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Spectral functions on Jordan algebras : differentiability and convexity properties

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  • BAES, Michel

Abstract

A spectral function on a formally real Jordan algebra is a real-valued function which depends only on the eigenvalues of its argument. One convenient way to create them is to start from a function f : Rexp.r [arrow] R which is symmetric in the components of its argument, and to define the function F(u) := f([delta](u)) where [delta](u) is the vector of eigenvalues of u. In this paper, we show that this construction preserves a number of properties which are frequently used in the framework of convex optimization: differentiability, convexity properties and Lipschitz continuity of the gradient for the Euclidean norm with the same constant as for f.

Suggested Citation

  • BAES, Michel, 2004. "Spectral functions on Jordan algebras : differentiability and convexity properties," LIDAM Discussion Papers CORE 2004016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2004016
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    Cited by:

    1. BAES, Michel, 2006. "Smoothing techniques in Euclidean Jordan algebras," LIDAM Discussion Papers CORE 2006013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    3. Sangho Kum & Yongdo Lim, 2010. "Penalized complementarity functions on symmetric cones," Journal of Global Optimization, Springer, vol. 46(3), pages 475-485, March.

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