Abstract

With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functions. In particular, we show that matrix valued functions inherit semismooth properties of the corresponding real valued functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexander Shapiro

School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, U.S.A.

ashapiro@isye.gatech.edu

Keywords

  • Matrix function
  • eigenvalues and eigenvectors
  • directional derivatives
  • semismooth mappings

Mathematics Subject Classification

15A18, 15A16, 90C30

References

  1. R. Bellman: Introduction to Matrix Analysis, McGraw-Hill, New York (1960).Google Scholar
  2. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  3. X. Chen, H. Qi, P. Tseng: Analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems, SIAM J. Optimization 4 (2003) 960–985.https://doi.org/10.1137/S1052623400380584Google Scholar
  4. F. Facchinei, J. S. Pang: Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, Springer, New York (2003).https://doi.org/10.1007/b97544Google Scholar
  5. T. Kato: Perturbation Theory for Linear Operators, Sringer, New York (1984).Google Scholar
  6. P. Lancaster: On eigenvalues of matrices dependent on a parameter, Numer. Math. 6 (1964) 377–387.https://doi.org/10.1007/BF01386087Google Scholar
  7. R. Mifflin: Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim. 33 (1977) 957–972.https://doi.org/10.1137/0315061Google Scholar
  8. J. S. Pang, D. Sun, J. Sun: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Math. Oper. Research 28 (2003) 39–63.https://doi.org/10.1287/moor.28.1.39.14258Google Scholar
  9. H. Poincaré: Sur les groupes continus, Trans. Cambridge Phil. Soc. 18 (1899) 220–225.Google Scholar
  10. L. Qi, J. Sun: A nonsmooth version of Newton’s method, Math. Programming 58 (1993) 353–367.https://doi.org/10.1007/BF01581275Google Scholar
  11. A. Shapiro, M. K. H. Fan: On eigenvalue optimization, SIAM J. Optimization 5 (1995) 552–569.https://doi.org/10.1137/0805028Google Scholar
  12. D. Sun, J. Sun: Semismooth matrix-valued functions, Math. Oper. Research 27 (2002) 150–169.https://doi.org/10.1287/moor.27.1.150.342Google Scholar
  13. D. Sun, J. Sun: Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue, SIAM J. Numer. Analysis 6 (2003) 2352–2367.https://doi.org/10.1137/S0036142901393814Google Scholar

Suggested citation

A. Shapiro. “On the Differentiability of Symmetric Matrix Valued Functions.” Journal of Convex Analysis 30 (2023), No. 4, 1307–1317. https://doi.org/10.68381/jca30058

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.