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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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