Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras

Sun, Defeng; Sun, Jie

doi:10.1287/moor.1070.0300

Cited by 108 publications

(83 citation statements)

References 49 publications

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“…More detailed expositions of Euclidean Jordan algebras can be found in the monograph by Faraut and Korányi [9]. Besides, one can find excellent summaries in the articles [2,12,24,26].…”

Section: Preliminariesmentioning

confidence: 99%

A one-parametric class of merit functions for the second-order cone complementarity problem

Chen

Pan

2008

Comput Optim Appl

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Abstract. In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [18] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R 02 -property. All of these results generalize some recent important works in [4,25,28] under a unified framework.

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“…More detailed expositions of Euclidean Jordan algebras can be found in the monograph by Faraut and Korányi [9]. Besides, one can find excellent summaries in the articles [2,12,24,26].…”

Section: Preliminariesmentioning

confidence: 99%

A one-parametric class of merit functions for the second-order cone complementarity problem

Chen

Pan

2008

Comput Optim Appl

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“…Lemma2.1. [ [18],Theorem13] For any x = ∑ r j=1 λ j (x)c j , let g sc be defined by (1). Then g sc is (continuously) differentiable at x if and only if g is (continuously) differentiable at all λ j (x).…”

Section: [[11] Theorem III 12] Suppose Thatmentioning

confidence: 99%

On the Convergence of Central Path and Generalized Proximal Point Method for Symmetric Cone Linear Programming

Yu¹,

Zhu²,

Qian-qian³

2013

Appl. Math. Inf. Sci.

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“…It turns out that this is always the case as we prove in Theorem 7.1. A generalization of Theorem 7.1 and Theorem 7.2 to the setting of formally real Jordan algebras can be found in [25]. Our approach is direct and first appeared in [21].…”

Section: Proof Use Again the Fact That (F • β) • ((X T); (Y R)) Ismentioning

confidence: 99%

Nonsmooth Analysis of Lorentz Invariant Functions

Sendov¹

2007

SIAM J. Optim.

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Abstract. A real valued function g(x, t) on R n ×R is called Lorentz invariant if g(x, t) = g(U x, t) for all n×n orthogonal matrices U and all (x, t) in the domain of g. In other words, g is invariant under the linear orthogonal transformations preserving the Lorentz cone:It is easy to see that every Lorentz invariant function can be decomposed as g = f • β, where f : R 2 → R is a symmetric function and β is the root map of the hyperbolic polynomial p(x, t) = t 2 −x 2 1 −· · ·−x 2 n . We investigate variety of important variational and non-smooth properties of g and characterize them in terms of the symmetric function f .

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Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras

Cited by 108 publications

References 49 publications

A one-parametric class of merit functions for the second-order cone complementarity problem

A one-parametric class of merit functions for the second-order cone complementarity problem

On the Convergence of Central Path and Generalized Proximal Point Method for Symmetric Cone Linear Programming

Nonsmooth Analysis of Lorentz Invariant Functions

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