Study On The Properties Of The Optimal Value Functions For A Class Of Sparse Nonconvex Optimization Problems

The optimization problem,a significant branch of mathematics,tries to take certain actions to obtain the optimal state under certain restricted conditions.In recent years,a type of optimization problem—sparse optimization,has quickly become a research hotspot in the field of optimization.This kind of problem is widely applied in compressed sensing,computer vision,image processing,statistical modeling and other fields.As this problem owns huge practical value in real life and the research of non-convex problem is very challenging at present,it has attracted researchers'increasing attentions in recent years.This paper mainly studies the properties of the optimal value functions of the L_p norm-constrained optimization problem,and conducts researches on monoton-icity,unevenness and smoothness of the optimal value functions.In recent years,the study of the L₁ regularization problem has made prominent advancement.Inspired by the research methods of the L₁ regularization problem,this paper studies the optimality conditions of L_p norm-constrained optimization problem and uses the variational analysis tools like(horizontal)subdifferential,tangent cone,normal cone,etc.to study the properties of the optimal value functions for L_p norm-constrained optimization problem.In the end,by comparing the propert-ies of the optimal value functions between L₁ norm-constrained optimization pro-blem and the L_p norm-constrained optimization problem,it is concluded that the biggest difference between the two optimal value functions lies in the concavity and convexity and smoothness.