Optimality Theory And Algorithm For Nonconvex Regular Group Sparse Problem

Group sparse optimization is a kind of special sparse optimization.It has wide applications in variable selection,gene expression data,image restoration,neuroimaging and other fields.The group structure is an important piece of a prior knowledge about the problem.Recent years,it has attracted much attention and is a hot research topic.In practice,the use of group structure yields solutions that are easier to interpret the signals in terms of the chose structures.Moreover,the group sparse optimizations allow to significantly reduce the number of required measurements for perfect recovery in the noiseless case and be more stable in the presence of noise.Sparse optimization with nonconvex regular has been extensively theory and algo-rithm studied in the last few decades.Due to the complexity of group structure and nonconvex regular,nonconvex regular group sparse optimization is still short of study.In this paper,we study the optimality theory and algorithm for nonconvex regular group sparse problem.The details are given as follows,?1?We consider nonconvex regular group sparse problem,capped-?₁group sparse problem.For the problem,we give the specific features of lifted stationary point,d-stationary point and critical point,and analyze the relationship between the three sta-tionary points.?2?We analyze the relationship between the lifted stationary point and the local solutions of the capped-?₁group sparse problems.We prove the lower bound property of the capped-?₁group sparse problems,and we prove the original problem and relaxation problem is equal in value with the lower bound property of the problem.?3?For capped-?₁group sparse problem,we design an effective algorithm.We design the smoothing proximal gradient algorithm for group sparse program and analyze its convergence property.?4?Finally,by our algorithm,we introduce two kinds of numerical experiments:stochastic problem simulation and sparse signal recovery.