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.

Keywords:

• Nonlinear programming
• sparsity constrained optimization
• stationarity
• optimality conditions
• constraint qualification

AMS Subject Classifications:

• 90C26
• 90C46
• 90C90

Acknowledgments

The second-named author was partially supported by a grant from IPM [No. 99900416]. The authors would like to thank the anonymous referees for their careful reading and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second-named author was partially supported by a grant from the Institute for Research in Fundamental Sciences (IPM) [grant number 99900416].