S. BURER and R.D.C. MONTEIRO. A projected gradient algorithm for solving the maxcut SDP relaxation. Optimization Methods and Software, 15(3-4):175--200, 2001.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are several algorithms to solve the SDP relaxation of MAX CUT problems other than the primal dual interior point methods. The dual scaling method by Benson, Ye, and Zhang ( 5] the spectral bundle method by Helmberg and Rendl ( 13] and nonlinear programming formulation by Burer and Monteiro ([6]) are such algorithms. Most of such algorithms are said to be more e#cient for solving the SDP relaxation of MAX CUT problems, mainly because by exploiting sparsity of the coe#cient matrices. However, their interest seems to solve as large SDP problems as possible, and not to solve the MAX CUT ....

Burer, S., R. Monteiro. (2001) A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Optimization Methods and Software 15, 175-200.

....was introduced in [20] A scaled variant was shown 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 ....

....Moreover, thedualfeasibility equation is sparse if the matrix of the quadratic form Q is sparse. Therefore each iteration is inexpensive. However, these are not primal dual methods and do not easily obtain high accuracy approximations of optimal solutions without many iterations. The methods in [5, 32] use Cholesky type transformations X = VV and discard the semidefinite constraint. For example, the method in [5] reduces to a first order gradient projection method. They argue for using a search direction based on first order information only (rather than second order information as used in ....

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S. BURER and R.D.C. MONTEIRO. A projected gradient algorithm for solving the maxcut SDP relaxation. Optimization Methods and Software, 15(3-4):175-200, 2001.

....presented for the MAX CUT semidefinite programming relaxation [24, 25] Homer and Peinado [25] refor mutated the constrained problem as an unconstrained one and used the standard steepest ascent method on the latter. A variant of the Homer and Peinado algorithm was proposed by Burer and Monteiro [7]. Their idea is based on the constrained nonlinear programming reformutation of the MAX CUT semidefinite programming relaxation obtained by a change of variables. More recently, Burer, Monteiro, and Zhang [8] proposed a rank 2 relaxation heuristic for MAX CUT and described a computer code, called ....

....randomized heuristics on test problems G1, 62, 63, 611, 612, 613, 614, 615, 616, 622, 623, 624, 632, 633, 634, 635, 636, 637, 643, 644, 645, 648, 649, and 650. These test problems were created by Helmberg and Rendl [24] using a graph generator written by Rinaldi and were used by Burer and Monteiro [7], Benson et al. 5] and Burer, Monteiro, and Zhang [8] for testing their algorithms. They consist of toroidal, planar, and randomly generated graphs of varying sparsity and sizes. These graphs vary in size from 800 to 3000 nodes and in density from 0.17 to 6.12 . We first ran the randomized ....

S. Burer and R.D.C. Monteiro. A projected gradient algorithm for solving the Max-Cut SDP relaxation. Optimization Methods and Software, 15:175-200, 2001.

....for larger problems this approach still encounters formidable difficulty because it, like the related primal dual methods, requires the storage and factorization of a dense matrix. In the past few years, several nonlinear programming methods have been proposed for solving large SDP problems (see[2,9,10]) and a common feature of these methods is that each relies only on gradient based information and consequently avoids costly matrix operations and linear solves. The approach by Helmberg and Rendl [9] for solving a special class of linear SDPs is to optimize a certain partial Lagrangian dual ....

....[10] use the change of variables X = V V T , V 2 n Thetan , where X is the primal matrix variable of the MAXCUT SDP relaxation (see Section 2) to transform it into an unconstrained, differentiable nonlinear programming problem in the new variable V . More recently, Burer and Monteiro [2] proposed a variant of Homer and Peinado s method by using the change of variable X = LL T , where L is a lower triangular matrix. We note that the substitutions X = V V T and X = LL T can be viewed as a matrix analogy to the square slack variable substitution for scalar inequality ....

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S. Burer and R. D. C. Monteiro. A projected gradient algorithm for solving the MAXCUT SDP relaxation. Optimization Methods and Software, 15:175-200, 2001.

....methods increases dramatically with the problem size. In fact, on many problems these methods are inappropriate for obtaining even low accuracy solutions. In contrast, first order methods have proven capable of obtaining moderate accuracy in a reasonable amount of time for large scale problems [3, 16, 17]. Based on a nonlinear transformation, we recently proposed a first order, log barrier method for solving a class of large scale SDPs and established its global convergence [4] The main purpose of this paper is to study the implementation issues for this algorithm and to report our computational ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998. (To appear in Optimization Methods and Software.)

....relaxation, i.e. not doing the change of variables X = V V T , where V 2 n Thetan is the original variable, they show how the maxcut SDP can be reformulated as an unconstrained maximization problem for which a standard steepest ascent method can be used. Burer and Monteiro [3] improved upon the idea of Homer and Peinado by simply noting that, without loss of generality, V can be required to be lower triangular. More recently, Vavasis [14] has shown that the gradient of the classical log barrier function of the dual maxcut SDP can be computed in time and space ....

...., where n is the size of matrix variable X , m is a problem dependent, nonnegative integer, and n is the positive orthant of n . The reformulation is based on the idea of eliminating the positive definiteness constraint on X by first applying the substitution X = LL T as was done in [3], where L is a lower triangular matrix, and then using a novel elimination scheme to reduce the number of variables and constraints. We also showed how to compute the gradient of the resulting nonlinear objective function efficiently, hence enabling the application of existing nonlinear ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998. (To appear in Optimization Methods and Software.)

....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 bounds produced by the SDP relaxation is quite high. For ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998. (To appear in Optimization Methods and Software.)

....shown in [13] how the change of variables X = V V T , where V is a real square matrix having the same size as X, allows one to recast the SDP as an unconstrained optimization problem for which any standard nonlinear method in particular, a first order method can be used. Burer and Monteiro [4] improved upon the idea of Homer and Peinado by simply noting that, without loss of generality, V can be required to be lower triangular in accordance with the Cholesky factorization. Then, in a series of papers [6, 7, 5] Burer, Monteiro, and Zhang showed how one could apply the idea of Cholesky ....

....an interior point technique based on Lagrangian duality which solves the class of SDPs studied in [12] and allows the use of first order methods in the unrestricted space of Lagrange multipliers. The current paper follows the path laid by these alternative methods and is specifically motivated by [13, 4], that is, we consider the use of first order methods for solving the nonlinear reformulation of an SDP obtained by replacing the positive semidefinite variable with an appropriate factorization. We work with the standard form primal SDP min C . X : A i . X = b i , i = 1, m, X # 0 , ....

S. Burer and R.D.C. Monteiro. A projected gradient algorithm for solving the maxcut SDP relaxation. manuscript, School of ISyE, Georgia Tech, Atlanta, GA, 30332, USA, December 1998. To appear in Optimization Methods and Software.

....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 approach ever become competitive in approximating large scale ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998. 14

....of the SDP relaxation approach with respect to the problem size. There have been a great deal of research e#orts towards improving the e#ciency 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 e#ciency of SDP solvers has been improved significantly 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 approach ever become competitive in approximating large scale ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998. 14

....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, 8, 12, 13, 23] and works on alternative methods [3, 4, 5, 6, 15, 16, 24, 25]. Indeed, the efficiency of SDP solvers has been improved significantly 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 approach ever become competitive in approximating large scale ....

S. Burer and R. D. C. Monteiro. A Projected Gradient Algorithm for Solving the Maxcut SDP Relaxation. Working paper, School of ISyE, Georgia Tech, USA, December 1998.

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S. BURER and R.D.C. MONTEIRO. A projected gradient algorithm for solving the maxcut SDP relaxation. Optimization Methods and Software, 15(3-4):175--200, 2001.

No context found.

S. Burer and R.D.C. Monteiro, A projected gradient algorithm for solving the maxcut SDP relaxation, Optim. Methods Softw., 15(2001), 175-200.

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S. Burer and R.D.C. Monteiro. A projected gradient algorithm for solving the MaxCut SDP relaxation. Optimization Methods and Software, 15: 175-200, 2001.

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S. Burer and R. D. C. Monteiro. A projected gradient algorithm for solving the maxcut SDP relaxation. Optim. Methods Softw., 15(3-4):175--200, 2001.

No context found.

S. Burer and R. Monteiro. "A projected gradient algorithm for solving the maxcut SDP relaxation". Optimization Methods and Software, 15 (2001) 175-200.

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