Research On Group Sparse Constrained Optimization Problems

Group sparse optimization is a special type of sparse optimization with a group structure.It has a wide range of applications in the fields of variable selection,gene expression,image restoration,and neuroimaging and so on,so it has become a hot research topic in optimization and related fields in recent years.In many practical applications,the group sparse model is used for signal recovery for information with a group sparse structure,which can improve the accuracy of recovery in the absence of noise,and improve the stability of recovery in the presence of noise.At present,the research for the theory and algorithm of sparse constrained optimization problems has been relatively rich and mature.However,the group sparse constrained optimization problem is non-convex,non-smooth,and non-Lipschitz,which is an NP-hard problem.In addition,due to the complexity of the group sparse structure,the research for the group sparse constrained optimization problems is still very lacking,so this paper studies the optimality theory and algorithm of the group sparse constrained optimization(GSCO)problems.The details are given as follows.(1)The equivalent expressions of Bouligand tangent cone,Clarke tangent cone and normal cone of the group sparse set S are given.(2)By using tangent cones and normal cones,four types of stationary points for GSCO problems are given:NB-stationary point,NC-stationary point,TB-stationary point and TC-stationary point,which are used to characterize first-order optimality conditions for GSCO problems.Furthermore,this paper discusses the relationship among the local optimal solutions,NB-stationary point,NC-stationary point,TB-stationary point and TC-stationary point of the group sparse constrained optimization problems.(3)For the group sparse constrained optimization problems,we give its second-order necessary and sufficient optimality conditions,and prove that the NC-stationary point is a strictly local optimal solution under certain conditions,and satisfies the second-order growth condition.(4)Effective algorithms are designed for group sparse constrained optimization problems.An iterative hard threshold algorithm for group sparse constrained optimization is proposed,the sequence of points generated by the algorithm converges to the ?stationary point,and the ?-stationary point must be NB-stationary point.(5)Finally,we conduct two types of experiments of the above algorithm:randomly generated numerical experiment and sparse signal recovery.