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

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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.

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Authors and Affiliations

  1. Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA

    Amir Ali Ahmadi, Alex Olshevsky, Pablo A. Parrilo & John N. Tsitsiklis

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  1. Amir Ali Ahmadi
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  2. Alex Olshevsky
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  3. Pablo A. Parrilo
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  4. John N. Tsitsiklis
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Corresponding author

Correspondence to Pablo A. Parrilo.

Additional information

This research was partially supported by the NSF Focused Research Group Grant on Semidefinite Optimization and Convex Algebraic Geometry DMS-0757207 and by the NSF grant ECCS-0701623.

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Ahmadi, A.A., Olshevsky, A., Parrilo, P.A. et al. NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Program. 137, 453–476 (2013). https://doi.org/10.1007/s10107-011-0499-2

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  • DOI: https://doi.org/10.1007/s10107-011-0499-2

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