Sparsity constrained optimization problems via disjunctive programming

Abstract

In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results. © 2021 Informa UK Limited, trading as Taylor & Francis Group.