Lifted Polymatroid Inequalities for Mean-Risk Optimization with Indicator Variables

TL;DR

This paper develops new lifted polymatroid inequalities to strengthen convex relaxations in mean-risk optimization problems with indicator variables, leading to faster solution times.

Contribution

It introduces three classes of strong convex valid inequalities derived from polymatroid inequalities, enhancing solution efficiency for non-convex mean-risk problems.

Findings

Inequalities significantly improve relaxation strength.

Enhanced solution times for mean-risk problems.

Effective in modeling fixed charges and cardinality constraints.

Abstract

We investigate a mixed 0-1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalities by lifting the polymatroid inequalities on the binary variables. Computational experiments demonstrate the effectiveness of the inequalities in strengthening the convex relaxations and, thereby, improving the solution times for mean-risk problems with fixed charges and cardinality constraints significantly.