Newton Polytopes and Relative Entropy Optimization

Abstract

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 signomialswhich 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.

Sublinear circuits and the constrained signomial nonnegativity problem

“…The property of a set being convex under this logarithmic transformation is known by various names, including log convexity [1], geometric convexity [15,30], or multiplicative convexity [28]. This property has previously been considered in the literature on ordinary SAGE certificates (i.e., SAGE certificates for the special case X = R n ) [17,23],…”

“…The earliest results here are due to Reznick [36], with a resurgence marked by the works of Pantea, Koeppl, and Craciun [32], Iliman and de Wolff [14], and Chandrasekaran and Shah [4]. Whether considered for signomials or polynomials, such techniques have appealing forms of sparsity preservation in the proofs of nonnegativity [23,40].…”

“…The purpose of this article is to undertake the first structural analysis of the cones of X -SAGE signomials on exponents A, which we henceforth denote by C X (A). At the outset of this research, our goals were to find counterparts to the many convexcombinatorial properties known for the unconstrained case C R n (A) [12,17,23], and to understand conditional SAGE relative to techniques such as nonnegative circuit polynomials [14,32,36]. Towards this end we have introduced an analysis tool of sublinear circuits which we call the X -circuits of A.…”

Stability and Control of Biomolecular Circuits through Structure

“…Although the location in the polytope depends on k j and x, the polytope itself depends on reactant vectors α j alone. We note that newton polytopes are fruitful tools in analysis and optimization of polynomial equations, dynamical systems, and CRNs [21]- [23].…”

Sublinear Circuits for Polyhedral Sets

“…Among the class of polyhedra, polyhedral cones exhibit particularly nice properties and were in the focus of attention in earlier treatments. Note that, as a very particular case, the unconstrained setting X = R n , which is treated in [7,12,15], also falls into the class of polyhedral cones. The univariate case R + was studied in detail in [17].…”