Variational analysis of non-Lipschitz spectral functions

James V. Burke, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

We consider spectral functions f ○ λ, where f is any permutation-invariant mapping from Cn to R, and λ is the eigenvalue map from the set of n x n complex matrices to Cn, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f ○ λ is the spectral abscissa, while if f is "maximum modulus", then f ○ λ is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f ○ λ at X is that Y* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.

Original languageEnglish (US)
Pages (from-to)317-351
Number of pages35
JournalMathematical Programming
Volume90
Issue number2
StatePublished - Apr 2001

Fingerprint

Non-Lipschitz
Variational Analysis
Spectral Function
Abscissa
Eigenvalue
Subgradient
Semidefinite Programming
Necessary Conditions
Jordan Form
Subdifferential
Complementarity
Equality Constraints
Square matrix
Spectral Radius
Linear Constraints
Commute
Linear Function
Lipschitz
Modulus
Permutation

Keywords

  • Eigenvalue function
  • Nonsmooth analysis
  • Semistable program
  • Spectral abscissa
  • Spectral radius
  • Stability

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Software
  • Mathematics(all)
  • Applied Mathematics
  • Safety, Risk, Reliability and Quality
  • Management Science and Operations Research

Cite this

Variational analysis of non-Lipschitz spectral functions. / Burke, James V.; Overton, Michael L.

In: Mathematical Programming, Vol. 90, No. 2, 04.2001, p. 317-351.

Research output: Contribution to journalArticle

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