Lower Bounds for Polynomials Using Geometric Programming

We make use of a result of Hurwitz and Reznick [8] [19], and a consequence of this result due to Fidalgo and Kovacec [5], to determine a new sufficient condition for a polynomial f ∈ R[X1, . . . , Xn] of even degree to be a sum of squares. This result generalizes a result of Lasserre in [10] and a result of Fidalgo and Kovacec in [5], and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound fgp for f , and we explain how fgp can be computed using geometric programming. The lower bound fgp is generally not as good as the lower bound fsos introduced by Lasserre [11] and Parrilo and Sturmfels [15], which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of fgp is much faster. The computation is simplest when the highest degree term of f has the form ∑n i=1 aiX 2d i , ai > 0, i = 1, . . . , n. The lower bounds for f established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points.