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$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$.The paper deals with the general problem of computing a lower bound for $f$ onthe subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$,$i=1,...,m$. The paper shows that there is an algorithm for computing such alower bound, based on geometric programming, which applies in a large number ofcases. The algorithm extends and generalizes earlier algorithms of Ghasemi andMarshall, dealing with the case $m=0$, and of Ghasemi, Lasserre and Marshall,dealing with the case $m=1$ and $g_1= M-(x_1^d+\cdots+x_n^d)$. Here, $d$ isrequired to be an even integer $d \ge \max\{2,\deg(f)\}$. The algorithm isimplemented in a SAGE program developed by the first author. The bound obtainedis typically not as good as the bound obtained using semidefinite programming,but it has the advantage that it is computable rapidly, even in cases where thebound obtained by semidefinite programming is not computable.

Lower Bounds for a Polynomial on a basic closed semialgebraic set using geometric programming