The First-Order Necessary Conditions For Sparsity Constrained Optimization

Optimization problems is an important component of operational research.S-parsity constrained optimization is an important branch of optimization theory,playing a role in the applied mathematics,statistics and computer science.The application is applied in various aspects such as compression perception,noise pro-cessing,image processing and so on.And most of the work is focused on solving the sparsity problem of linear equations.Later,it is found that the development theory and its algorithm of sparsity optimization apply to a broader class of opti-mization problems,that is the nonlinear model studied under sparsity constraints.In this paper,we study optimization problems with the sparsity constraints in fi-nite dimensional spaces.We first give the characterization of the Mordukhovich(the limiting)normal cone of sparsity constraint,and then based on the expressions of the Mordukhovich normal cone of sparsity constraint,we present the first-order necessary conditions for sparsity constrained optimization.