S. J. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443--461 (electronic), 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....However, the root cause of this problem lies in the primal dual framework used to solve the SDP and it cannot be easily overcome by simply using an iterative method to compute the search direction. For such a problem, it is more appropriate to use methods, such as the dual scaling method in [5], that avoid the need to form X explicitly. In this paper, we will mainly focus on SDPs where m is large, but n is moderate, say, less than 1000. We propose an e#cient preconditioned iterative method to solve the augmented system (4a) 4b) Like the SCE, the augmented system also su#ers from ....

....Unfortunately, unlike the former, the eigenvalue decomposition of the latter is not readily available even if those of X and Z are known. Because of this reason, the augmented system (12) cannot be reduced to the form in (15) for the HRVW KSH M direction. However, for the dual scaling direction in [5], W 1 W 1 is replaced by Z Z and the corresponding reduced augmented system can be found readily once the eigenvalue decomposition of Z is known. 2.1 Nondegeneracy and condition number of the reduced augmented matrix Now we present some examples to illustrate the validity of Theorem 2.4, ....

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S. J. Benson, Y. Ye, and X. Zhang, Solving large-scale sparse semidefinite programs for combinatorial optimization, SIAM J. Optimization, 10 (2000), pp. 443461.

....adaptive choice of #, these methods could just as easily be viewed as potential reduction methods which do not check that the potential function is actually reduced. As a final remark on potential reduction methods, note that, for the more general area of semidefinite programming, Benson et al. [9,10] have shown that dual potential reduction methods can exploit the structure of certain classes of problems more e#ectively than path following methods. 5.4. Exponential gaps To conclude this section, I want to point out an interesting parallel between worst case and typical behaviors of the ....

S. J. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10(2):443--461, 2000.

....to converge in polynomial time in [7] see also [21] Several recent papers have concentrated on exploiting the special structure of the SDP relaxation for the Max Cut problem. A discussion of several of the methods is given in Burer Montreiro [5] See also [32] In particular, Benson et al. [3] present an interiorpoint method based on potential reduction and dual scaling; while, Helmberg Rendl [18] use a bundle trust approach on a nondifferentiable function arising from the Lagrangian dual. Both of these methods exploit the small dimension n of the dual problem compared to the dimension ....

S. J. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optira., 10(2):443-461 (electronic), 2000.

....matrix, to form U and V more efficiently. 2. Form the gradient and Hessian of as follows: These formulas are derived using equation (27) The structure exploited here is similar to the methods used in the dual scaling algorithm for large scale combinatorial opti mization problems, studied in [3]. The total costs of this step (number of flops) is on the order of m 2 (negligible compared with step I and 3) 3. Compute the Newton step H lg by Cholesky factorization and back substitution. The cost of this step is (1 3)m 3 flops. The primal barrier method usually finds the optimal solution ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combi- natoffal optimization. SIAM Journal Optimization, 10:443 461, 2000.

....helmberg zib.de, http: www.zib.de helmberg Technische Universitat Graz, Institut fur Mathematik, Steyrergasse 30, A 8010 Graz, Austria. rendl opt.math.tu graz.ac.at generically dense even if cost and coefficient matrices are very sparse. Very recently, a pure dual approach was proposed in [5] which offers some possibilities to exploit sparsity. It is too early to judge the potential of this method. In combinatorial optimization semidefinite relaxations where introduced in [25] At that time they were mainly considered a theoretical tool for obtaining strong bounds [12, 26, 36] With ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. Working paper, Department of Management Science, University of Iowa, IA, 52242, USA, Sept. 1997.

....problems have been devised using SDP relaxations [22, 26, 34] More recent algorithms for solving the semidefinite programming relaxation are particularly efficient, because they explore the structure of the MAX CUT prob lem. One approach along this line is the use of interior point methods [6, 16, 17]. In particular, Benson, Ye, and Zhang [6] used the semidefinite relaxation for ap proximating combinatoriat and quadratic optimization problems subject to linear, quadratic, and Boolean constraints. They proposed a dual potential reduction atgorithm that exploits the sparse structure of the ....

....[22, 26, 34] More recent algorithms for solving the semidefinite programming relaxation are particularly efficient, because they explore the structure of the MAX CUT prob lem. One approach along this line is the use of interior point methods [6, 16, 17] In particular, Benson, Ye, and Zhang [6] used the semidefinite relaxation for ap proximating combinatoriat and quadratic optimization problems subject to linear, quadratic, and Boolean constraints. They proposed a dual potential reduction atgorithm that exploits the sparse structure of the relaxation. Other nonlinear programming ....

[Article contains additional citation context not shown here]

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combi- natorial optimization. SIAM J. on Optimization, 10:443461, 2000.

....recently proposed in [6] A second observation is that our SDP relaxation has a very particular structure, as exemplified in Figure 1. The question of how to solve SDPs by taking advantage of their intrinsic structure (or sparsity pattern) is currently an area of active research; see for example [7, 15, 33, 22, 25, 26, 8]. Beyond studying ways to exploit the structure of the SDP within a more specialized algorithm, we intend to introduce e#ective use of bounds within our enumerative scheme, and generally improve the e#ciency of our algorithm. Further research work and computational experiments are ongoing. ....

S.J. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443--461 (electronic), 2000.

....success. We present an elegant application of semidefinite programming followed by randomised rounding to this problem, which yields a 0.878 approximation algorithm, whenever the edge weights are nonegative. This algorithm is due to [GW95] We also present the dual potential algorithm due to [BYZ00] which solves the semidefinite programming relaxation in O( # n ln( 1 # ) iterations. The algorithm is thus a polynomial algorithm which also exploits the structure and sparsity of practical Max Cut problems. Semidefinite Programming involves 1. Minimising the inner product of two n n ....

Steven J. Benson, Yinyu Ye, and Xiong Zhang. Solving large scale sparse semidefinite programs for combinatorial optimisation. SIAM Journal on Optimisation, 6 10(2):443--461, 2000.

....X, exactly as we found below (13) and then updating X while holding (y, S) unchanged also can be shown to give a constant decrease in the potential function. It follows that we can attain the iteration complexity bound given in Theorem 6.1. Details can be found in, for example, Benson et al. [8], which describes why this method is attractive for SDP problems arising in combinatorial optimization problems and gives some excellent computational results. Now let us consider a symmetric primal dual method. Suppose we have a strictly feasible point (X, y, S) In addition to the dual ....

S. J. Benson, Y. Ye, and X. Zhang, Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10:443--461, 2000.

....of using primal dual interior point methods is that the nn primal matrix variable X is fully dense in general even when all the data matrices C,A p # S n (p = 1, 2, m) are sparse. This is a disadvantage of primal dual interior point methods 2 compared to the dual scaling method [2] which generates iterates only in the dual space; note that the dual matrix variable S computed by S = # m p=1 A p y p C inherits the sparsity of the data matrices C,A p # S n (p = 1, 2, m) To overcome this disadvantage, Fukuda et al. 11] and Nakata et al. 21] recently ....

....bundle method [13] and nonlinear programming reformulations of SDPs [4, 5, 34] Numerical results on large scale SDPs have been reported. They include (i) SDP relaxations of the max cut problem and the graph bisection problem solved by the spectral bundle method [12, 13] the dual scaling method [2] and nonlinear programming reformulations [4] ii) an SDP relaxation of the max clique problem solved by the primal dual interior point method with the use of the CG method [22] and the CR method [29] However successful numerical results on large scale SDPs have been restricted so far to a few ....

S. J. Benson, Y. Ye, and X. Zhang, "Solving large-scale sparse semidefinite programs for combinatorial optimization," SIAM J. Optim. 10 (2000) 443--461. 38

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Benson, S. J., Ye, Y., and Zhang, X. 2000. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization 10, 2, 443--461.

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S. J. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10(2):443--461, 2000.

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S. J. Benson, Y. Ye, and X. Zhang. Solving large scale sparse semidefinite programs for combinatorial optimization. SIAM Journal of Optimization, 10:443--461, 2000.

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S. J. Benson, Y. Ye, and X. Zhang, Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization. SIAM Journal of Optimization, 10:443--461, 2000.

....optimal objective values of these two problems are equal. Various approaches have been tried to solve these positive semidefinite programs. These approaches include primal dual interior point methods (see Todd [23] for a survey) and a dual scaling interior point method of Benson, Ye, and Zhang [7]. Other approaches include the partial Lagrangian approach of Helmberg and Rendl [17] that uses a spectral bundle method to solve the nondi#erentiable convex program, a penalty approach by Kocvara and Stingl [19] low rank factorizations of Burer and Monteiro [10] and transformation to a ....

....variable does not appear in (1) it does not have to be given to begin the algorithm and it does not have to be computed at each iteration. For notational convenience, we denote M to be matrix on the left hand side of (1) A more detailed explanation and derivation of the algorithm can be found in [7] and [29] Computing the matrix M in (1) and solving the equations are the two most computationally expensive parts of the algorithm. For arbitrary matrices A i and C, 1) can be computed by using O(n m n ) operations and solved by using O(m ) operations, although sparsity in the ....

[Article contains additional citation context not shown here]

S. J. Benson, Y. Ye, and X. Zhang. Solving large scale sparse semidefinite programs for combinatorial optimization. SIAM Journal of Optimization, 10:443--461, 2000.

....from the use of Newton s method, which requires the solution of large, dense linear systems in each iteration. Even with efforts to exploit sparsity in the problem data (see [6] for example) the practical limitations of primal dual interior point methods still remain. Recently, Benson et al. [1] proposed a potential reduction dual scaling interior point method that can better take advantage of the special structure in SDP relaxations of certain combinatorial optimization problems, hence enabling it to solve problems with matrix dimension up to several thousands. Nonetheless, for larger ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10:443-461, 2000.

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S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10:443--461, 2000. 23

....and robust in practice on small to medium sized problems. Even though primal dual path following algorithms can in theory solve semidefinite programs very efficiently, they are unsuitable for solving large scale problems in practice because of their high demand for storage and computation. In [1], Benson et al. have proposed another type of interior point algorithm a polynomial time potential reduction dual scaling method that can better take advantage of the special structure of the SDP relaxations of certain combinatorial optimization problems. Moreover, the efficiency of the ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10:443--461, 2000.

....In other words, a key issue here is the scalability of the SDP relaxation approach with respect to the problem size. There have been a great deal of research efforts towards improving the efficiency of SDP solvers, including works on exploiting sparsity in more traditional interior point methods [1, 9, 16, 17, 29] and works on alternative methods [5, 6, 7, 20, 21, 30, 31] Indeed, the efficiency of SDP solvers has been improved significantly in the last few years. Nevertheless, the scalability problem still remains. On the other hand, computational studies have continued to affirm that the quality of ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization., 10:443--461, 2000.

No context found.

S. J. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443--461 (electronic), 2000.

No context found.

S. J. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443{ 461 (electronic), 2000.

No context found.

S. J. Benson, Y. Ye, and X. Zhang, "Solving large-scale sparse semidefinite programs for combinatorial optimization," SIAM J. Optim. 10 (2000) 443--461. 38

No context found.

S. J. Benson, Yinyu Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443--461 (electronic), 2000.

No context found.

S. J. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim., 10(2):443--461 (electronic), 2000.

No context found.

S.J. BENSON,Y.YE and X. ZHANG, "Solving large scale sparse semidefinite programs for combinatorial optimization", SIAM Journal of Optimization, vol. 10, 2000, pages 443-461.

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