconvex,nonsmooth and even non-Lipschitz.(2)For such an NP-hard problems,the introduction of constraints increases the difficulties for solving problems.(3)Since the problems in real applications are most very large in scale,the need of the efficiency of the algorithms is very high.In order to solve the above two problems,in this paper,by exploitation the characteristics of the model itself,we usethe subspace method to establish the first and second order optimality conditions for the target model.Then we obtain the lower bound of the absolute values of the solution vectors,which can be used to improve the sparsity of the numeric solutions.By constructing a uniformly smooth approximation to the object function,we propose the smoothing projected gradient algorithm to solve the target model,and prove the convergence of the algorithm.By large numbers of numeric experiments,the efficiency of the algorithm in practical application are validated.