Strong Formulations for Quadratic Optimization with M-matrices and Indicator Variables

TL;DR

This paper introduces polynomial-time solvable formulations for quadratic optimization problems involving M-matrices and indicator variables, using submodular minimization and convex-hull descriptions to enhance solution strength.

Contribution

It presents a novel polynomial-time approach for quadratic problems with M-matrices by linking them to submodular minimization and developing strong convex-hull and conic inequalities.

Findings

Proven polynomial-time solvability under mild conditions.

Decomposition of quadratic functions into simpler components.

Preliminary experiments show improved relaxation bounds.

Abstract

We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.