S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semide nite programs for combinatorial optimization. Technical report, Dept. of Mgmt. Sc., Univ. of Iowa,, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm. In the worst case, the complexity of this step is O(n 6 ) where n is the number of nodes in the graph. However, for problems where the constraint matrices F i are relatively sparse as is the case with max cut sparse matrix techniques and iterative methods can speed up this step [11, 15]. The results of Helmberg and Rendl [24] lead us to speculate that disjunctive cuts together with suitably designed interior point algorithms could yield an ecient cutting plane algorithm for solving max cut and related quadratic optimization problems. 4.2 Computational results for the traveling ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidenite programs for combinatorial optimization. Technical report, Dept. of Mgmt. Sc., Univ. of Iowa,, 1997.

....1 main attractions of semide nite programming. We present a small selection in Section 6. In Section 7, we sketch the algorithms which, in our opinion, dominate in current implementational e orts, namely, primal dual interior point algorithms [19, 26, 32, 1] a potential reduction algorithm[5], and the spectral bundle method[18] We also provide some guidelines on what classes of problems they might best be employed. We conclude by giving a short outlook in Section 8. 2 The Cone of Positive Semide nite Matrices We rst review some basic notions form linear algebra. Unless stated ....

....The factorization of M needs m 3 =3 operations and is usually the most expensive step in each iteration. If n is of the same order of magnitude as m, then the line search in step 3 of Algorithm 7.7 is as important as the factorization of M . For a large but structured Z, Benson, Ye, and Zhang [5] pointed out that a dual potential reduction method, i.e. a pure dual method, may help to save work. Linearizing X = Z 1 (instead of XZ = I) yields X X = Z 1 Z 1 ZZ 1 (21) in (14) For this choice and strictly feasible Z = C A T (y) 0, 20) transforms to A(Z 1 A T ( y)Z ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidenite programs for combinatorial optimization. Working paper, Department of Management Science, University of Iowa, IA, 52242, USA, Sept. 1997. To appear in SIAM J. Opt.

....is typically dominated by the factorization of a dense symmetric positive de nite matrix of order m, and by one or more factorizations of the variable X during the line search. The solution times for problems with m 5000 or n 500, say, are prohibitive. The recent purely dual approach of [2] is able to exploit the sparsity of C and A i for the max cut relaxation (see also [3] It is not yet clear whether this approach could be extended to problems with a huge number of less structured constraints. The alternative approach of [9] transforms (D) into the eigenvalue optimization ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidenite programs for combinatorial optimization. Working paper, Department of Management Science, University of Iowa, IA, 52242, USA, Sept. 1997. To appear in SIAM J. Opt.

....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 that avoid the need to form X explicitly. One such method is the dual scaling interior point method proposed in [3], where the method is able to solve large SDPs (with m;n 2000) arising from MAXCUT problems. Another method that avoids the explicit formation of X is the spectral bundle method proposed in [10] This is a rst order method that is designed for SDPs where Trace(X) is a constant. The authors ....

.... on the earlier work, a preconditioned conjugate residual method was proposed to solve the SCE in [17] Recently, Choi and Ye [5] also reported computational results for large SDPs arising from MAXCUT problems (with n up to 14000) They used the dual scaling interior point method described in [3], but solve the associated SCE in each iteration by a preconditioned conjugate gradient method. Earlier works on using preconditioned conjugate gradient methods to solve the SCE in interior point methods for SDP include [14] and [24] As far as we are aware, all the earlier works mentioned above ....

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S. J. Benson, Y. Ye, and X. Zhang, Solving large-scale sparse semidenite programs for combinatorial optimization, SIAM J. Optimization, 10 (2000), pp. 443-461.

....they are too expensive for solving larger instances of (QK) to optimality. However, the approach gives some insight on important classes of inequalities that may help to set up good initial relaxations of more complex problems. This is of particular importance in combination with new methods [5, 17] that can compute approximate solutions to semide nite relaxations for large problem instances as well. 2 Modeling Linear Constraints for Semide nite Relaxations A common approach for designing relaxations for quadratic 0 1 programs is to linearize the quadratic cost function by switching to ....

S. BENSON, Y. YE, and X. ZHANG. Solving large-scale sparse semidenite programs for combinatorial optimization. Working paper, Department of Management Science, University of Iowa, IA, 52242, USA, Sept. 1997.

....results for relaxations of MC instances with up to n = 3000 nodes. A detailed survey of their work and related results appears in [46] The min max eigenvalue approach for more general SDPs is discussed in Section 3.1. Finally, back in the realm of interior point methods, Benson, Ye and Zhang [16] derived and implemented an ecient and promising potentialreduction ane scaling algorithm to solve DSDP1. This polynomial time algorithm generates the Newton system very quickly by virtue of the special structure of the n linear constraints of SDP1. This approach is further improved by Choi and Ye ....

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

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

....Real applications of these problems can be particularly large. Various approaches have been tried to solve these problems. These approaches include primal dual interior point methods (see Todd [18] for a survey of these methods) and a dual scaling interior point method of Benson, Ye, and Zhang [3]. Other types of methods that have been developed and applied to combinatorial problems, such as the maximum cut problem, include the partial Lagrangian approach of Helmberg and Rendl [12] which uses a spectral bundle method to solve the nondi erentiable convex program, and transformation to a ....

.... nonlinear program which was rst proposed by Homer and Peinado [13] and further developed by Burer and Monteiro [7] and Burer, Monteiro, and Zhang [8] 9] Details concerning the convergence of the feasible start dual scaling algorithm and its advantages over primal dual methods can be found in [3] and [21] The advantages of the algorithm are as follows 1. The cost of each iteration is relatively low. For many problems, the computation of the primal matrix X requires considerable computational e ort, which is unnecessary in this algorithm. 2. The memory requirements of the dual algorithm ....

[Article contains additional citation context not shown here]

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

....In a nutshell, the central issue is the scalability of the SDP relaxation approach with respect to the problem size. There have been a great deal of research e orts towards improving the eciency of SDP solvers, including works on exploiting sparsity in more traditional interior point methods [1, 8, 12, 13, 22] and works on alternative methods [3, 4, 5, 6, 15, 16, 23, 24] Indeed, the eciency of SDP solvers has been improved signi cantly in the last few years. Nevertheless, the scalability problem still remains. Can the scalability problem of the SDP relaxation be overcome Can the SDP relaxation ....

....of the theorem can be proved in a similar way using the second equivalence of Lemma 1. Hence, the result follows. The converses of the two implications in the above theorem do not hold. Indeed, consider the unweighted graph K 3 (the complete graph with three nodes) for which the cut x = [1 1 1] T is maximum. From (12) we have M( x) 2 4 2 1 1 1 0 1 1 1 0 3 5 ; which is indeed nonnegatively summable, but not positive semi de nite. Hence, the corresponding angular representation is not a local minimum of the function f( in view of the fact that M( x) r 2 f( ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidenite programs for combinatorial optimization. SIAM J. Optim., Vol. 10, No. 2, pp. 443-461, 2000.

....ecient and robust in practice on small to medium sized problems. Even though primal dual path following algorithms can in theory solve semide nite programs very eciently, 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 eciency of the algorithm ....

S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidenite programs for combinatorial optimization. Research Report, Department of Management Science, University of Iowa, Iowa, 1998.

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S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semide nite programs for combinatorial optimization. Technical report, Dept. of Mgmt. Sc., Univ. of Iowa,, 1997.

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