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

  Published:2009-02-01
 Printed: Feb 2009
  • M. Marshall
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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, 12Y05, 13P99, 14P10, 90C22 show english descriptions Real algebra [See also 12D15, 14Pxx]
Computational aspects of field theory and polynomials
None of the above, but in this section
Semialgebraic sets and related spaces
Semidefinite programming
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
 

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