Real Algebraic and Analytic Geometry |

Lower bounds for polynomials using geometric programming.

e-mail: ,

Submission: 2011, June 25.

*Abstract:
We make use of a result of Hurwitz and Reznick, and a consequence of this result
due to Fidalgo and Kovacec, to determine a new sufficient condition for a
polynomial f in R[X_1,...,X_n] of even degree to be a sum of squares. This
result generalizes a result of Lasserre and a result of Fidalgo and Kovacec,
and it also generalizes recent improvements of these results due to the authors.
We apply this result to obtain a new lower bound f_{gp} for f, and we explain
how f_{gp} can be computed using geometric programming. The lower bound f_{gp}
is generally not as good as the lower bound f_{sos} introduced by Lasserre and
Parrilo and Sturmfels, which is computed using semidefinite programming, but a
run time comparison shows that, in practice, the computation of f_{gp} is much
faster. The computation is simplest when the highest degree term of f has the
form a_1X_1^{2d}+...+a_nX_n^{2d}, a_i>0, i=1,...,n. The lower bounds for f
established earlier by the authors are obtained by evaluating the objective
function of the geometric program at the appropriate feasible points.*

Mathematics Subject Classification (2000): 12D15, 14P99, 90C25.

Keywords and Phrases: Positive polynomials, sums of squares, optimization, geometric programming.

**Full text**, 14p.:
dvi 75k,
pdf 362k.

Server Home Page