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

Abstract

We consider the quadratic optimization problem \(\max _{x \in C}\ x^{\mathrm {T}}Q x + q^{\mathrm {T}}x\), 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.