Symmetry Reduction in AM/GM-Based Optimization

Moustrou, Philippe; Naumann, Helen; Riener, Cordian; Theobald, Thorsten; Verdure, Hugues

doi:10.1137/21m1405691

Cited by 7 publications

(3 citation statements)

References 44 publications

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“…Let us mention some further research directions on the SAGE cone and the conditional SAGE cone. Symmetry reduction for AM/GM-based optimization has been studied and also computationally evaluated by Moustrou, Naumann, Riener et al [21]. Recently, extensions of the conditional SAGE approach towards hierarchies and Positivstellensätze [35] and to additional non-convex constraints [10] have been given.…”

Section: Further Developmentsmentioning

confidence: 99%

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%

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Theobald

2023

Springer Optimization and Its Applications

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“…, x n ] over a field K by permuting variables. In particular, it has been observed in different computational tasks that the understanding of this action can lead to substantial algorithmic improvements (see for example [14,29,28,18,6,23,5,19]). These improvements mostly build on the fact that in this situation, both representation and invariant theory are classically understood, and are closely related to the combinatorics of partitions and Young Tableaux.…”

Section: Introductionmentioning

confidence: 99%

The poset of Specht ideals for hyperoctahedral groups

Debus,

Moustrou,

Riener

et al. 2024

Algebraic Combinatorics

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Specht polynomials classically realize the irreducible representations of the symmetric group. The ideals defined by these polynomials provide a strong connection with the combinatorics of Young tableaux and have been intensively studied by several authors. We initiate similar investigations for the ideals defined by the Specht polynomials associated to the hyperoctahedral group Bn. We introduce a bidominance order on bipartitions which describes the poset of inclusions of these ideals and study algebraic consequences on general Bn-invariant ideals and varieties, which can lead to computational simplifications.

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“…Symmetries are common in both natural and engineering systems [18], [19], [20], [21], [22], [23]. The analysis of symmetries has been exploited in various disciplines such as semi-definite programming [24], [25], [26], network synchronization [27], [28], arithmetic optimization methods [29], the backward computation of reachable sets for nonlinear discrete-time control systems [30], and stability of interconnected subsystems [31], thus making it a popular approach for reducing the computation complexity associated with high dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

Nazerian

Bhatta

Sorrentino

2023

IEEE Open J. Control. Syst.

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In this paper, we consider optimal control problems (OCPs) applied to large-scale linear dynamical systems with a large number of states and inputs. We attempt to reduce such problems into a set of independent OCPs of lower dimensions. Our decomposition is 'exact' in the sense that it preserves all the information about the original system and the objective function. Previous work in this area has focused on strategies that exploit symmetries of the underlying system and of the objective function. Here, instead, we implement the algebraic method of simultaneous block diagonalization of matrices (SBD), which we show provides advantages both in terms of the dimension of the subproblems that are obtained and of the computation time. We provide practical examples with networked systems that demonstrate the benefits of applying the SBD decomposition over the decomposition method based on group symmetries.

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Symmetry Reduction in AM/GM-Based Optimization

Cited by 7 publications

References 44 publications

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

Relative Entropy Methods in Constrained Polynomial and Signomial Optimization

The poset of Specht ideals for hyperoctahedral groups

Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

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