Eigenvalue optimization and subdifferentials
Optimization problems involving a symmetric matrix variable X are becoming increasingly prominent, and appear in many applications. Very often the objective or constraint functions are spectral functions, that is symmetric functions of the eigenvalues of X. Common examples are the maximum eigenvalue of X, log det X, and the indicator function of the cone of positive semidefinite matrices. The study of first-order conditions, duality, and sensitivity results necessitates the calculation of derivatives, subdifferentials (convex or Clarke) or Fenchel conjugates of spectral functions on the space of symmetric matrices (endowed with the trace inner product). Three cases will be considered in turn: convex, differentiable, and locally Lipschitz. Surprisingly, in each case simple and unified calculations are possible for completely general spectral functions. The resulting formulae are elegant, powerful and illuminating.
- OSTI ID:
- 36216
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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