M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 13:41-45, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(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 ....

M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, pages 41--45, 1992.

....oldest form of semidefinite programming is the evaluation of eigenvalues of a symmetric matrix. In fact, one can reformulate the classical theorems of Rayleigh Ritz for the largest eigenvalue, and of Fan for the sum of the first few eigenvalues of a symmetric matrix, as semidefinite programs, see [OW91, OW92] and section 4 below. However, for these special cases, techniques of this paper do not seem to be appropriate as there exist better algorithms from both theoretical and practical points of view. Most nontrivial semidefinite programs (those that are not simply equivalent to evaluation of ....

....most known interior point algorithms for LP into similar algorithms for SDP. In this section we also go over some differences between SDP and LP as far as interior point methods and polynomial time algorithms in general are concerned. In section 4 we build on the results of Overton and Womersley [OW91, OW92] and derive semidefinite programming formulation for various eigenvalue optimization problems. We also state complementary slackness results for these problems. Finally, in section 5 we study some applications of SDP interior point methods to various combinatorial optimization problems. Notation ....

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M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl., 13:41--45, 1992.

.... 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 (see, e.g. Fiacco and McCormick ....

....provides an elegant way to derive this result. It is well known that the sum of the r largest eigenvalues of a matrix A = A T 2 R p Thetap can be expressed as maximum TrW T AW subject to W 2 R p Thetar W T W = I: 41) This result is attributed to Ky Fan [Fan49] Overton and Womersley [OW92] have observed that (41) can be expressed as the semidefinite program maximize TrAZ 11 subject to TrZ 11 = r Z 11 Z 22 = I Z 11 Z 12 Z T 12 Z 22 # 0: 42) The equivalence can be seen as follows. The matrix Z 12 can be assumed to be zero without loss of generality. The matrix Z 22 acts ....

M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, pages 41--45, 1992.

....5. diag(Y Y t ) e (rows of Y are unit vectors) Historically, the first tractable relaxations for equipartition utilize eigenvalue optimization. Donath and Hoffman [13] use the orthogonality property of the columns of Y and the following result, often called Fan s Theorem; see for instance [47]. We denote by 1 (L) n (L) the eigenvalues of a symmetric matrix L. Theorem 3 Let L be a symmetric matrix of order n, with eigenvalues i (L) and corresponding normalized and pairwise orthogonal respective eigenvectors p i (i = 1; n) Let k n be given, and define P = p 1 ; ....

....property, and Y Y t e = me together with the fact that rows of Y are unit vectors, and get the following basic semidefinite relaxation: minf 1 2 tr LX : X 0; diag(X) e; Xe = meg: The following result can be used to rewrite the Donath Hoffman bound as a semidefinite program. Lemma 4 ([16, 47]) convfY Y t : Y t Y = I k g = fX : tr X = k; I X 0g = M k : If A and B are symmetric matrices of the same size, then A B stands for A Gamma B 0: A discussion of various ways to prove this result can be found in [47] This relation can be used as follows: min Y t Y =I k tr Y ....

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M.L. OVERTON and R.S. WOMMESLEY. On the sum of the largest eigenvalues of a symmetric matrix, SIAM J. Matrix Anal. Appl., 13:41--45, 1992.

....of Y satisfies Sigma = I = Z 1 (1 Gamma )Z 2 , where Z i = U t Y i V; i = 1; 2. Since Z i ; i = 1; 2, are still feasible and so have norm 1, we conclude that Z i = I; i = 1; 2, i.e. Y = Y i ; i = 1; 2. Thus Y is an extreme point. 2 Equivalent formulations of Lemma 3. 1 can be found in [11, 37] for gaining a better insight. Although the formulations are slightly different, the resulting feasible set and its extreme points are the same. If the objective function is concave, then the optimum of (P ) is attained at an extreme point of F and is then orthogonal. It is easy to see that f(X) ....

M.L. OVERTON and R.S. WOMERSLEY. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Analysis and Applications, 13:41--45, 1992.

....this parametrization the bound simplifies to z RW : max u t e=0 min f m 2 tr Z t V t (L diag(u) V Z : Z t Z = I k Gamma1 g: 3. 3) These bounds can be expressed as eigenvalue optimization problems using Fan s Theorem minftr X t AX : X t X = I k g = k X i=1 i (A) see e.g. Overton and Womersley [1992]. We tacitly assume that the eigenvalues of A are in nondecreasing order, i.e. 1 (A) 2 (A) n (A) This was in fact the form in which these bounds were used by Donath and Hoffman [1973] Rendl and Wolkowiz [1995] and Falkner, Rendl and Wolkowiz [1994] We are now going to reformulate ....

....Y t LY = tr L(Y Y t ) and replacing Y Y t by a new variable X . In case of (k GPDH ) we can optimize over S k : convfY Y t : Y t Y = I k g: In order to do so, we need a nice description of this set. Fortunately, such a description exists. Lemma 3. 1 (Fillmore and Williams [1971] Overton and Womersley [1992]) S k = fX : X = X t ; tr X = k; I X 0g: Semidefinite Programming and Graph Equipartition 5 As observed by Alizadeh [1995] the eigenvalue bound z DH can now be reformulated as follows. z DH = max u t e=0 min X2Sk m 2 tr (L diag(u) X = min X2Sk max u t e=0 m 2 tr LX m 2 u ....

Overton, M.L and Womersley, R.S. [1992] On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl., 13: 41--45.

....result is implicitly contained in [6] where a similar theorem is proved for the case k = n, and both matrices positive definite. The proof easily generalizes to the present situation. For the sake of completeness we include the following argument and note that a similar result is also proved in [13]. We use the spectral decompositions of A and B, A = PEP t ; B = QFQ t ; with orthogonal matrices P; Q and diagonal matrices E; F of appropriate sizes. Note that the matrix Y : P t XQ has orthonormal columns, so we get: tr AXBX t = tr EY FY t (11) n X i=1 k X j=1 i (A) j (B)y ....

M. J. OVERTON and R. S. WOMMERSLEY. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl., 13:41--45, 1992.

....is, minimizing P i m i i , where 1 2 Delta Delta Delta n and m 1 m 2 Delta Delta Delta m n 0. The equivalence of the semidefinite program we consider and the eigenvalue bound of Delorme and Poljak was established by Poljak and Rendl [58] Building on work by Overton and Womersley [54, 53], Alizadeh [1] has shown that eigenvalue minimization problems can in general be formulated as semidefinite programs. This is potentially quite useful, since there is an abundant literature on eigenvalue bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak [49] ....

M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41--45, 1992.

....(u) q u (x) Gamma (x t x Gamma n) n Gamma 1 2 c t x. 2 Similar relations between trust region subproblems and eigenvalue problems are presented in [30, 23] The problem (4.3) is equivalent to minimizing the maximum eigenvalue of a matrix. These type of problems are treated in e.g. [20, 21], where efficient algorithms are presented as well as optimality conditions. The above theorem shows that these problems can also be treated using efficient trust region subproblem algorithms. We can now combine the above equivalences between the three given bounds with a fourth bound to get: ....

M.L. OVERTON and R.S. WOMERSLEY. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Analysis and Applications, 13:41--45, 1992.

....oldest form of semidefinite programming is the evaluation of eigenvalues of a symmetric matrix. In fact, one can reformulate the classical theorems of Rayleigh Ritz for the largest eigenvalue, and of Fan for the sum of the first few eigenvalues of a symmetric matrix, as semidefinite programs, see [54, 53] and section 4 below. However, for these special cases, techniques of this paper do not seem to be appropriate as better algorithms from both theoretical and pragmatic points of view Received by the editors October 25, 1991; accepted for publication (in revised form) August 30, 1993. This ....

....many known interior point algorithms for LP into similar algorithms for SDP. In this section we also go over some differences between SDP and LP as far as interior point methods and polynomial time algorithms in general are concerned. In section 4 we build on the results of Overton and Womersley [54, 53] and derive semidefinite programming formulation for various eigenvalue optimization problems. We also state complementary slackness results for these problems. Finally, in section 5 we study some applications of SDP interior point methods to various combinatorial optimization problems. These ....

[Article contains additional citation context not shown here]

M. L. Overton and R. S. Womersley, On the sum of the largest eigenvalues of a symmetric matrix, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 41--45.

....derivative of x 7 fm (x) oe m [A(x) when x 7 A(x) is C 1 . We end with an illustrative example in Sect. 5. The present work is based upon the third chapter of the second author s thesis [27] While completing this thesis ( 16] we became aware of preprints by Overton and Womersley [21], 22] where the first order sensitivity analysis of the oe m function has also been considered independently; in particular, a new way to get at Ky Fan s variational formulation is presented in [21] and the sensitivity analysis results on oe m proposed in [22, Sect. 3] are, to a great extent, ....

.... [27] While completing this thesis ( 16] we became aware of preprints by Overton and Womersley [21] 22] where the first order sensitivity analysis of the oe m function has also been considered independently; in particular, a new way to get at Ky Fan s variational formulation is presented in [21], and the sensitivity analysis results on oe m proposed in [22, Sect. 3] are, to a great extent, equivalent to ours. 2 Preliminaries 2.1 Variational formulations for the sums of eigenvalues of a (real) symmetric matrix Let E denote the space of n by n real symmetric matrices, equipped with the ....

Overton, M.L., Womersley, R.S. (June 1991): On the sum of the largest eigenvalues of a symmetric matrix. Preprint, New York university. To appear in SIAM J. on Matrix Analysis and Applications

....be solved in polynomial time by the ellipsoid algorithm [18] since the objective function can be seen to be convex. There is an abundant literature on spectral bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak [35] Building on work by Overton and Womersley [40, 39], Alizadeh [1] has shown that eigenvalue minimization problems can be formulated as semidefinite programs. For MAX CUT, the nonlinear relaxation we consider is equivalent to a spectral bound proposed by Delorme and Poljak [8, 7] This equivalence to the semidefinite program we consider was ....

M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41--45, 1992.

.... this parametrization the bound simplifies to z RW : max u t e=0 min f m 2 tr Z t V t (L diag(u) V Z : Z t Z = I k Gamma1 g: 4) These bounds can be expressed as eigenvalue optimization problems using Fan s Theorem minftr X t AX : X t X = I k g = k X i=1 i (A) see e.g. [25]. We tacitly assume that the eigenvalues of A are in nondecreasing order, i.e. 1 (A) 2 (A) n (A) This was in fact the form in which these bounds were used in [9, 10, 27] We are now going to reformulate (k Gamma GPDH ) and (k Gamma GPRW ) as semidefinite programs. The key step is ....

....cost function by using tr Y t LY = tr L(Y Y t ) and replacing Y Y t by a new variable X . In case of (k Gamma GPDH ) we can optimize over S k : convfY Y t : Y t Y = I k g: In order to do so, we need a nice description of this set. Fortunately, such a description exists. Lemma 3. 1 [25, 11] S k = fX : X = X t ; tr X = k; I X 0g: As observed by Alizadeh [1] the eigenvalue bound z DH can now be reformulated as follows. z DH = max u t e=0 min X2S k m 2 tr (L diag(u) X = min X2S k max u t e=0 m 2 tr LX m 2 u t diag(X) min f 1 2 tr LX : diag(X) e; mI X ....

M.L. OVERTON, and R.S. WOMERSLEY. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis, 13: 41--45, 1992.

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M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 13:41-45, 1992.

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M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 13:41-45, 1992.

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M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41--45, 1992.

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M. L. Overton and R. S. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM Journal on Matrix Analysis and Applications, 13:41--45, 1992. 25

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M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, 13:41-45, 1992.

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M. Overton and R. Womersley. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. on Matrix Analysis and Applications, pages 41--45, 1992.

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M.L. OVERTON and R.S. WOMERSLEY. On the sum of the largest eigenvalues of a symmetric matrix. SIAM J. Matrix Anal. Appl., 13(1):41--45, 1992. 53

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