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On minimizing the maximum eigenvalue of a symmetric matrix. (English) Zbl 0669.65052

Linear algebra in signals, systems, and control, Proc. SIAM Conf., Boston/Mass. 1986, 150-169 (1988).
[For the entire collection see Zbl 0661.00019.]
The paper presents a new algorithm for the minimization of the function f(x) over all \(x\in R^ m\), where f(x) is the maximum absolute value of the eigenvalues of a real symmetric \(n\times n\) matrix-valued affine function of x. Such minimization problems arise in control engineering. The function f(x) is convex, but not differentiable in general, since the eigenvalues are not differentiable quantities at points where they coalesce. Furthermore, one can usually expect the solution to be at a nondifferentiable point.
The algorithm presented in the paper solves the minimization problem with an asymptotic quadratic rate of convergence. Moreover, second derivatives are not always required to obtain the quadratic convergence. An important feature of the algorithm is the ability to obtain a descent direction from any point which is not optimal, even if this requires splitting eigenvalues which are currently equal.
Reviewer: F.Flandoli

MSC:

65K05 Numerical mathematical programming methods
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F30 Other matrix algorithms (MSC2010)

Citations:

Zbl 0661.00019

Software:

EISPACK