J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the dimension of the subspace is bounded by (roughly) the square root of the number of constraints. If this is still considered too large the introduction of an aggregate subgradient guarantees convergence even in case of one dimensional subspaces. In contrast to classical algorithms [7, 32, 31] for eigenvalue optimization our method does not require the knowledge of the correct multiplicity of the maximal eigenvalue in the optimal solution. On the other hand [32, 31] show that their algorithms are quadratically convergent in the limit. A similar property cannot be expected to hold for ....

J. Cullum, W. E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

.... denominator matrix is constant, the problem reduces to minimizing the maximum eigenvalue of a symmetric matrix that depends affinely on x. In this case, the problem is in fact convex (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 ....

J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

....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 ....

J. Cullum, W.E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....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 ....

J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

....METHOD 3 is still considered too large the introduction of an aggregate subgradient guarantees convergence for restricted bundle sizes. In the extreme the bundle may consist of one new eigenvector to the maximal eigenvalue only. In contrast, the classical algorithms of Cullum, Donath, and Wolfe [6] and Polak and Wardi [38] require in each iteration the computation of all eigenvectors to eigenvalues within an distance of the maximal eigenvalue, thus close to the optimal solution this number is at least as large as the multiplicity of the maximal eigenvalue in the optimal solution. In the ....

....point here is that instead of optimizing over all semidefinite matrices W , we constrain ourselves to a small subset. Remark 3.1. If we set W = 0 and use for the matrix P a set of eigenvectors to the r largest eigenvalues at y, we would end up with a model closely related to the approach from [6]. In this case it would be important to select r at least as large as the multiplicity of the largest eigenvalue. In our present approach this is not necessary. 6 C. HELMBERG AND F. RENDL 3.2. Proximal point idea. The next goal is to minimize f instead of f . Since f is built from local ....

J. Cullum, W. E. Donath, and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programming Study, 3 (1975), pp. 35-- 55.

....example of such problems were studied by Donath and Hoffman in connection with graph bisection and graph partitioning problems [DH72, DH73] see section 5 below. Cullum, Donath and Wolfe studied the problem of minimizing the sum of the first few eigenvalues of a linearly constrained matrix in [CDW75]. They analyzed this problem from the nonsmooth optimization point of view. Also Fletcher studied 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 ....

J. Cullum, W. E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalue problems. Mathematical Programming Study, 3:35--55, 1975. 34

.... Fletcher [Fle85] Allwright [All88] Wolkowicz [Wol81] and Kojima, 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 ....

J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

....in recent control conferences. Special classes of the SDP have a long history in optimization as well. For example, certain eigenvalue minimization problems that can be cast as SDPs have been used for obtaining bounds and heuristic solutions for combinatorial optimization problems (see [7, 8] and [9, Chapter 9] 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 ....

.... 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 ....

J. Cullum, W. Donath, and P. Wolfe, "The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices", Math. Programming Study, vol. 3, pp. 35--55, 1975.

....function of the matrix elements. Because of this fact, it has been recognized for some time that the techniques of convex analysis (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 ....

....(x)i; 7) for some U 2 S t Thetat ; U 0; tr U = 1g: 6 Proof. By the chain rule just cited, 1 (x) fv 2 m : v k = hG; A k (x)i for some G 2 1 (A)g: The proof is completed by using (6) and noting that hQ 1 UQ T 1 ; A k i = hU; Q T 1 A k Q 1 i:2 Equation (4) is well known; see [CDW75,PW83,Cla83] The equivalent form (6) is much less known and much more useful, as we shall see shortly; the earliest reference we know for this explicit form is Fletcher [Fle85] where a different proof was given. Equation (7) was given in the case that A(x) is affine in [Ove88] using a proof ....

J. Cullum, W.E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

.... where f k (A(x) is the sum of the k largest eigenvalues of A(x) The problem (EV k ) can be formulated as a semidefinite program, as it was shown by Alizadeh [1] and Nesterov and Nemirovskii [24] In fact, it is the earliest instance of SDP that has been the subject of a computational study; see [12, 14] In these studies, and in many more recent papers dealing with eigenvalue optimization, the following phenomenon was observed. At optimal solutions of (EV k ) the eigenvalues of the optimal matrix tend to coalesce; if x achieves the minimum, then frequently k (A(x ) k 1 (A(x ) and ....

....then frequently k (A(x ) k 1 (A(x ) and k (A(x ) often has multiplicity even larger than two. GEOMETRY OF CONE LP s: in HANDBOOK OF SEMIDEFINITE PROGRAMMING 26 The clustering phenomenon plays a central role in eigenvalue optimization. As proven by Cullum, Donath, and Wolfe [12] the function f k is differentiable at a fixed symmetric matrix a if and only if k (a) k 1 (a) If this condition fails to hold, then the dimension of the subgradient of f k at a grows quadratically with the multiplicity of k (a) Furthermore, if f k is nonsmooth at A(x ) then generally ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

.... Gamma V )y e t v : y t y = n 1g = n 1) max (Q c Gamma V ) e t v: So we get B eo : minf(n 1) max (Q c Gamma V ) e t v : v 2 IR n 1 g: This is a nonsmooth but convex optimization problem, and has been studied thoroughly in nonsmooth optimization; see for instance [9, 46]. We can also apply semidefinite programming relaxations, as described at the beginning. The semidefinite relaxation of the homogenized problem is B sd : maxftr Q c Y : Y 0; diag(Y ) eg: Finally we can use the Minimax inequality to get the Lagrangian Dual B ld : max x min u q u (x) ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....i.e. zero, variables) However, it is not possible to formulate separate differentiable bounds on individual eigenvalues. Min max eigenvalue problems can be formulated as an SDP, i.e. one can maximize a variable subject to A Gamma I 0: An early example of applications of min max eigenvalues is [15]. The converse is not true in general, but is true when the trace is a constant function on the feasible set, see e.g. 26] This can be used to exploit structure and solve large scale problems. For more on eigenvalue optimization, see [37] 4.3 Engineering Many early applications of SDP ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....requirements, i.e. between total complexity and speed of convergence. Starting directly from problem (P ) itself, we will use a recent second order theory, namely the U Lagrangian theory [24] to speed up the asymptotic convergence of a first order method developed by Cullum, Donath and Wolfe [6] for a particular instance of (P ) A diagonal) and by Polak and Wardi [37] in a more general framework. Using the terminology of [16, Chap.XIII] the method can be seen as a Markovian dual bundle method: at each iteration an approximation of the subdifferential is computed, via a bundling ....

....infinite max function. Then a first idea could be to consider the enlargement proposed in [8, Chap. VI] the convex hull of the gradients of active functions. Here the functions are linear and it is easy to see, via (7) that the obtained enlargement is exactly the subdifferential of 1 . In [6], J. Cullum, W. E. Donath and P. Wolfe introduced a smaller set: they considered the eigenvectors of eigenvalues at a distance from 1 (X) and the convex set generated by the associated rank one matrices. Definition 2.6 For X 2 S n and 0 we define ffl the set of indices of largest ....

[Article contains additional citation context not shown here]

J. Cullum, W. E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

....: Gamma ln det (X) is a self concordant barrier function for the cone of symmetric semidefinite matrices. Some applications of semidefinite optimization were already known for a long time, for instance, minimizing the maximal eigenvalue of a linear combination of matrices (see Cullum et al. [10]) New applications are found in system and control theory (see e.g. Boyd and Vandenberghe [8] and Boyd et al. 8] structural optimization (e.g. Ben Tal and Bendsoe [7] geometrical problems involving quadratic forms or ellipsoids (e.g. Boyd et al. 8] statistics (e.g. minimum trace factor ....

J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....f . This observation will be used later on. For computational purposes it is preferable to have a smaller dimensional set over which to minimize. From a theoretical viewpoint it seems easier to work with an unconstrained problem. The function f has several well known and often used properties, see [9, 21], which we summarize below. Proposition 3.1 The function f is (a) convex, b) bounded from below, c) continuous, d) differentiable if and only if k ( k Gamma1 ( In this case d j (f) e t j V n ZZ t V t n e j Gamma k Gamma 1 n ; where Z contains an orthonormal set of ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....more complicated) Let A(d) A V t n diag(d)V n ; and g(d) k Gamma1 X j=1 j ( A(d) Donath and Hoffman [6] point out that k X j=1 j (A B) is a convex function of B for A fixed, provided both A and B are symmetric. Therefore g(d) is convex. Moreover, using Theorem 4. 6 in [15], it is easy to verify that g(d) is differentiable for all d such that k Gamma1 ( A(d) 6= k ( A(d) Note also that under the assumption s(d) 0 it is easily shown that lim kdk 1 g(d) 1: 5.1) The above discussion is summarized as follows. Lemma 5.3 Suppose m = n k u k : Then jE ....

....most of the iterations are spent finding a subgradient of small norm. Moreover, it turns out that at the final perturbation d, the largest eigenvalue has multiplicity larger than 1, see also Table 2, therefore g(d) is nondifferentiable for this d. This coincides with the experiences reported in [15]. Finally we point out that the maximizer X , producing the bound, is already very close to a 0; 1 matrix, see Table 3. We conclude this section with a perturbation of the main diagonal of A that allows an application of Corollary 4.2. Theorem 5.1 Let A and m describe a graph partitioning ....

[Article contains additional citation context not shown here]

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....generically the dimension of the subspace is bounded by (roughly) the square root of the number of constraints. If this is still considered too large the introduction of an aggregate subgradient guarantees convergence even in case of one dimensional subspaces. In contrast to classical algorithms [7, 32, 31] for eigenvalue optimization our method does not require the knowledge of the correct multiplicity of the maximal eigenvalue in the optimal solution. On the other hand [32, 31] show that their algorithms are quadratically convergent in the limit. A similar property cannot be expected to hold for ....

J. Cullum, W. E. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

....An early example of such problems were studied by Donath and Hoffman in connection with graph bisection and graph partitioning problems [17, 18] see section 5 below. Cullum, Donath and Wolfe studied the problem of minimizing the sum of the first few eigenvalues of a linearly constrained matrix in [15]. They analyzed this problem from the point of view of nonsmooth optimization. Also Fletcher studied a similar problem and derived expressions for the subgradients of the sum of the first few eigenvalues of a symmetric matrix and formulated optimality conditions for this problem. In the same ....

J. Cullum, W. E. Donath, and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalue problems, Math. Programming Stud., 3 (1975), pp. 35--55.

....The objective function in (6.12) is not differentiable when the multiplicity of the largest eigenvalue exceeds one. In fact, a singleton eigenvalue characterizes differentiability. Since the largest eigenvalue is a convex function, subgradient approaches can be used to solve (6. 12) see, e.g. [8]) More recently, it has been shown that Newton based algorithms with local quadratic convergence exist (see, e.g. 23] but the local convergence depends on correctly identifying the multiplicity of the largest eigenvalue. We present computational experiments showing that our interior point ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....note: an influential volume in the early history of Math Programming Study, the predecessor of Math Programming Series B, was that on nondifferentiable optimization (Volume 3, 1975, M.L. Balinski and P. Wolfe, eds) That volume included one of the early papers on matrix eigenvalue optimization [4]. SDP is an eigenvalue optimization problem in the sense that the semidefinite constraint is equivalent to nonnegative bounds on eigenvalues. Solutions to SDP generally have multiple zero eigenvalues (just as solutions to LP have many nonbasic, i.e. zero, variables) However, it is not possible to ....

J. CULLUM, W.E. DONATH, and P.WOLFE. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Mathematical Programming Study, 3:35--55, 1975.

....Indeed, spectrahedra may be considered next natural successors to polyhedra, as one moves beyond linear constraints in optimization theory. 1. 1 Background Historically, semidefinite programming has been studied in more general contexts such as convex and cone programming (see [BCK69] BW81] [CDW75] and [Wol81] See also [Fle85] and [Ove92] Further references can be found in [Ali94] However, the more recent surge of interest in SDP was primarily inspired by the work of [GLS84] see [GLS88] Chapter 9) In this work, the authors associate with every graph G, a convex set denoted by TH(G) ....

J. Cullum, W.E. Donath and P. Wolfe, The Minimization of Certain Nondifferentiable Sums of Eigenvalue Problems, Math. Prog. Study, 3(1975), pp. 35-55.

....or (some) intermediate eigenvalues arise in many applications, among others structural optimization problems (cf. Haug et al. 13] Haug and Rousselet [14] Laborde [19] Overton [20] Rousselet and Chenais [25] Polak and Wardi [23] etc. graph partitioning problems (cf. Cullum et al. [4]) inverse eigenvalue problems (see Friedland et al. 11] and references therein) More traditional areas of applications of eigenvalue sensitivity results are systems engineering (where controlling 1 [A(x) is of utmost importance) and statistics (analysis of data) Rather recently there has ....

....in studying generalized differentiability (in whatever sense) and (possible) directional derivatives of either fm or m , but not in the most general setting. The first paper dealing with the sensitivity of fm (using tools from convex analysis) seems to be the one by Cullum, Donath and Wolfe [4], where the (varying) parameter consisted of the diagonal elements of a given symmetric matrix; in particular, they provided a formula for the subdifferential (in the sense of convex analysis) of fm . In a truly interesting paper (which partly motivated our study) Fletcher [10, Appendix] ....

Cullum, J., Donath, W.E., Wolfe, P. (1975): The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study 3, 35--55

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J. Cullum, W. Donath, and P. Wolfe. The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Programming Study, 3:35--55, 1975.

No context found.

J. E. Cullum, W. E. Donath and P. Wolfe (1975), "The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices", Mathematical Programming Study 3, pp. 35-55.

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Cullum, J., W.E. Donath, P. Wolfe (1975). The minimization of certain nondifferentiable sums of eigenvalue problems. Math. Programming Stud. 3 35-55

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