A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

TL;DR

This paper introduces a graph-based decomposition approach for convex quadratic problems with indicator variables, exploiting sparsity in the quadratic matrix to enable efficient solutions and improved formulations.

Contribution

It presents a polynomial-time solution for problems where the quadratic matrix's support graph is a path and develops a novel decomposition method for general sparse cases.

Findings

Efficient polynomial-time algorithm for path-structured problems

Compact extended formulations for convex hulls of the problem

Computational results show improved performance over existing solvers

Abstract

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix $Q$ defining the quadratic term in the objective is sparse. We use a graphical representation of the support of $Q$, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This enables us to construct a compact extended formulation for the closure of the convex hull of the epigraph of the mixed-integer convex problem. Furthermore, we propose a novel decomposition method for general (sparse) $Q$, which leverages the efficient algorithm for the path case. Our computational experiments demonstrate the effectiveness of the proposed method compared to state-of-the-art mixed-integer optimization solvers.