Sublinear Circuits for Polyhedral Sets

Naumann, Helen; Theobald, Thorsten

doi:10.1007/s10013-021-00528-1

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…The combinatorial structure of the sublinear circuits is also rather open. For some results concerning polyhedral sets X see [28]. That work gives some necessary and sufficient conditions for an element ν to be a (reduced) X-circuit and discusses distinguished classes, such as X = R n + and X = [−1, 1] n .…”

Section: Further Developmentsmentioning

confidence: 99%

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Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Theobald

2023

Springer Optimization and Its Applications

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Relative entropy programs belong to the class of convex optimization problems. Within techniques based on the arithmetic-geometric mean inequality, they facilitate to compute nonnegativity certificates of polynomials and of signomials. While the initial focus was mostly on unconstrained certificates and unconstrained optimization, recently, Murray, Chandrasekaran and Wierman developed conditional techniques, which provide a natural extension to the case of convex constrained sets. This expository article gives an introduction into these concepts and explains the geometry of the resulting conditional SAGE cone. To this end, we deal with the sublinear circuits of a finite point set in R n , which generalize the simplicial circuits of the affine-linear matroid induced by a finite point set to a constrained setting.

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Section: Further Developmentsmentioning

confidence: 99%

“…Sublinear circuits of polyhedral sets have specifically been studied in [28]. Meanwhile, the conditional SAGE approaches has also been extended towards hierarchies and Positivstellensätze for conditional SAGE [35] and to additional non-convex constraints [10].…”

Section: Introductionmentioning

confidence: 99%

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Theobald

2023

Springer Optimization and Its Applications

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Sublinear circuits and the constrained signomial nonnegativity problem

2022

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Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset X of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets X. We introduce the X-circuits of a finite subset $${\mathcal {A}}\subset {\mathbb {R}}^n$$ A ⊂ R n , which generalize the simplicial circuits of the affine-linear matroid induced by $${\mathcal {A}}$$ A to a constrained setting. The X-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral X, in which case the set of X-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of X-circuits transparently reveals when an X-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler X-nonnegative signomials. We develop a duality theory for X-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when X is polyhedral. In conjunction with a notion of reduced X-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.

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Algebraic Perspectives on Signomial Optimization

Dressler

Murray

2022

SIAM J. Appl. Algebra Geometry

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Sublinear Circuits for Polyhedral Sets

Cited by 3 publications

References 21 publications

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Sublinear circuits and the constrained signomial nonnegativity problem

Algebraic Perspectives on Signomial Optimization

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