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buWe present two connections between the two topics in the title. The first is the use of semidefinite programming for deciding positivity and global minimization of polynomials, as developed in Pablo Parrilo's dissertation (Caltech 2000, http://www.cds.caltech.edu/~pablo/). The second concerns the moment map which identifies the real positive part of a toric variety with the underlying convex polytope. Inverting the moment map is a fundamental problem with applications ranging from statistics to symplectic geometry. We show that this problem is equivalent to geometric programming and hence dual to entropy minimization subject to linear constraints.