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

Author(s)
Ahmadi, Amir Ali; Olshevsky, Alexander; Parrilo, Pablo A.; Tsitsiklis, John N.
Thumbnail
DownloadParrilo_NP-hardness.pdf (219.9Kb)
OPEN_ACCESS_POLICY
Terms of use
Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/
Metadata
Show full item record
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.
Date issued
2011-11
URI
http://hdl.handle.net/1721.1/73527
Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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
ISSN
0025-5610
1436-4646
Collections