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 $f_{\mathrm{g}\mathrm{p},M}$ for a multivariate polynomial $f(x)\in \mathbb{R}[x]$ of degree $\le 2d$ in $n$ variables $x = (x_1,…,x_n)$ontheclosedball${x \in $\mathbb{R}$^n : \sum x_i^{2d} \le M}$, computable by geometric programming, for any real $M$. We compare this bound with the (global) lower bound $f_{\mathrm{g}\mathrm{p}}$ 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 $f_{\mathrm{g}\mathrm{p},M}$ improves on the bound $f_{\mathrm{g}\mathrm{p}}$ and that the computation of $f_{\mathrm{g}\mathrm{p},M}$, like that of $f_{\mathrm{g}\mathrm{p}}$, 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.