### 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.

Original language | English (US) |
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Pages (from-to) | 453-476 |

Number of pages | 24 |

Journal | Mathematical Programming |

Volume | 137 |

Issue number | 1-2 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Keywords

- 68Q25 Analysis of algorithms & problem complexity
- 90C60 Abstract computational complexity for mathematical programming problems
- Mathematics Subject Classification (2000): 90C25 Convex programming

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## Cite this

*Mathematical Programming*,

*137*(1-2), 453-476. https://doi.org/10.1007/s10107-011-0499-2