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.
It has been shown in REF that testing if a polynomial of degree four is convex is NP-hard in the strong sense.
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems
Mathematical Programming, Vol. 137, Issue 1-2, pp 453-476, 2013
Mathematics Computer Science