## Optimization of polynomial functions

Venue: | Can. Math. Bull |

Citations: | 17 - 4 self |

### BibTeX

@ARTICLE{Marshall_optimizationof,

author = {M. Marshall},

title = {Optimization of polynomial functions},

journal = {Can. Math. Bull},

year = {},

pages = {575--587}

}

### OpenURL

### Abstract

Recently progress has been made in the development of algorithms for optimizing polynomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing methods. The idea traces back to work of Shor [16][17] and is further developed by Parrilo [10] and by Parrilo and Sturmfels [11] and by Lasserre [7][8]. In [7][8] Lasserre describes an extension of the method to minimizing a polynomial on an arbitrary basic closed semialgebraic set and uses a result due to Putinar [13] to prove that the method produces the exact minimum in the compact case. In the general case it produces a lower bound for the minimum. The ideas involved come from three branches of mathematics: algebraic geometry (positive polynomials), functional analysis (the moment problem) and optimization. This makes the area an attractive one not only from the computational but also from the theoretical point of view. In Section 1 we define three lower bounds for a polynomial and point out relationships