SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

Shaohua Pan; Yungyen Chiang; Jein Shan Chen

doi:10.1016/j.laa.2012.04.030

SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

Shaohua Pan, Yungyen Chiang, Jein Shan Chen^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

Original language	English
Pages (from-to)	1264-1284
Number of pages	21
Journal	Linear Algebra and Its Applications
Volume	437
Issue number	5
DOIs	https://doi.org/10.1016/j.laa.2012.04.030
Publication status	Published - 2012 Sept 1

Keywords

Hilbert space
SOC-convexity
SOC-monotonicity
Second-order cone

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2012.04.030

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title = "SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions",

abstract = "The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.",

keywords = "Hilbert space, SOC-convexity, SOC-monotonicity, Second-order cone",

author = "Shaohua Pan and Yungyen Chiang and Chen, {Jein Shan}",

note = "Funding Information: This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds Funding Information: for∗ Correspondingauthor.theCentral Universities. E-mail addresses: [email protected] (S. Pan), [email protected] (Y. Chiang), [email protected] (J.-S. Chen). 1 The author{\textquoteright}s work is supported by National Science Council of Taiwan, Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan. 2 Also a Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.",

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AU - Pan, Shaohua

AU - Chiang, Yungyen

AU - Chen, Jein Shan

N1 - Funding Information: This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds Funding Information: for∗ Correspondingauthor.theCentral Universities. E-mail addresses: [email protected] (S. Pan), [email protected] (Y. Chiang), [email protected] (J.-S. Chen). 1 The author’s work is supported by National Science Council of Taiwan, Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan. 2 Also a Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

PY - 2012/9/1

Y1 - 2012/9/1

N2 - The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

AB - The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

KW - Hilbert space

KW - SOC-convexity

KW - SOC-monotonicity

KW - Second-order cone

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JF - Linear Algebra and Its Applications

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ER -

SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

Abstract

Keywords

ASJC Scopus subject areas

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