Lower bounds on the global minimum of a polynomial

We extend the method of Ghasemi and Marshall (SIAM J. Optim. 22(2):460–473, <CitationRef CitationID="CR1">2012</CitationRef>), to obtain a lower bound f <Subscript>gp,M </Subscript> for a multivariate polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(\mathbf{x}) \in\mathbb{R}[\mathbf {x}]$</EquationSource> </InlineEquation> of degree ≤2d in n variables <Emphasis Type="Bold">x=(x <Subscript>1</Subscript>,…,x <Subscript> n </Subscript>) on the closed ball <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{ \mathbf{x} \in\mathbb{R}^{n} : \sum x_{i}^{2d} \le M\}$</EquationSource> </InlineEquation>, computable by geometric programming, for any real M. We compare this bound with the (global) lower bound f <Subscript>gp</Subscript> 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, <CitationRef CitationID="CR2">2001</CitationRef>). Our computations show that the bound f <Subscript>gp,M </Subscript> improves on the bound f <Subscript>gp</Subscript> and that the computation of f <Subscript>gp,M </Subscript>, like that of f <Subscript>gp</Subscript>, 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. Copyright Springer Science+Business Media New York 2014