Theoryand Algorithm Analysis Of Sparse Optimization Model With Cardinality Constraint

The sparse optimization model with cardinality constraint refers to the optimization problem with l₀ norm on the constraint.It has wide applications in signal processing,regression analysis,compressed sensing and image processing.With the deepening of the research,it has been applied to target detection,human face recognition,computer vision and other problems,which has attracted great attention and become one much research topic interesting in recent years.In this paper,the nonconvex constraint relaxation method is used to solve a class of sparse optimization models with cardinality constraint.We first construct a nonconvex continuous relaxation function of the l₀ norm by using the capped-l₁ function,and obtain the continuous relaxation model of the sparsity optimization with cardinality constraint.Under some suitable assumptions,we analyze the relationships of the global optimal solutions of the original model and the continuous relaxation model,and illustrate the relationships between the local minimizers and the stationary points of the two models.We then use SCAD function to obtain another continuous relaxation model of the sparse optimization with cardinality constraint,and analyze the relationships between the original model and the new continuous relaxation model.Under some assumptions,we give the relationships between the global optimal solutions of the two models,and also show the relationships between their local minimizers and stationary points.Finally,we introduce the algorithms for solving the two continuous relaxation models respectively,and then obtain the optimal solutions of the sparse optimization problem with cardinality constraint by solving the continuous relaxation models.