Lower bounds on the global minimum of a polynomial

We extend the method of Ghasemi and Marshall (SIAM J. Optim. 22(2):460–473, 2012), to obtain a lower bound fgp,M for a multivariate polynomial $f(\mathbf{x}) \in\mathbb{R}[\mathbf {x}]$ of degree ≤2d in n variables x=(x1,…,xn) on the closed ball $\{ \mathbf{x} \in\mathbb{R}^{n} : \sum x_{i}^{2d} \le M\}$, computable by geometric programming, for any real M. We compare this bound with the (global) lower bound fgp obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre (SIAM J. Optim. 11(3):796–816, 2001). 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.