NP-hardness of deciding convexity of quartic polynomials and related problems
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Parrilo_NP-hardness.pdf
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Author(s)
Ahmadi, Amir Ali
Olshevsky, Alexander
Parrilo, Pablo A.
Tsitsiklis, John N.
Date Issued
November 2011
Journal
Mathematical Programming
Publisher
Springer-Verlag
Citation
Ahmadi, Amir Ali et al. “NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems.” Mathematical Programming (2011).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s10107-011-0499-2
