Abstract:

Based on the Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones, the optimality theories of the non-negative group sparse constrained optimization problem are studied. This paper defines the Bouligand tangent cone and its normal cone and the Clarke tangent cone and its normal cone of the non-negative group sparse constraint set, and presents their equivalent characterizations. Under the assumption that the objective function is continuously differentiable, with the help of the tangent cone and the normal cone of the sparse constrained set of the non-negative group, the definitions of four types of stable points for the optimization problem are given and the relationships between these four types of stable points are discussed. Finally, the first-order and second-order optimality conditions for the optimization problem of sparse constraint of non-negative groups are established.

Key words: Non-negative group sparse constrained optimization, Optimality conditions, Tangent cone, Normal cone