On the complexity of finding a local minimizer of a quadratic function over a polytope

TL;DR

This paper proves that finding a local minimizer of a quadratic function over a polytope is NP-hard, indicating no polynomial-time algorithm can approximate it within exponential distance unless P=NP.

Contribution

It resolves a long-standing open problem by showing the NP-hardness of approximating local minima of quadratic functions over polytopes.

Findings

NP-hardness of finding a local minimizer within exponential distance

NP-hardness of deciding local minimizers for quadratic functions over polyhedra

NP-hardness of deciding local minimizers for quartic polynomials

Abstract

We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c \ge 0$) of a local minimizer of an $n$-variate quadratic function over a polytope. This result (even with $c=0$) answers a question of Pardalos and Vavasis that appeared in 1992 on a list of seven open problems in complexity theory for numerical optimization. Our proof technique also implies that the problem of deciding whether a quadratic function has a local minimizer over an (unbounded) polyhedron, and that of deciding if a quartic polynomial has a local minimizer are NP-hard.