Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Gowda, M. Seetharama; Jeong, Ju Young

doi:10.1137/16m1071006

Cited by 16 publications

(21 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, Items (ii)− (iv) in Corollary 1 improve known operator commutativity relations ([13], Theorem 2 and Proposition 8) for linear h and variational inequalities. We also note that this Corollary is similar to Theorem 1.3 in [6], which is applicable to simple Euclidean Jordan algebras.…”

Section: Proofssupporting

confidence: 63%

See 1 more Smart Citation

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

Gowda

2021

Optim Lett

Self Cite

View full text Add to dashboard Cite

The commutation principle of Ramírez, Seeger, and Sossa [13] proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function Φ is minimized/maximized over a spectral set E, any local optimizer a at which h is Fréchet differentiable operator commutes with the derivative h ′ (a). In this paper, assuming the existence of a subgradient in place the derivative (of h), we establish 'strong operator commutativity' relations: If a solves the problem max E (h+Φ), then a strongly operator commutes with every element in the subdifferential of h at a; If E and h are convex and a solves the problem min E h, then a strongly operator commutes with the negative of some element in the subdifferential of h at a. These results improve known (operator) commutativity relations for linear h and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.

show abstract

Section: Proofssupporting

confidence: 63%

“…In [6], Gowda and Jeong extended the above result by assuming that E and Φ are invariant under the automorphisms of V and stated an analogous result in the setting of normal decomposition systems. Subsequently, certain modifications (such as replacing the sum by other combinations) and applications were given by Niezgoda [12].…”

Section: Introductionmentioning

confidence: 88%

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

Gowda

2021

Optim Lett

Self Cite

View full text Add to dashboard Cite

show abstract

“…Based on this, they present a commutation principle for a variational inequality problem and consider the problem of describing the distance to a spectral set. See [12] for a slight weakening of the conditions and a similar result proved in the setting of normal decomposition systems. See also [30] for certain elaborations and applications.…”

Section: Introductionmentioning

confidence: 69%

“…Then, the variational inequality problem VI(G, E) [4] is to find an a ∈ E such that We now state a result that is similar to (actually generalizes) Theorem 1.3 in [12].…”

Section: Optimizing a Combination Of A Linear Function And A Spectral...mentioning

confidence: 99%

Optimizing certain combinations of spectral and linear$/$distance functions over spectral sets

Gowda

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In the settings of Euclidean Jordan algebras, normal decomposition systems (or Eaton triples), and structures induced by complete isometric hyperbolic polynomials, we consider the problem of optimizing a certain combination (such as the sum) of spectral and linear/distance functions over a spectral set. To present a unified theory, we introduce a new system called Fan-Theobald-von Neumann system which is a triple (V, W, λ), where V and W are real inner product spaces and λ : V → W is a norm preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. In this general setting, we show that optimizing a certain combination of spectral and linear/distance functions over a set of the form E = λ −1 (Q) in V, where Q is a subset of W, is equivalent to optimizing a corresponding combination over the set λ(E) and relate the attainment of the optimal value to a commutativity concept. We also study related results for convex functions in place of linear/distance functions. Particular instances include the classical results of Fan and Theobald, von Neumann, results of Tam, Lewis, and Bauschke et al., and recent results of Ramírez et al. As an application, we present a commutation principle for variational inequality problems over such a system.

show abstract

“…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.

View full text Add to dashboard Cite

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.

show abstract

Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Cited by 16 publications

References 15 publications

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

Optimizing certain combinations of spectral and linear$/$distance functions over spectral sets

When a maximal angle among cones is nonobtuse

Contact Info

Product

Resources

About