Optimality Conditions for Cardinality-Constrained Programs and a SCA Method

TL;DR

This paper investigates optimality conditions for cardinality-constrained optimization problems with nonnegative variables, introduces a reformulation, and proposes a successive convex approximation method with convergence guarantees, demonstrating promising numerical results.

Contribution

It provides new properties of optimal solutions, a reformulation approach, and a convergent SCA method for cardinality-constrained problems with practical applications.

Findings

The SCA method converges to a KKT point of the reformulated problem.

Under certain conditions, KKT points are local optimizers of the original problem.

Numerical results show promising performance in portfolio selection.

Abstract

We study a cardinality-constrained optimization problem with nonnegative variables in this paper. This problem is often encountered in practice. Firstly we study some properties on the optimal solutions of this optimization problem under some conditions. An equivalent reformulation of the problem under consideration is proposed. Based on the reformulation, we present a successive convex approximation method for the cardinality constrained optimization problem. We prove that the method converges to a KKT point of the reformulation problem. Under some conditions, the KKT points of the reformulation problem are local optimizers of the original problem. Our numerical results on a limited diversified mean-variance portfolio selection problem demonstrate some promising results.