location:  Publications → journals → CJM
Abstract view

# Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization

Published:2009-02-01
Printed: Feb 2009
• M. Marshall
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability $1$. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.
 MSC Classifications: 13J30 - Real algebra [See also 12D15, 14Pxx] 12Y05 - Computational aspects of field theory and polynomials 13P99 - None of the above, but in this section 14P10 - Semialgebraic sets and related spaces 90C22 - Semidefinite programming