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On minimizing the maximum eigenvalue of a symmetric matrix. (English) Zbl 0647.65044
Let A(x) be a real symmetric \(n\times n\) matrix-valued affine function of \(x\in R_ m\), \(\phi (x)=\max \{| \lambda_ i(A(x))|: 1\leq i\leq n\},\) where \(\lambda_ i\) are the eigenvalues of A. The paper is devoted to the problem of finding min\(\{\phi\) (x): \(x\in R\) \(m\}\). Since the function is affine, \(\phi\) (x) is convex. However, it is not differentiable at points, where they coalesce. An algorithm is presented that converges to the minimum of \(\phi\) (x) at a quadratic rate. In cases where the solution is strongly unique second derivatives are not required to obtain quadratic convergence. The algorithm is able to split a multiple eigenvalue, if necessary, in order to obtain a descent direction.
Numerical examples illustrating applicability of the algorithm are presented. It is a significant improvement of the first-order methods of E. Polak and Y. Wardi [Automatica 18, 267-283 (1982; Zbl 0532.49017)] and of J. Doyle [Proc. IEE-D 129, 242-250 (1982; M.R. 84a:93035)]. It has much in common with the recent work of R. Fletcher [SIAM J. Control Optimization 23, 493-513 (1985; Zbl 0567.90088)] on semidefinite constraints and of S. Friedland, J. Nacedal, and the author [SIAM J. Numer. Analysis 24, 634-667 (1987; Zbl 0622.65030)] on inverse eigenvalue problems.
Reviewer: I.Dvořák

MSC:
65K10 Numerical optimization and variational techniques
90C25 Convex programming
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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