Sublinear circuits for polyhedral sets

Naumann, Helen; Theobald, Thorsten

doi:10.48550/arxiv.2103.09102

Cited by 2 publications

(2 citation statements)

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“…Conditional SAGE is a relatively new concept. Progress has been made in understanding this technique through a convex-combinatorial structural analysis of "X-SAGE cones" [51,53], group-theoretic dimension reduction techniques [44], and a Positivstellensatz for signomial nonnegativity over compact convex sets [66]. These methods also have demonstrated applications in engineering [64].…”

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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Section: Related Workmentioning

confidence: 99%

Algebraic perspectives on signomial optimization

Dressler¹,

Murray²

2021

Preprint

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“…Chapter 4 is based on parts of [Mou+21] and also partially on [Dre+20], the former is joint work with Philippe Moustrou, Cordian Riener, Thorsten Theobald, and Hugues Verdure, the latter is joint work with Mareike Dressler, Janin Heuer, and Timo de Wolff. Chapter 5 is based on joint work with Thorsten Theobald and contained in [NT21b], and Chapter 6 is based on the two works [NT21a] and on selected parts of [MNT20], where the former is joint work with Thorsten Theobald, and the latter is joint work with Riley Murray and Thorsten Theobald. Chapter 7 is my own work, except for the results on symmetries for the X-SAGE-cone, which come from [Mou+21].…”

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confidence: 99%

AM/GM-based optimization : gheometry and generalizations

Naumann¹

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The problem of unconstrained or constrained optimization occurs in many branches of mathematics and various fields of application. It is, however, an NP-hard problem in general. In this thesis, we examine an approximation approach based on the class of SAGE exponentials, which are nonnegative exponential sums. We examine this SAGE-cone, its geometry, and generalizations. The thesis consists of three main parts: 1. In the first part, we focus purely on the cone of sums of globally nonnegative exponential sums with at most one negative term, the SAGE-cone. We ex- amine the duality theory, extreme rays of the cone, and provide two efficient optimization approaches over the SAGE-cone and its dual. 2. In the second part, we introduce and study the so-called S-cone, which pro- vides a uniform framework for SAGE exponentials and SONC polynomials. In particular, we focus on second-order representations of the S-cone and its dual using extremality results from the first part. 3. In the third and last part of this thesis, we turn towards examining the con- ditional SAGE-cone. We develop a notion of sublinear circuits leading to new duality results and a partial characterization of extremality. In the case of poly- hedral constraint sets, this examination is simplified and allows us to classify sublinear circuits and extremality for some cases completely. For constraint sets with certain conditions such as sets with symmetries, conic, or polyhedral sets, various optimization and representation results from the unconstrained setting can be applied to the constrained case.

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Sublinear circuits for polyhedral sets

Cited by 2 publications

References 20 publications

Algebraic perspectives on signomial optimization

Algebraic perspectives on signomial optimization

AM/GM-based optimization : gheometry and generalizations

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