Commutation Principle for Variational Problems on Euclidean Jordan Algebras

Ramírez, Héctor; Seeger, Alberto; Sossa, David

doi:10.1137/120879397

Cited by 18 publications

(16 citation statements)

References 7 publications

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“…Extending an earlier result of Iusem and Seeger [7] for real symmetric matrices, Ramirez, Seeger, and Sossa [13] prove the following commutation principle.…”

Section: Introductionsupporting

confidence: 60%

“…Proof. The proof is similar to the one given in [13], Proposition 1.9. If a solves VI(G, Ω, F ), then…”

Section: Euclidean Jordan Algebrasmentioning

confidence: 57%

“…A number of important and interesting applications are mentioned in [13]. The proof of the above result in [13] is somewhat intricate, deep, and long. In our paper we extend the above result by assuming only the automorphism invariance of Ω and F , and at the same time provide (perhaps) a simpler and shorter proof.…”

Section: Introductionmentioning

confidence: 94%

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Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Gowda¹,

Jeong²

2017

SIAM J. Optim.

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The commutation principle of Ramirez, Seeger, and Sossa [13] proved in the setting of Euclidean Jordan algebras says that when the sum of a Fréchet differentiable function Θ(x) and a spectral function F (x) is minimized over a spectral set Ω, any local minimizer a operator commutes with the Fréchet derivative Θ ′ (a). In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.

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“…Extending an earlier result of Iusem and Seeger [7] for real symmetric matrices, Ramirez, Seeger, and Sossa [13] prove the following commutation principle.…”

Section: Introductionsupporting

confidence: 60%

“…Proof. The proof is similar to the one given in [13], Proposition 1.9. If a solves VI(G, Ω, F ), then…”

Section: Euclidean Jordan Algebrasmentioning

confidence: 57%

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Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Gowda¹,

Jeong²

2017

SIAM J. Optim.

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“…In the case of R n , spectral sets/cones/functions are related to permutation invariance, and in S n (H n ) they are precisely those that are invariant under linear transformations of the form X → U XU * , where U ∈ R n×n is an orthogonal (respectively, unitary) matrix. In the general setting of Euclidean Jordan algebras, they have been studied in several works, see [1], [9], [14], [15], [16], [20], [23], and [24].…”

Section: Introductionmentioning

confidence: 99%

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

Gowda

Jeong

2018

Linear Algebra and its Applications

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Let V be a Euclidean Jordan algebra of rank n. A set E in V is said to be a spectral set if there exists a permutation invariant set Q in R n such that E = λ −1 (Q), where λ : V → R n is the eigenvalue map that takes x ∈ V to λ(x) (the vector of eigenvalues of x written in the decreasing order). If the above Q is also a convex cone, we say that E is a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit [x] := {y : λ(y) = λ(x)} is arcwise connected, we show that if a permutation invariant set Q is connected (arcwise connected), then λ −1 (Q) is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.

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“…These sorts of problems are still an active area of research. For example, the commutation principle used by Iusem and Seeger was later shown by Ramírez et al (2013) to hold in a general Euclidean Jordan Algebra, and by Gowda and Jeong (2017) to hold in a normal decomposition system.…”

mentioning

confidence: 99%

When a maximal angle among cones is nonobtuse

Orlitzky

2020

Comp. Appl. Math.

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Principal angles between linear subspaces have been studied for their application to statistics, numerical linear algebra, and other areas. In 2005, Iusem and Seeger defined critical angles within a single convex cone as an extension of antipodality in a compact set. Then, in 2016, Seeger and Sossa extended that notion to two cones. This was motivated in part by an application to regression analysis, but also allows their cone theory to encompass linear subspaces which are themselves convex cones. One obstacle to computing the maximal critical angle between cones is that, in general, the maximum will not occur at a pair of generators of the cones. We show that in the special case where the maximal angle between the cones is nonobtuse, it does suffice to check only the generators. This special case can be checked at essentially no extra cost, and we incorporate that information into an improved algorithm to find the maximal angle. Keywords Critical angle • Maximal angle • Nash angle • Convex cone • Polyhedral cone • Principal angle • Nonconvex optimization Mathematics Subject Classification 90C26 • 49M05 • 52B55 1 Background The idea of a principal angle between linear subspaces goes back at least to Afriat (1957), who set out to construct a linear-algebraic framework that would encompass multivariate statistical analysis. A newer reference whose terminology is closer to our own is Miao and Ben-Israel (1992). The first principal angle between a pair of subspaces P and Q is defined by cos θ 1 := max { u, v | u ∈ P, v ∈ Q, and u = v = 1}, (i) and since the cosine function is decreasing on [0, π], we can think of θ 1 as being a minimal angle between the spaces P and Q. A pair (u 1 , v 1) achieving the minimal angle θ 1 is called a pair of principal vectors. Communicated by Jinyun Yuan.

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Commutation Principle for Variational Problems on Euclidean Jordan Algebras

Abstract: This paper establishes a commutation result for variational problems involving spectral sets and spectral functions. The discussion takes places in the context of a general Euclidean Jordan algebra.

Cited by 18 publications

References 7 publications

Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

When a maximal angle among cones is nonobtuse

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