## Abstract

A multivariate polynomial p(x) = p(x _{1},⋯, x _{n}) is sos-convex if its Hessian H(x) can be factored as H(x) = M ^{T} (x) M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it is natural to study whether sos-convexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex.

Original language | English (US) |
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Pages (from-to) | 275-292 |

Number of pages | 18 |

Journal | Mathematical Programming |

Volume | 135 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 2012 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

## Keywords

- Convexity
- Semidefinite programming
- Sos-convexity
- Sum of squares