Abstract
One usually tries to raise the efficiency of optimization techniques by changing the dynamics of local optimization. In contrast to the above approach, we propose changing the surface of the problem rather than the dynamics of local search. The Mix-Matrix algorithm proposed by the authors previously [1] realizes such transformation and can be applied directly to a max-cut problem and successfully compete with other popular algorithms in this field such as CirCut and Scatter Search.
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References
Karandashev, I., Kryzhanovsky, B.: The Mix-Matrix Method in the Problem of Binary Quadratic Optimization. In: Villa, A.E.P., Duch, W., Érdi, P., Masulli, F., Palm, G. (eds.) ICANN 2012, Part I. LNCS, vol. 7552, pp. 41–48. Springer, Heidelberg (2012)
Liers, F., Junger, M., Reinelt, G., Rinaldi, G.: Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch-and-Cut. In: New Optimization Algorithms in Physics, pp. 47–68. Wiley (2004)
Goemans, M.X., Williamson, D.P.: 878-approximation Algorithms for MAXCUT and MAX2SAT. In: ACM Symposium on Theory of Computing (STOC) (1994)
Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and nonapproximability-towards tight results. In: Proc. of 36th IEEE Symp. on Foundations of Computer Science, pp. 422–431 (1995)
Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to Optimality by Intersecting Semidefinite and Polyhedral Relaxations. Math. Programming 121(2), 307 (2010)
Wiegele, A.: Nonlinear Optimization Techniques Applied to Combinatorial Optimization Problems. Dissertation, i-x, pp. 1-131 (October 2006)
Hopfield, J.J.: Neural Networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79, 2554–2558 (1982)
Hopfield, J.J., Tank, D.W.: Neural computation of decisions in optimization problems. Biological Cybernetics 52, 141–152 (1985)
Fu, Y., Anderson, P.W.: Application of statistical mechanics to NP-complete problems in combinatorial optimization. Journal of Physics A 19, 1605–1620 (1986)
Poggio, T., Girosi, F.: Regularization algorithms for learning that are equivalent to multilayer networks. Science 247, 978–982 (1990)
Mulder, S., Wunsch II, D.: A Million City Traveling Salesman Problem Solution by Divide and Conquer Clustering and Adaptive Resonance Neural Networks. Neural Networks 16(5-6), 827–832 (2003)
Wu, F., Tam, P.K.S.: A neural network methodology of quadratic optimization. International Journal of Neural Systems 9(2), 87–93 (1999)
Pinkas, G., Dechter, R.: Improving Connectionist Energy Minimization. Journal of Artificial Intelligence Research 3(195), 23–48 (1995)
Kryzhanovsky, B.V., Magomedov, B.M., Mikaelyan, A.L.: A Relation Between the Depth of a Local Minimum and the Probability of Its Detection in the Generalized Hopfield Model. Doklady Mathematics 72(3), 986–990 (2005)
Kryzhanovsky, B.V., Magomedov, B.M.: Application of domain neural network to optimization tasks. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds.) ICANN 2005. LNCS, vol. 3697, pp. 397–403. Springer, Heidelberg (2005)
Hartmann, A.K., Rieger, H. (eds.): New Optimization Algorithms in Physics. Wiley-VCH, Berlin (2004)
Duch, W., Korczak, J.: Optimization and global minimization methods suitable for neural networks. KMK UMK Technical Report 1/99; Neural Computing Surveys (1998), http://www.is.umk.pl/~duch/cv/papall.html
Hartmann, A., Rieger, H.: Optimization Algorithms in Physics. Wiley-VCH, Berlin (2001)
Litinskii, L.B.: Eigenvalue problem approach to discrete minimization. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds.) ICANN 2005. LNCS, vol. 3697, pp. 405–410. Springer, Heidelberg (2005)
Smith, K.A.: Neural Networks for Combinatorial Optimization: A Review of More Than a Decade of Research. INFORMS Journal on Computing 11(1), 15–34 (1999)
Joya, G., Atencia, M., Sandoval, F.: Hopfield Neural Networks for Optimization: Study of the Different Dynamics. Neurocomputing 43(1-4), 219–237 (2002)
Litinskii, L.B., Magomedov, B.M.: Global Minimization of a Quadratic Functional: Neural Networks Approach. Pattern Recognition and Image Analysis 15(1), 80–82 (2005)
Boettecher, S.: Extremal Optimization for Sherrington-Kirkpatrick Spin Glasses. Eur. Phys. Journal B. 46, 501 (2005)
Kryzhanovsky, B.V., Magomedov, B.M., Fonarev, A.B.: On the Probability of Finding Local Minima in Optimization Problems. In: Proc. of Int. Joint Conf. on Neural Networks IJCNN 2006, pp. 5888–5892 (2006)
Kryzhanovsky, B.V., Kryzhanovsky, V.M.: The shape of a local minimum and the probability of its detection in random search. In: Filipe, J., Ferrier, J.-L., Andrade-Cetto, J. (eds.) Informatics in Control, Automation and Robotics. LNEE, vol. 24, pp. 51–61. Springer, Heidelberg (2009)
Houdayer, J., Martin, O.C.: Hierarchical approach for computing spin glass ground states. Phys. Rev. E 64, 56704 (2001)
Marti, R., Duarte, A., Laguna, M.: Advanced Scatter Search for the Max-Cut Problem. INFORMS Journal on Computing 01(21), 26–38 (2009)
Burer, S., Monteiro, R.D.C., Zhang, Y.: Rank-Two Relaxation Heuristics for Max-Cut and Other Binary Quadratic Programs. SIAM Journal on Optimization 12, 503–521 (2000)
Festa, P., Pardalos, P.M., Resende, M.G.C., Ribeiro, C.C.: Randomized heuristics for the max-cut problem. Optim. Methods Software 7, 1033–1058 (2002)
Krishnan, K., Mitchell, J.E.: A Semidefinite Programming Based Polyhedral Cut and Price Approach for the Maxcut Problem. Comput. Optim. Appl. 33(1), 51–71 (2006)
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Karandashev, I., Kryzhanovsky, B. (2014). Mix-Matrix Transformation Method for Max-Сut Problem. In: Wermter, S., et al. Artificial Neural Networks and Machine Learning – ICANN 2014. ICANN 2014. Lecture Notes in Computer Science, vol 8681. Springer, Cham. https://doi.org/10.1007/978-3-319-11179-7_41
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DOI: https://doi.org/10.1007/978-3-319-11179-7_41
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