Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm

Simone, Caterina De; Diehl, M.; Jünger, Michael; Mutzel, Petra; Reinelt, Gerhard; Rinaldi, Giovanni

doi:10.1007/bf02178370

Cited by 145 publications

(102 citation statements)

References 16 publications

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“…As test instances we generated Downloaded by [New York University] at 11:49 09 April 2014 several general random sparse SDP problems satisfying the constant trace property as well as semidefinite relaxations of max-cut problems corresponding to Ising spin glasses [27] on three dimensional toroidal grids with edge weights chosen randomly from {−1, 1}.…”

Section: Numerical Resultsmentioning

confidence: 99%

The spectral bundle method with second-order information

Helmberg

Overton

Rendl

2014

Optimization Methods and Software

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The spectral bundle (SB) method was introduced by Helmberg and Rend [A spectral bundle method for semidefinite programming. SIAM J. Optim. 10 (2000), pp. 673-696] to solve a class of eigenvalue optimization problems that is equivalent to the class of semidefinite programs with the constant trace property. We investigate the feasibility and effectiveness of including full or partial second-order information in the SB method, building on work of Overton [On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9(2) (1988), pp. 256-268] and Overton and Womersley [Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM J. Matrix Anal. Appl. 16 (1995), pp. 697-718]. We propose several variations that include second-order information in the SB method and describe efficient implementations. One of these, namely diagonal scaling based on a low-rank approximation of the second-order model for λ max , improves the standard SB method both with respect to accuracy requirements and computation time.

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Section: Numerical Resultsmentioning

confidence: 99%

The spectral bundle method with second-order information

Helmberg

Overton

Rendl

2014

Optimization Methods and Software

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“…For these type of instances the LP based approach appears to be, by far, the method of choice. Choosing the weights randomly from a Gaussian distribution, in [15] the solutions of instances up to 22 500 nodes are reported. For instances with ±1 objective function coefficients drawn from a uniform distribution, solutions are reported for sizes up to 12 100 nodes in [4].…”

Section: Cutting-plane Algorithm For (Mc)mentioning

confidence: 99%

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

2008

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We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a "nearly optimal" solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process.We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem.The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. The Max-Cut ProblemThe Max-Cut problem is one of the basic NP-hard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,

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“…Then, the problem of minimizing the energy of a map corresponds to the problem of finding an exact ground state of a spin glass model (see e.g. [2]). Of course, there are also relations to other fields and problems with a two-dimensional structure, e.g.…”

Section: Introductionmentioning

confidence: 99%

Evolutionary Search for Smooth Maps in Motor Control Unit Calibration

Poland

Knödler

Mitterer

et al. 2001

Lecture Notes in Computer Science

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Abstract. We study the combinatorial optimization task of choosing the smoothest map from a given family of maps, which is motivated from motor control unit calibration. The problem is of a particular interest because of its characteristics: it is NP-hard, it has a direct and important industrial application, it is easy-to-state and it shares some properties of the wellknown Ising spin glass model. Moreover, it is appropriate for the application of randomized algorithms: for local search heuristics because of its strong 2-dimensional local structure, and for Genetic Algorithms since there is a very natural and direct encoding which results in a variable alphabet. We present the problem from two points of view, an abstract view with a very simple definition of smoothness and the real-world application. We run local search, Genetic and Memetic Algorithms. We compare the direct encoding with unary and binary codings, and we try a 2-dimensional encoding. For a simple smoothness criterion, the Memetic Algorithm clearly performs best. However, if the smoothness citerion is more complex, the local search needs many function evaluations. Therefore we prefer the pure Genetic Algorithm for the application.

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Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm

Cited by 145 publications

References 16 publications

The spectral bundle method with second-order information

The spectral bundle method with second-order information

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

Evolutionary Search for Smooth Maps in Motor Control Unit Calibration

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