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

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 comple...

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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0499-2