Lower bounds on the global minimum of a polynomial

We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound fgp,M for a multivariate polynomial f(x) ∈ R[x] of degree ≤ 2d in n variables x = (x1, . . . , xn) on the closed ball {x ∈ Rn : ∑ x2d i ≤ M}, computable by geometric programming, for any real M . We compare this bound with the (global) lower bound fgp obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our computations show that the bound fgp,M improves on the bound fgp and that the computation of fgp,M , like that of fgp, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down.