A new polynomially solvable class of quadratic optimization problems with box constraints

TL;DR

This paper introduces a new class of quadratic optimization problems with box constraints that can be solved in polynomial time when the rank of the matrix is fixed, expanding the scope beyond previous positive semidefinite cases.

Contribution

The paper generalizes previous polynomial solvability results to include arbitrary matrices with fixed rank and linear terms, broadening the class of efficiently solvable quadratic problems.

Findings

Polynomial-time solution for fixed-rank quadratic problems with box constraints.

Reduction to enumeration of zonotope faces in low dimension.

Generalization beyond positive semidefinite matrices.

Abstract

We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary $Q$ and $q$. The idea is based on a reduction of the problem to enumeration of faces of a certain zonotope in dimension $O(r)$. This paper generalizes previous results where $Q$ had been assumed to be positive semidefinite and no linear term was allowed in the objective function. Positive definiteness was a strong restriction and it is now relaxed. Generally, the problem is NP-hard; this paper describes a new polynomially solvable class of instances, larger than those known previously.