A unified framework of SAGE and SONC polynomials and its duality theory

Katthän, Lukas; Naumann, Helen; Theobald, Thorsten

doi:10.1090/mcom/3607

Cited by 18 publications

(39 citation statements)

References 17 publications

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“…One step further, a reducibility concept for sublinear circuits provides a non-redundant decomposition of the conditional SAGE cone in terms of reduced circuits. This reducibility notion generalizes the reducibility notion for the unconstrained situation which was introduced in [12], see also [7]. The reduced R n -circuits are the key concept to characterize the extremal rays of the unconstrained SAGE cone, since the reduced R n -circuits induce extremal rays.…”

Section: Introductionmentioning

confidence: 91%

“…For the classical case of affine circuits supported on a finite set A, the following exact characterization in terms of the support is known. Proposition 6.1 ([12,Corollary 4.7], [7, Theorem 3.2]) A vector ν is a reduced R n -circuit if and only if A ∩ relint conv ν + = {ν − }.…”

Section: Reducibility and Extremalitymentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…It remains to show that the 3-term AGE function f cannot be decomposed in terms of 3-term AGE functions. However, since f has a zero in [−1, 1] and thus in R, it induces an extremal ray of the cone C R (A) and cannot be decomposed using only 3-term AGE functions by [12,Proposition 4.4]. Example 6.8 The reduced sublinear circuits for the cube [−1, 1] 2 .…”

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confidence: 99%

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Sublinear Circuits for Polyhedral Sets

Naumann

Theobald

2021

Vietnam J. Math.

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Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.

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Section: Introductionmentioning

confidence: 91%

Section: Reducibility and Extremalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Sublinear Circuits for Polyhedral Sets

Naumann

Theobald

2021

Vietnam J. Math.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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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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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A unified framework of SAGE and SONC polynomials and its duality theory

Cited by 18 publications

References 17 publications

Sublinear Circuits for Polyhedral Sets

Sublinear Circuits for Polyhedral Sets

Sublinear circuits and the constrained signomial nonnegativity problem

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

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