A new polynomially solvable class of quadratic optimization problems with box constraints
Milan Hladík, Michal Černý, Miroslav Rada
TL;DR
The paper proves that the quadratic maximization problem over a box with a fixed-rank matrix $Q$ can be solved in polynomial time, even for arbitrary $Q$ and linear term $q^T x$, by reducing the problem to enumeration of faces of a zonotope in dimension $2r$, where $r=\mathrm{rank}(Q)=O(1)$. The method hides the linear term, performs a dimension reduction via a rank factorization $Q=U^T V$, and reformulates the problem as a maximization of a quadratic form over a zonotope, leveraging the duality between zonotopes and hyperplane arrangements to enumerate faces efficiently. For each face, a small linear program yields a stationary point, and the maximum over all faces is the solution; the approach scales as $O(n^{2r-1})$ face enumerations and polynomial-time LPs per face when $q=0$ (with an additional factor when $q\neq 0$). A key contribution is extending polynomial-time solvability beyond PSD $Q$ and absent linear terms, providing a larger class of tractable instances and insights into low-rank representations of quadratic forms. The results are memory-intensive due to storing the face lattice but establish a clear, constructive route to polynomial solvability for fixed-rank cases with potential impacts on related binary/psd settings.
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.