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

• Published in: Mathematical programming Vol. 137; no. 1-2; pp. 453 - 476
• Main Authors: Ahmadi, Amir Ali, Olshevsky, Alex, Parrilo, Pablo A., Tsitsiklis, John N.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2013; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Complexity; Convex analysis; Convexity; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Multivariate analysis; Numerical Analysis; Optimization; Polynomials; Studies; System theory; Theoretical; 90C25 Convex programming; 90C60 Abstract computational complexity for mathematical programming problems; 68Q25 Analysis of algorithms & problem complexity
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0499-2

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.