Newton Polytopes and Relative Entropy Optimization

Newton polytopes play a prominent role in the study of sparse polynomial systems, where they help formalize the idea that the root structure underlying sparse polynomials of possibly high degree ought to still be "simple." In this paper we consider sparse polynomial optimization problems, and we seek a deeper understanding of the role played by Newton polytopes in this context. Our investigation proceeds by reparametrizing polynomials as signomials -- which are linear combinations of exponentials of linear functions in the decision variable -- and studying the resulting signomial optimization problems. Signomial programs represent an interesting (and generally intractable) class of problems in their own right. We build on recent efforts that provide tractable relative entropy convex relaxations to obtain bounds on signomial programs. We describe several new structural results regarding these relaxations as well as a range of conditions under which they solve signomial programs exactly. The facial structure of the associated Newton polytopes plays a prominent role in our analysis. Our results have consequences in two directions, thus highlighting the utility of the signomial perspective. In one direction, signomials have no notion of "degree"; therefore, techniques developed for signomial programs depend only on the particular terms that appear in a signomial. When specialized to the context of polynomials, we obtain analysis and computational tools that only depend on the particular monomials that constitute a sparse polynomial. In the other direction, signomials represent a natural generalization of polynomials for which Newton polytopes continue to yield valuable insights. In particular, a number of invariance properties of Newton polytopes in the context of optimization are only revealed by adopting the viewpoint of signomials.

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