The Truncated Moment Problem for Unital Commutative R-Algebras

Curto, Raul E.; Ghasemi, Mehdi; Infusino, Maria; Kuhlmann, Salma

doi:10.48550/arxiv.2009.05115

Cited by 1 publication

(1 citation statement)

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“…We particularly rely on a generalized Riesz-Haviland Theorem proven by Marshall almost two decades ago [42]. More recent work along these lines includes [26,33] and especially [11] -which may have computational implications for our hierarchy of lower bounds. Moment problems involving univariate power functions have been studied in [34,35] for purposes of estimating probability density functions.…”

Section: Related Workmentioning

confidence: 99%

Algebraic perspectives on signomial optimization

Dressler¹,

Murray²

2021

Preprint

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Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of "degree" that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity and Archimedeanity of the feasible set. Through worked examples we illustrate the practicality of this hierarchy in areas such as chemical reaction network theory and chemical engineering. These examples include comparisons to direct global solvers (e.g., BARON and ANTIGONE) and the Lasserre hierarchy (where appropriate). The completeness of our hierarchy of upper bounds follows from a generic construction whereby a Positivstellensatz for signomial nonnegativity over a compact set provides for arbitrarily strong outer approximations of the corresponding cone of nonnegative signomials. While working toward that result, we prove basic facts on the existence and uniqueness of solutions to signomial moment problems.

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