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1、SIAM J. MATRIX ANAL. APPL.Vol. 9, No. 2, April 1988(C) 1988 Society for Industrial and Applied Mathematics011ON MINIMIZING THE MAXIMUM EIGENVALUE OF A SYMMETRIC MATRIX*MICHAEL L. OVERTON’fAbstract. An important optimizat

2、ion problem that arises in control is to minimize o(x), the largest eigenvalue(in magnitude) of a symmetric matrix function of x. If the matrix function is affine, 9(x) is convex. However,9(x) is not differentiable, sinc

3、e the eigenvalues are not differentiable at points where they coalesce. In this paper an algorithm that converges to the minimum of 9(x) at a quadratic rate is outlined. Second derivatives are notrequired to obtain quadr

4、atic convergence in cases where the solution is strongly unique. An important feature of the algorithm is the ability to split a multiple eigenvalue, if necessary, to obtain a descent direction. In theserespects the new

5、algorithm represents a significant improvement on the first-order methods ofPolak and Wardi and ofDoyle. The new method has much in common with the recent work ofFletcher on semidefinite constraints and Friedland, Noceda

6、l, and Overton on inverse eigenvalue problems. Numerical examples are presented.Key words, nonsmooth optimization, nondifferentiable optimization, convex programming, semidefiniteconstraints, minimizing maximum singular

7、valueAMS(MOS) subject classifications. 65F99, 65K10, 90C251. Introduction. Many important optimization problems involve eigenvalue con-straints. For example, in structural engineering we may wish to minimize the cost of

8、some structure subject to constraints on its natural frequencies. A particularly common problem, which arises in control engineering, is(1.1) min qg(x) XEmwhere(1.2) o(x) max [Xi(A(x))l, l_i_nA(x) is a real symmetric n n

9、 matrix-valued affine function of x, and?i(A(x)), 1, n)are its eigenvalues. Since A(x) is an affine function, it may be writtenA(x) Ao + , x,A,.k=lThe function o(x) is convex, since the largest eigenvalue of a matrix is

10、a convex function of the matrix elements. An important special case is(1.3) A eeReceived by the editors February 1, 1987; accepted for publication October 1, 1987. This work was supported in part by the National Science

11、Foundation under grant DCR-85-02014. Some of the computer program development was performed at Stanford University, Stanford, California with support from the Office ofNaval Research under contract ONR N00014-82-K-0335.

12、This paper was presented at the SIAM Conference on Linear Algebra in Signals, Systems, and Control, which was held in Boston, Massachusetts on August 12- 14, 1986.“Courant Institute of Mathematical Sciences, New York Uni

13、versity, New York, New York 10012. This work was completed while the author was on sabbatical leave at the Centre for Mathematical Analysis and Mathematical Sciences Research Institute, Australian National University, Ca

14、nberra, Australia.256Downloaded 04/28/14 to 222.205.5.133. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php258 MICHAEL L. OVERTONwhere “>=“ in (2.4), (2.5) indicates a mat

15、rix positive semidefinite constraint. The secondformulation immediately suggests that the work of Fletcher (1985) is applicable. Fletchergives a quadratically convergent algorithm to solve(2.6) max xk i=(2.7) s.t. A0- Di

16、ag (x) >= 0, x >= 0and many of the components of his algorithm are therefore applicable to solving(2.3)-(2.5). However, the algorithm is not directly applicable and there are several reasons why it is possible to i

17、mprove on Fletcher’s method in this case. One reason is that Fletcher’s method solves a sequence of subproblems, each defined by a guess of the nullity of A0 + Diag (x), until the correct nullity is identified. Such a st

18、rategy cannot easily be extended to the case oftwo (or more) semidefinite constraints. One goal ofour algorithmis to be able to adjust multiplicity estimates while always obtaining a reduction of o(x)at each iteration. W

19、e are able to do this by computing an eigenvalue-eigenvector factor-ization of A(x) at each iteration. By contrast, Fletcher’s method uses a block Choleskifactorization ofA0 + Diag (x), together with an exact penalty fun

20、ction to impose (2.7). Also, because of the special form of (2.6), (2.7), Fletcher’s method does not require a technique for splitting eigenvalues. In other words, given a matrix A0 + Diag (x),satisfying (2.7), with null

21、ity t, it cannot be advantageous, in the sense of increasing (2.6),to reduce the multiplicity t. On the other hand it may be necessary to split a multipleeigenvalue in our case, and the ability to recognize this situatio

22、n and obtain an appropriate descent direction is an important part of our algorithm. Because we use an eigenvalue factorization of the matrix A(x) at each iterate x, our method has much in common with the methods describ

23、ed by Friedland, Nocedal, and Overton (1987). In the latter paper several quadratically convergent methods are given to solve(2.8) X(A(x)) 0, 1, t,(2.9) Xi(A(x)) ti, i= + 1, ..., [where (o, {ui}) are given distinct value

24、s and t, t“ (and m, the number of variables) areappropriately chosen. One of the contributions of that paper was to explain that the condition (2.8), although apparently only conditions, actually genetically imposest(t +

25、 1)/2 linearly independent constraints on the parameter space, and that effective numerical methods must be based on this consideration. The present paper may be viewed as generalizing the methods of Friedland, Nocedal,

26、and Overton to solve(2.10) min ,X [m(2.11) s.t. X(A(x)) w, 1, ..., t,(2.12) Xi(A(x)) -oo, n s + 1, nwhere, as a product ofthe minimization process, it is established that 0 max (Xl, with(2.13) oa= h Xt> kt+ kn-s> k

27、n-s+ kn We shall subsequently refer to and s as the upper and lower (eigenvalue) multiplicities of A(x). Note that it is possible that either or s is zero. The following notation will beuseful subsequently: let {ql(x), q

28、n(X)} be any orthonormal set of eigenvectors of A(x) corresponding to {,i}, and let Q [q, qt], Q2 [qn-s+, qn].Downloaded 04/28/14 to 222.205.5.133. Redistribution subject to SIAM license or copyright; see http://www.siam

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