M. Overton, "On minimizing the maximum eigenvalue of a symmetric matrix," SIAM J. Matrix Analysis and Appl., vol. 9, no. 2, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and self contained proof of local quadratic convergence of the SSP method under the assumption that the optimal solution is locally unique, strictly complementary, and satisfies a second order sufficient condition. One of the first numerical approaches for solving a class of NLSDPs was given in [16, 17]. Other recent approaches for solving NLSDPs are the program package LOQO [26] based on a primal dual approach; see also [27] Another promising approach for solving large scale semidefinite programs is the modified barrier approach proposed in [12] This modified barrier approach does not require ....

Overton, M.L. (1988): On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9, 256--268

....(but still nondifferentiable) Many researchers have considered this problem. Relevant work includes Cullum et al. [CDW75] Craven and Mond [CM81] Polak and Wardi [PW82] Fletcher [Fle85] Shapiro [Sha85] Friedland et al. FNO87] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92, OW93, OW92], Ringertz [Rin91] Fan and Nekooie [FN92] and Fan [Fan92] In [BY89] Boyd and Yang use the cutting plane algorithm and Shor s subgradient method [Sho85] to solve eigenvalue minimization problems that arise in control theory. They also describe a saddle point method for eigenvalue mimimization ....

.... more precisely, it identifies the branches of the eigenvalue functions that are active at the optimal point x ) We presume that once these active eigenvalues are identified, an optimum point can be computed more rapidly by switching to a quadratically convergent method such as Overton s (see [Ove88]; the extension to the generalized eigenvalue case is considered in [Hae91] We have not given a complete complexity analysis (worst case operation count) of the algorithm, since we have not given any bound on the number of Newton steps required to reach (in some appropriate approximate sense) ....

M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 9(2):256--268, 1988.

....maximum eigenvalue #(a) of the matrix (58) then g i = u Q i u yields g ##(a) So it is very easy to find elements of the set ##(a) From Polak and Wardi s characterization of the subgradient we can readily derive conditions for a = 0 to be optimal. These conditions can be found in Overton [Ove87, OW87], but with a completely di#erent proof. Theorem 7.1 A is not SSLS, or equivalently, 0 is a global minimizer of #, if and only if there is a nonzero R = R 0 such that TrQ i R = 0, i = 1, r. 26 First suppose that A is not SSLS, so that that 0 ##(0) By Polak and Wardi s ....

M. L. Overton. On Minimizing the Maximum Eigenvalue of a Symmetric Matrix. Technical Report, Center for Mathematical Analysis, Australian National University, 1987. to appear, Linear Algebra in Signals, Systems and Control (SIAM, Philadelphia, 1987).

....as possible. It is of some interest that in recent numerical analysis and mathematical programming literature there is a great deal of interest in minimzing the largest eigenvalue of a parameter dependent matrix. Compare, for example, various papers by Michael Overton and co workers [Overton, 1988; Haeberly and Overton, 1994; Overton and Womersley, 1993] In fact, it might be interesting in general to look at the range of aspects, i.e. to compute both the minimum and the maximum over all monotonic transformations. To compute the derivatives of the aspect, we need a general result on the ....

Overton, M. (1988). On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl., 9, 256--268.

.... Analyticity is discussed in [26] thus our result lies in some sense between the results in [9] and [26] Smoothness properties of some special spectral functions (such as the largest eigenvalue) on certain manifolds are helpful in perturbation theory and Newton type methods: see for example [15, 16, 18, 17, 22, 21, 14]. We show that a spectral function is twice (continuously) di erentiable at a matrix if and only if the corresponding symmetric function is twice (continuously) di erentiable at the vector of eigenvalues. Thus in particular, a spectral function is C 2 if and only if its restriction to the ....

M.L. OVERTON. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 9:256-268, 1988.

....n be a convex cone and W ae H n be a compact set. Then exactly one of the following holds: a) There is a matrix A 2 K such that tr AB 0 for all B 2 W . b) There are i 0 and B i 2 W such that k X i=1 i B i 2 K : Optimality conditions similar to those in the next lemma are given in [12] and [3] Lemma 1.4. Let S be a closed convex subset of H n , and let A 2 H n and X 0 2 S be given. Let K be the cone of feasible directions at X 0 . Then the following are equivalent: a) min (A X 0 ) min (A X) for all X 2 S. 4 roy mathias (b) There is a nonzero positive semidefinite ....

M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl., 9(2):256--268, 1988.

....61 vector x. Then x determines a cut of size (G; c) and hence mc(G; c) G; c) However, it is NP hard to determine the minimum size of an optimality certificate ( 6] The optimality certificate from [5] recalled in Theorem 4. 2 is related to an optimality criterion of Overton, formulated in [12] for a more general problem. We will re phrase this criterion. Let (G; c) be a weighted graph, and u 2 U . Let q 1 ; q t be an orthonormal basis of the eigenspace Eig( max (L U) where t denotes the dimension of the eigenspace. Let r 1 ; r n denote the rows of the matrix Q = q 1 ....

M. L. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 256-268.

....yet powerful unifying framework in which to study a wide variety of important results. Examples include Schur convexity (see for example [22] the convexity of eigenvalue functions ( 10, 6, 11, 3, 13, 19] calculations of Fenchel conjugates and subdifferentials of convex eigenvalue functions [24, 5, 12, 30, 28, 25, 26, 27, 15, 16, 1, 17, 19], von Neumann s original result [33] and generalizations (for example [4, 20] subdifferentials of unitarily invariant norms [34, 35, 36, 37, 38, 8, 7, 9, 20] and characterizations of extreme, exposed and smooth points of unit balls [2, 37, 38, 8, 7, 9, 20] This paper concentrates on convexity ....

M.L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 9:256--268, 1988.

....solution of these problems, for example, the ellipsoid algorithm (see e.g. 24, 25] The ellipsoid method has polynomial time complexity, and works in practice for smaller problems, but can be slow for larger problems. Other algorithms specifically for LMI based problems are discussed in, e.g. [26, 27]. Recently, various researchers [28, 1, 29, 30] have developed interior point methods for solving LMIbased problems, based on the work of Nesterov and Nemirovsky [31] Numerical experience shows that these algorithms solve LMI problems with extreme efficiency. In some specific cases (one is ....

M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 9(2):256--268, 1988.

....precisely what occurs in the above problem since the second eigenvalue of the Laplacian on a disk has multiplicity two. Methods for the optimization of the largest eigenvalue of a symmetric matrix that can handle the lack of smoothness at the solution have recently been devised by Michael Overton [16,15] who implemented them in an algorithm which has been successfully applied to an extensive collection of problems [16,8] We extend these techniques to apply in our context. The main steps are as follows. First we use finite elements to describe a family Omega Gamma x) of perturbations of a disk ....

M. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM Journal on Matrix Analysis and Applications, 9 (1988), 256-268.

....a similar problem from the point of view of nondifferentiable optimization. In particular, he derives some expressions for the subgradients of the sum of the first few eigenvalues of a symmetric matrix and formulates optimality conditions for this problem. In the same spirit as Fletcher, Overton [Ove88] studies the largest eigenvalue of a symmetric matrix as a convex, but nondifferentiable function. Based on earlier work [FNO87] in [Ove88] he derives a quadratically convergent algorithm for the problem of minimizing the largest eigenvalue of an affinely constrained matrix. This work is further ....

....of the sum of the first few eigenvalues of a symmetric matrix and formulates optimality conditions for this problem. In the same spirit as Fletcher, Overton [Ove88] studies the largest eigenvalue of a symmetric matrix as a convex, but nondifferentiable function. Based on earlier work [FNO87] in [Ove88] he derives a quadratically convergent algorithm for the problem of minimizing the largest eigenvalue of an affinely constrained matrix. This work is further extended in [Ove92] where both second order methods based on sequential quadratic programming, and first order methods based on sequential ....

M. L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl., 9(2), 1988. 36

.... Kojima and Hara [KKH94] Many researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast as a semidefinite program (see x2) See, for instance, Cullum, Donath and Wolfe [CDW75] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92], Overton and Womersley [OW93, OW92] Ringertz [Rin91] Fan and Nekooie [FN92] Fan [Fan93] Hiriart Urruty and Ye [HUY95] Shapiro and Fan [SF94] and Pataki [Pat94] Interior point methods for LPs were introduced by Karmarkar in 1984 [Kar84] although many of the underlying principles are older ....

M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 9(2):256--268, 1988.

.... The efficiency of recent interior point methods for SDP, which is directly responsible for the popularity of SDP in control, has therefore also attracted a great deal of interest in optimization circles, overshadowing earlier solution methods based on techniques from nondifferentiable optimization [8, 10, 11, 12, 13]. At every major optimization conference, there are workshops and special sessions devoted exclusively to SDP, and a special issue of Mathematical Programming has recently been devoted to SDP [14] This interest was primarily motivated by applications of SDP in combinatorial optimization but, more ....

M. Overton, "On minimizing the maximum eigenvalue of a symmetric matrix", SIAM J. on Matrix Analysis and Applications, vol. 9, pp. 256--268, 1988.

....Hence y # R m is a solution to (4.4) if and only if 0 # #h(y) where #h(y) is the subdi#erential of f at y (see Section 4.2) In the sequel, we assume without any loss of generality that a = 1. Eigenvalue optimization problems such as (4. 4) have been extensively studied in the literature [21, 85, 86, 88]; see [67] for a tutorial survey. Details on subgradient bundle methods can be found in [50, 56, 57, 98] which develop the original pioneering work of Lemarechal [61, 62] see also [63] Bundle methods specifically tailored for eigenvalue optimization are discussed in [47, 48, 64, 83, 84] The ....

M. L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 9(2), 1988.

....approach to primal dual interior point methods for SDP is given in [10] 4. Quadratically Convergent Local Methods In [2] the authors derived a quadratically convergent local method for optimizing eigenvalues of pencils. This method extended earlier work on optimizing eigenvalues of matrices [11,12]. Note that each step of this algorithm requires only the solution of a linear system of equations, though the form of the equations is quite complicated. Even in the case of matrix eigenvalue optimization, the proof of quadratic convergence is nontrivial [13] since the method cannot be described ....

M. L. Overton, "On minimizing the maximum eigenvalue of a symmetric matrix," SIAM J. Matrix Anal. Appl., 9:256--268,1988.

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M. L. Overton (1988), "On minimizing the maximum eigenvalue of a symmetric matrix", SIAM Journal on Matrix Analysis and Application 9, pp. 256-268.

....(e.g. Roc70] are applicable to eigenvalue optimization problems; optimality conditions and or first order algorithms for various problem classes have been given by [CDW75,PW83,Doy82,Sha85b,Gol87,All89] See also [OT83,Bra86] for discussion of problems arising in structural engineering. In [Ove88] a quadratically convergent algorithm was given to solve the model problem, using a dual matrix formulation of the optimality conditions to fully exploit the nonsmooth problem structure. Two papers which greatly influenced this work were [FNO87,Fle85] Numerical examples were given, ....

....eigenvalues (algebraically or in absolute value) are given by [OWa] These optimality conditions are derived by characterizing Clarke s generalized gradient ( Cla83] in terms of a dual matrix. Proofs of the local quadratic convergence of the successive quadratic programming algorithm used in [Ove88] are being developed in [OWb] There are three main purposes of the present paper. The first is to present a clear and self contained derivation of the generalized gradient of the max eigenvalue functional in terms of a dual matrix. An understanding of this is essential for the appreciation of ....

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M.L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 9:256--268, 1988.

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M. Overton, "On minimizing the maximum eigenvalue of a symmetric matrix," SIAM J. Matrix Analysis and Appl., vol. 9, no. 2, 1988.

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M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 9(2):256--268, 1988.

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M. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 9(2):256--268, 1988.

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M. L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Marix Anal. Appl., 9:256--268, 1988.

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Overton M.L., "On minimizing the maximum eigenvalue of a symmetric matrix," SIAM Journal on Matrix Analysis and Applications, Vol.9, No.2, pp.256-268, 1988. 14

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M.L. OVERTON. On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl., 9:256--268, 1988.

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M. L. Overton# #On minimizing the maximum eigenvalue of a symmetric matrix## SIAM J. Matrix Anal. Appl.#vol. 9# pp. 256#268# Apr. 1988.

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M. L. Overton# #On minimizing the maximum eigenvalue of a symmetric matrix## SIAM J. Matrix Anal. Appl.#vol. 9# pp. 256#268# Apr. 1988.

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