Optimality conditions and constraint qualifications for cardinality constrained optimization problems

TL;DR

This paper develops precise optimality conditions and constraint qualifications for cardinality constrained optimization problems by leveraging the structure of subspaces, leading to new insights and sufficient conditions for stability and penalization.

Contribution

It introduces specialized formulas for tangent and normal cones in CCOP, and establishes new optimality conditions and constraint qualifications specific to subspace-structured disjunctive constraints.

Findings

RCPLD is a sufficient condition for metric subregularity in CCOP

Exact penalization holds under the proposed constraint qualifications

New formulas for tangent and normal cones in subspace disjunctive sets

Abstract

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.