NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

TL;DR

This paper proves that deciding convexity and related properties of multivariate quartic polynomials is NP-hard, resolving an open problem since 1992, and highlights complexity differences between even and odd degree polynomials.

Contribution

It establishes NP-hardness of convexity decision problems for quartic polynomials and identifies polynomial-time decidability for certain properties of odd degree polynomials.

Findings

Deciding convexity of quartic polynomials is NP-hard.

Deciding strict, strong, quasiconvexity, and pseudoconvexity for quartic polynomials is strongly NP-hard.

Quasiconvexity and pseudoconvexity of odd degree polynomials are decidable in polynomial time.

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.