Global optimization via the dual SONC cone and linear programming

Dressler, Mareike; Heuer, Janin; Naumann, Helen; Wolff, Timo de

doi:10.1145/3373207.3404043

Cited by 8 publications

(12 citation statements)

References 18 publications

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“…In this article we extend and generalize the observations made in [DHNdW20] by defining the DSONC cone as the conic closure of the set of circuit functions with coefficient vector in the dual SONC cone; see Definition 3.1 for a formal definition. On the one hand, the DSONC cone D A is a subset of the primal SONC cone S A (Corollary 3.3), which contains more functions than the certificates introduced in [DHNdW20]. On the other hand, as for the dual SONC cone, one can check membership in the DSONC cone by linear programming, which makes it a very interesting object with high potential for applications.…”

Section: Introductionmentioning

confidence: 87%

“…In this subsection we study the dual of the SONC cone S A + ,A − . This cone was previously studied in, e.g., [KNT21,DHNdW20]. The following theorem gives three different representations of the dual SONC cone using the natural duality pairing…”

Section: The Dual Sonc Conementioning

confidence: 99%

“…Recently, the dual cone ŠA corresponding to these objects has been studied in, e.g., [DHNdW20,DNT21,KNT21]. This cone is particularly interesting since first, circuit functions whose coefficients are contained in the dual SONC cone are also contained in the primal SONC cone; see [DHNdW20, Proposition 3.6], and second, optimizing over the dual SONC cone can be done via linear programming; see [DHNdW20, Proposition 4.1].…”

Section: Introductionmentioning

confidence: 99%

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The Duality of SONC: Advances in Circuit-based Certificates

Heuer¹,

Wolff²

2022

Preprint

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The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years. In this article, we construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone can be seen as an extension of the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC cone cannot have real zeros, which yields that DSONC cone does not intersect the boundary of the SONC cone. Furthermore, we discuss the intersection of the DSONC cone with the SOS and SDSOS cones. Finally, we show that circuit functions in the boundary of the DSONC cone are determined by points of equilibria, which hence are the analogues to singular points in the primal SONC cone, and relate the DSONC cone to tropical geometry.

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Section: Introductionmentioning

confidence: 87%

Section: The Dual Sonc Conementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Duality of SONC: Advances in Circuit-based Certificates

Heuer¹,

Wolff²

2022

Preprint

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“…the signed SAGE cone, which allows negative coefficients only in a certain subset B of the support A ∪ B. This a common notation in optimization viewpoints [10,11,16,20,21].…”

Section: Preliminariesmentioning

confidence: 99%

Symmetry reduction in AM/GM-based optimization

Moustrou,

Naumann,

Riener

et al. 2021

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The arithmetic mean/geometric mean-inequality (AM/GM-inequality) facilitates classes of non-negativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and develop techniques to reduce the size of the resulting relative entropy programs. We study in more detail the complexity gain in the case of the symmetric group. In this setup, we can show in particular certain stabilization results. We exhibit several sequences of examples in growing dimensions where the size of the problem stabilizes. Finally, we provide some numerical results, emphasizing the computational speed-up. m i=1 (exp( α i , x )) λ i . Clearly, sums of such exponential sums are non-negative as well. Note that exponential sums can be seen as a generalization

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“…This condition is satisfied by SONC, sums of arithmetic-mean/geometric-mean (SAG) [5,20], diagonally dominant SOS (DSOS), and scaled diagonally dominant SOS (SDSOS) polynomials. Thus, non-SOS Schmüdgen-type Positivstellensätze exist for all these classes of polynomials, and some of their numerical characteristics have been investigated [9,11,22,24].…”

Section: Introductionmentioning

confidence: 99%

Sparse non-SOS Putinar-type Positivstellensätze

Roebers¹,

Vera²,

Zuluaga³

2021

Preprint

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Recently, non-SOS Positivstellensätze for polynomials on compact semialgebraic sets, following the general form of Schmüdgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of polynomials. An open question in the literature is how to obtain similar results following the general form of Putinar's Positivstellensatz. Extrapolating the algebraic geometry tools used to obtain this type of result in the SOS case fails to answer this question, because algebraic geometry methods strongly use hallmark properties of the class of SOS polynomials, such as closure under multiplication and closure under composition with other polynomials. In this article, using a new approach, we show the existence of Putinar-type Positivstellensätze that are constructed using non-SOS classes of non-negative polynomials, such as SONC, SDSOS and DSOS polynomials. Even not necessarily non-negative classes of polynomials such as sums of arithmetic-mean/geometric-mean polynomials could be used. Furthermore, we show that these certificates can be written with inherent sparsity characteristics. Such characteristics can be further exploited when the sparsity structure of both the polynomial whose non-negativity is being certified and the polynomials defining the semialgebraic set of interest are known. In contrast with related literature focused on exploiting sparsity in SOS Positivstellensätze, these latter results show how to exploit sparsity in a more general setting in which non-SOS polynomials are used to construct the Positivstellensätze.

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Global optimization via the dual SONC cone and linear programming

Cited by 8 publications

References 18 publications

The Duality of SONC: Advances in Circuit-based Certificates

The Duality of SONC: Advances in Circuit-based Certificates

Symmetry reduction in AM/GM-based optimization

Sparse non-SOS Putinar-type Positivstellensätze

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