Optimality Conditions for Group Sparse Constrained Optimization Problems

Wenying Wu,Dingtao Peng

doi:10.3390/math9010084

Abstract

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

Highlights

This paper focuses on the following group sparse constrained optimization (GSCO)
We use Bouligand tangent cones, Clarke tangent cones and their corresponding normal cones to describe the N-stationary points and T-stationary points of Problem (1), based on the descriptions, we investigate the relationship among the stationary points and the relationship between stationary points and local minimizers
The first-order optimality conditions are built for group sparsity constrained optimization problems by use of Bouligand tangent cone, Clarke tangent and their corresponding normal cones, and the relationship among the local minimizers and the four types of stationary points of Problem (1) is investigated

Summary

Introduction

One of them is the basic feasibility which is a generation of the necessary optimality condition for zero gradient in unconstrained optimization Another one of them is the L-stationary point which is based on the fixed point condition and can be used to derive the iterative hard thresholding algorithm for solving sparse constrained optimization problems. We will use Boligand tangent cone, Clarke tangent cone and the corresponding normal cones of the group sparse set to describe optimality conditions for Problem (1).

Notations and Definitions

Tangent Cones and Normal Cones of the Group Sparse Set S

Concluding Remarks