Relating Homogeneous Cones and Positive Definite Cones via <i>T</i>-Algebras

Chua, Chek Beng

doi:10.1137/s1052623402406765

Cited by 42 publications

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“…We call [ω] the source homogeneous cone for corresponding to the source ω. Accordingly we call V [ω] the source subclan of V corresponding to ω. Chua's description of in [8] corresponding to our Theorem 4.6 amounts to …”

Section: Lemma 43 Suppose That There Is An Arcmentioning

confidence: 99%

“…Thus, the set of source homogeneous cones is uniquely determined up to a linear equivalence, and in this sense our realization of is unique. Chua's realization in [8] is merely in the outer direct sum space i∈I V [i] , and does not take the actual overlapping parameters into consideration. With the above stapling process to gather up the overlaps in the direct sum vector space, our realization 0…”

Section: Remark 415mentioning

confidence: 99%

“…A similar idea is found in Graczyk and Ishi [12] by using some system of vector spaces with bigraded indices. Another way of such representation as a slice of P(N, R) is due to Chua [8,Corollary 4.3] as well as Xu [25,Theorem 4.12], Ishi and Nomura [16,Proposition 5.3] (see also Ishi [15,Theorem 4]). However, in general, these descriptions demand bigger matrices than actually needed, and some parameters appear with unnecessarily more repetitions.…”

Section: Introductionmentioning

confidence: 99%

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Realization of Homogeneous Cones Through Oriented Graphs

Yamasaki

Nomura

2015

Kyushu J. Math.

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Abstract. In this paper, we realize any homogeneous cone by assembling uniquely determined subcones. These subcones are realized in the cones of positive-definite real symmetric matrices of minimal possible sizes. The subcones are found through the oriented graphs drawn by using the data of the given homogeneous cones. We also exhibit several interesting examples of our realizations of homogeneous cones. These are of rank 5, of dimension 19, of dimension 11 of continuously many inequivalent homogeneous cones, and some of the low-dimensional homogeneous cones.

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Section: Lemma 43 Suppose That There Is An Arcmentioning

confidence: 99%

Section: Remark 415mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Realization of Homogeneous Cones Through Oriented Graphs

Yamasaki

Nomura

2015

Kyushu J. Math.

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“…In this case the representability question was recently answered affirmatively in [17] (see also [21]): any homogeneous cone is a semidefinite slice.…”

Section: Hyperbolic Polynomialsmentioning

confidence: 99%

The mathematics of eigenvalue optimization

Lewis

2003

Math. Program., Ser. B

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Abstract. Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.Key words. Eigenvalue optimization -convexity -nonsmooth analysis -duality -semidefinite program -subdifferential -Clarke regular -chain rule -sensitivity -eigenvalue perturbation -partly smooth -spectral function -unitarily invariant norm -hyperbolic polynomial -stability -robust control -pseudospectrum -H∞ norm PART I: INTRODUCTION Von Neumann and invariant matrix normsBefore outlining this survey, I would like to suggest its flavour with a celebrated classical result. This result, von Neumann's characterization of unitarily invariant matrix norms, serves both as a historical jumping-off point and as an elegant juxtaposition of the central ingredients of this article.Von Neumann [64] was interested in unitarily invariant norms · on the vector space M n of n-by-n complex matrices:where U n denotes the group of unitary matrices. The singular value decomposition shows that the invariants of a matrix X under unitary transformations of the form X → U XV are given by the singular values σ 1 (X) ≥ σ 2 (X) ≥ · · · ≥ σ n (X), the eigenvalues of the matrix √ X * X. Hence any invariant norm · must be a function of the vector σ(X), so we can write X = g(σ(X)) for some function g : R n → R. We ensure this last equation if we define g(x) = Diag x for vectors x ∈ R n , and in that case g is a symmetric gauge function: that is, g is a norm on R n whose value is invariant under permutations and sign changes of the components. Von Neumann's beautiful insight was that this simple necessary condition is in fact also sufficient.Theorem 1.1 (von Neumann, 1937). The unitarily invariant matrix norms are exactly the symmetric gauge functions of the singular values.Von Neumann's proof rivals the result in elegance. He proceeds by calculating the dual norm of a matrix Y ∈ M n ,where the real inner product X, Y in M n is the real part of the trace of X * Y . Specifically, he shows that any symmetric gauge function g satisfies the simple duality relationshipwhere g * is the norm on R n dual to g. From this, the hard part of his result follows:...

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“…The cone P n of positive definite n × n real symmetric matrices is a typical example of homogeneous cones. It is known [12][13][14][15][16] that every homogeneous cone is linearly isomorphic to a cone P n ∩ Z with an appropriate subspace Z of the vector space Sym(n, R) of all n × n real symmetric matrices, where Z admits a specific block decomposition. Based on such results, our matrix realization method [15,17,18] has been developed for the purpose of the efficient study of homogeneous cones.…”

Section: Introductionmentioning

confidence: 99%

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

Ishi

2016

Entropy

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Abstract:The Koszul-Vinberg characteristic function plays a fundamental role in the theory of convex cones. We give an explicit description of the function and related integral formulas for a new class of convex cones, including homogeneous cones and cones associated with chordal (decomposable) graphs appearing in statistics. Furthermore, we discuss an application to maximum likelihood estimation for a certain exponential family over a cone of this class.

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Relating Homogeneous Cones and Positive Definite Cones via T-Algebras

Abstract: Relating homogeneous cones and positive definite cones via T-algebras Chua, Chek Beng. 2003 Chua, C. B., (2003). Relating homogeneous cones and positive definite cones via T-algebras.

Cited by 42 publications

References 4 publications

Realization of Homogeneous Cones Through Oriented Graphs

Realization of Homogeneous Cones Through Oriented Graphs

The mathematics of eigenvalue optimization

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

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