Nonsmooth analysis of eigenvalues

The eigenvalues of a symmetric matrix depend on the matrix nonsmoothly. This paper describes the nonsmooth analysis of these eigenvalues. In particular, I present a simple formula for the approximate (limiting Frechet) subdifferential of an arbitrary function of the eigenvalues, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdifferential of the k'th largest eigenvalue.

[1]  加藤 敏夫 A short introduction to perturbation theory for linear operators , 1982 .

[2]  B. Gollan EIGENVALUE PERTURBATIONS AND NONLINEAR PARAMETRIC OPTIMIZATION , 1987 .

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

[4]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[5]  M. Overton On minimizing the maximum eigenvalue of a symmetric matrix , 1988 .

[6]  R. Fletcher Semi-Definite Matrix Constraints in Optimization , 1985 .

[7]  A. S. Lewis,et al.  Group Invariance and Convex Matrix Analysis , 1996, SIAM J. Matrix Anal. Appl..

[8]  Alexander Shapiro,et al.  On Eigenvalue Optimization , 1995, SIAM J. Optim..

[9]  C. Theobald An inequality for the trace of the product of two symmetric matrices , 1975 .

[10]  Michael L. Overton,et al.  Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices , 2015, Math. Program..

[11]  B. Mordukhovich Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[12]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[13]  T. Willmore AN INTRODUCTION TO DIFFERENTIABLE MANIFOLDS AND RIEMANNIAN GEOMETRY (Second edition) , 1987 .

[14]  S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

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

[16]  N. S. Barnett,et al.  Private communication , 1969 .

[17]  A. Ioffe Approximate subdifferentials and applications. I. The finite-dimensional theory , 1984 .

[18]  W. Boothby An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[19]  A. Lewis Eigenvalue-constrained faces☆ , 1998 .

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

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

[22]  Adrian S. Lewis,et al.  Convex Analysis on the Hermitian Matrices , 1996, SIAM J. Optim..

[23]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[24]  J. Hiriart-Urruty,et al.  Sensitivity Analysis for a Class of Convex Functions Defined Over a Space of Symmetric Matrices , 1992 .