NP-hardness of deciding convexity of quartic polynomials and related problems

Abstract

We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.

Task Scheduling Algorithm based on Greedy Strategy in Cloud Computing

“…In cloud computing, the problem of task scheduling has become a hot issue. At the same time, it is also a NP hard problem [3,4].…”

On Cones of Nonnegative Quartic Forms

“…There are some substantial recent progresses along this direction. As we mentioned earlier, e.g., the question of Shor [28] regarding the complexity of deciding the convexity of a quartic polynomial was nicely settled by Ahmadi et al [2]. It is also natural to inquire if the Hessian matrix of a convex polynomial is sos-convex.…”

“…Recall the recent breakthrough [2] mentioned in Section 1, that checking the convexity of a quartic form is strongly NP-hard. However, if we are given more information, that the quartic form to be considered is a sum of squares, will this make the membership easier?…”

“…On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al [2] proved that checking the convexity of a quartic polynomial function is strongly NP-hard in general, which settles a long-standing open question. In this paper we proceed to studying six fundamentally important convex cones of quartic functions in the space of symmetric quartic tensors, including the cone of nonnegative quartic polynomials, the sum of squared polynomials, the convex quartic polynomials, and the sum of fourth powered polynomials.…”

“…For almost two decades, the question remained open. Only until recently Ahmadi et al [2] proved that checking the convexity of a general quartic polynomial function is actually strongly NP-hard. Their result not only settled this particular open problem, but also helped to highlight a crucial difference between quartic and quadratic polynomials, which makes the study of quartic polynomials all the more compelling and interesting.…”