Abstract
LetA(·) be ann × n symmetric affine matrix-valued function of a parameteru∈R m, and let λ (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating ∂λ(u), the subdifferential of λ atu, which is useful for both the construction of efficient algorithms for the minimization of λ (u) and the sensitivity analysis of λ (u), namely, the perturbation theory of λ (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of λ (·) and the ɛ-subdifferential ∂ɛλ (u) of λ atu. Then, we discuss relations between the set ∂ɛλ (u) and some perturbation bounds for λ (u).
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Communicated by O. L. Mangasarian
The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.
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Ye, D.Y. Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form. J Optim Theory Appl 76, 287–304 (1993). https://doi.org/10.1007/BF00939609
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DOI: https://doi.org/10.1007/BF00939609