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

Part of: Computational aspects of field theory and polynomials Computational aspects and applications Topological rings and modules

Published online by Cambridge University Press:  20 November 2018

M. Marshall
M. Marshall*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, S7N 5E6, marshall@snoopy.usask.ca
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Abstract

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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 classification

Primary: 13J30: Real algebra
Secondary: 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
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 1 , 01 February 2009 , pp. 205 - 221
Copyright
Copyright © Canadian Mathematical Society 2009

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