Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras

We study analyticity, differentiability, and semismoothness of Lowner's operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra is shown to be strongly semismooth. The research also raises several open questions, whose answers would be of strong interest for optimization research.

[1]  Jong-Shi Pang,et al.  Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps , 1996, Math. Oper. Res..

[2]  M. Koecher,et al.  The Minnesota Notes on Jordan Algebras and Their Applications , 1999 .

[3]  Heinz H. Bauschke,et al.  Hyperbolic Polynomials and Convex Analysis , 2001, Canadian Journal of Mathematics.

[4]  Ė. B. Vinberg,et al.  A course in algebra , 2003 .

[5]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[6]  M. Seetharama Gowda,et al.  Some P-Properties for Nonlinear Transformations on Euclidean Jordan Algebras , 2005, Math. Oper. Res..

[7]  Xiaoqi Yang,et al.  Semismoothness of Spectral Functions , 2003, SIAM J. Matrix Anal. Appl..

[8]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[9]  M. Seetharama Gowda,et al.  Inverse and implicit function theorems for H-differentiable and semismooth functions , 2004, Optim. Methods Softw..

[10]  Chek Beng Chua.  Relating Homogeneous Cones and Positive Definite Cones via T-Algebras , 2003, SIAM J. Optim..

[11]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[12]  Farid Alizadeh,et al.  Extension of primal-dual interior point algorithms to symmetric cones , 2003, Math. Program..

[13]  C. Davis.  Notions generalizing convexity for functions defined on spaces of matrices , 1963 .

[14]  P. Lancaster.  On eigenvalues of matrices dependent on a parameter , 1964 .

[15]  L. Faybusovich.  Euclidean Jordan Algebras and Interior-point Algorithms , 1997 .

[16]  Paul Tseng,et al.  Non-Interior continuation methods for solving semidefinite complementarity problems , 2003, Math. Program..

[17]  W. Hager.  Lipschitz Continuity for Constrained Processes , 1979 .

[18]  Masao Fukushima,et al.  Smoothing Functions for Second-Order-Cone Complementarity Problems , 2002, SIAM J. Optim..

[19]  Defeng Sun,et al.  Strong Semismoothness of Eigenvalues of Symmetric Matrices and Its Application to Inverse Eigenvalue Problems , 2002, SIAM J. Numer. Anal..

[20]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[21]  O. Mangasarian.  On Concepts of Directional Differentiability , 2004 .

[22]  Adam Korányi,et al.  Monotone functions on formally real Jordan algebras , 1984 .

[23]  C. Lemaréchal,et al.  ON A BUNDLE ALGORITHM FOR NONSMOOTH OPTIMIZATION , 1981 .

[24]  Paul Tseng,et al.  Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems , 2003, SIAM J. Optim..

[25]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[26]  Karl Löwner.  Über monotone Matrixfunktionen , 1934 .

[27]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[28]  L. Gårding.  An Inequality for Hyperbolic Polynomials , 1959 .

[29]  B. Kummer.  NEWTON's METHOD FOR NON-DIFFERENTIABLE FUNCTIONS , 1988, Advances in Mathematical Optimization.

[30]  A. S. Lewis,et al.  Derivatives of Spectral Functions , 1996, Math. Oper. Res..

[31]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[32]  Defeng Sun,et al.  Semismooth Matrix-Valued Functions , 2002, Math. Oper. Res..

[33]  J. Warga,et al.  Fat homeomorphisms and unbounded derivate containers , 1981 .

[34]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[35]  W. Donoghue.  Monotone Matrix Functions and Analytic Continuation , 1974 .

[36]  S. Scholtes,et al.  Structural Analysis of Nonsmooth Mappings, Inverse Functions, and Metric Projections , 1994 .

[37]  Alexander Shapiro,et al.  On differentiability of symmetric matrix valued functions , 2002 .

[38]  Adrian S. Lewis,et al.  Twice Differentiable Spectral Functions , 2001, SIAM J. Matrix Anal. Appl..

[39]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[40]  Defeng Sun,et al.  A Squared Smoothing Newton Method for Nonsmooth Matrix Equations and Its Applications in Semidefinite Optimization Problems , 2003, SIAM J. Optim..

[41]  Adrian Lewis,et al.  The mathematics of eigenvalue optimization , 2003, Math. Program..

[42]  E. Verriest,et al.  On analyticity of functions involving eigenvalues , 1994 .

[43]  R. Sznajder,et al.  Some P-properties for linear transformations on Euclidean Jordan algebras , 2004 .

[44]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[45]  Franz Rellich,et al.  Perturbation Theory of Eigenvalue Problems , 1969 .

[46]  Defeng Sun,et al.  Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems , 2003, Math. Oper. Res..

[47]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[48]  Defeng Sun,et al.  Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems , 2003, Comput. Optim. Appl..

[49]  Michel Baes,et al.  Spectral Functions on Jordan Algebras: Differentiability and Convexity Properties , 2004 .

[50]  Paul Tseng,et al.  Analysis of nonsmooth vector-valued functions associated with second-order cones , 2004, Math. Program..