The $${\mathcal {S}}$$-cone and a primal-dual view on second-order representability

Naumann, Helen; Theobald, Thorsten

doi:10.1007/s13366-020-00512-9

Cited by 7 publications

(8 citation statements)

References 14 publications

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“…Theorem 4.4 also motivates a basis identification technique where an approximate relative entropy certificate of f ∈ C X (A) may be refined by power cone programming. Combining Theorems 3.7 and 4.4 yields a corollary that when X is a polyhedron, cones of X -SAGE signomials are (in principle) power cone representable; this generalizes results by several authors in the unconstrained case [2,26,33,41].…”

Section: Main Contributionssupporting

confidence: 77%

“…That aspect of the corollary has uses in computational optimization when applied judiciously. The second part of Corollary 4.5 generalizes results by Averkov [2] and Wang and Magron [41] for ordinary SAGE polynomials, and recent results by Naumann and Theobald for several types of ordinary SAGE-like certificates [26]. We have deliberately framed the second part of the corollary in abstract terms (semidefinite extension degree), because that aspect of the corollary seems not useful for computational optimization.…”

Section: Corollary 45 If X Is a Polyhedron Then C X (A) Is Power Cone...mentioning

confidence: 80%

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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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Section: Main Contributionssupporting

confidence: 77%

Section: Corollary 45 If X Is a Polyhedron Then C X (A) Is Power Cone...mentioning

confidence: 80%

Sublinear circuits and the constrained signomial nonnegativity problem

2022

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“…For further recent work on the techniques for certifying non-negativity of signomials and polynomials based on the SAGE cone and its variants, see [1,5,18,23,24].…”

Section: Contributionsmentioning

confidence: 99%

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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“…For further recent work on the techniques for certifying non-negativity of signomials and polynomials based on the SAGE cone and its variants, see [1,5,18,23,24].…”

Section: Introductionmentioning

confidence: 99%

Sublinear circuits for polyhedral sets

Naumann¹,

Theobald²

2021

Preprint

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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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The $${\mathcal {S}}$$-cone and a primal-dual view on second-order representability

Cited by 7 publications

References 14 publications

Sublinear circuits and the constrained signomial nonnegativity problem

Sublinear circuits and the constrained signomial nonnegativity problem

Sublinear Circuits for Polyhedral Sets

Sublinear circuits for polyhedral sets

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