Abstract
We show how to solve network combinatorial optimization problems using a randomized algorithm based on the cross-entropy method. The proposed algorithm employs an auxiliary random mechanism, like a Markov chain, which converts the original deterministic network into an associated stochastic one, called the associated stochastic network (ASN). Depending on a particular problem, we introduce the randomness in ASN by making either the nodes or the edges of the network random. Each iteration of the randomized algorithm based on the ASN involves the following two phases:
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1.
Generation of trajectories using the random mechanism and calculation of the associated path (objective functions) and some related quantities, such as rare-event probabilities.
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2.
Updating the parameters associated with the random mechanism, like the probability matrix P of the Markov chain, on the basis of the data collected at first phase.
We show that asymptotically the matrix P converges to a degenerated one P* d in the sense that at each row of the MC P* d only a single element equals unity, while the remaining elements in each row are zeros. Moreover, the unity elements of each row uniquely define the optimal solution. We also show numericaly that for a finite sample the algorithm converges with very high probability to a very small subset of the optimal values. We finally show that the proposed method can also be used for noisy networks, namely where the deterministic edge distances in the network are replaced by random variables with unknown expected values. Supporting numerical results are given as well. Our numerical studies suggest that the proposed algorithm typically has polynomial complexity in the size of the network.
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References
Aarts, E.H.L. and J.H.M. Korst (1989), Simulated Annealing and Boltzmann Machines,John Wiley and Sons.
Aarts, E.H.L. and J.K. Lenstra (1997), Local Search in Combinatorial Optimization, Wiley, Chichester.
Abdullah, W.A.T.W. (1994), “Seeking global minima”, Journal of Computational Physics, 110.
Ahuja, R.K., Magnanti, T.L. and J.B. Orlin (1993), Network Flows Theory, Algorithms and Applications,Prentice Hall.
Ali, S.M. and S.D. Silvey (1966), “A general class of coefficients of divergence of one distribution from another”, Journal of the Royal Statistical Society, Series B, 28, 131–142.
Andradóttir, S. (1995), “A method for discrete stochastic optimization”, Management Science, 41, 1946–1961.
Andradóttir, S. (1996), “A global search method for discrete stochastic optimization”, SIAM J. Optimization, 6, 513–530.
Asmussen, S. (1987), Applied Probability and Queues,John Wiley and Sons.
Bertsekas and Galager (1992), Data Networks,Prentice Hall.
Cohn, H. and M. Fielding (1999), “Simulated annealing searching for an optimal temperature schedule”, SIAM Journal of Optimization, 9 (3), 779–802.
Colorni, A., Dorigo, M., Maffioli, F., Maniezzo, V., Righini, G. and M. Trubian (1996), “Heuristics from nature for hard combinatorial problems”, International Transactions in Operational Research, 3 (1), 1–21.
Di Caro, G. and M. Dorigo (1998), “AntNet: Distributed stigmergetic control for communications networks”, Journal of Artificial Intelligence Research, 9, 317–365.
Dorigo, M. and G. Di Caro (1999), “The ant colony optimization meta-heuristic”, in D. Corne, M. Dorigo and F. Glover (eds.), New Ideas in Optimization, McGraw-Hill, 11–32.
Dorigo, M., Di Caro, G. and L.M. Gambardella (1999), “Ant algorithms for discrete optimization”, Artificial Life, 5 (2), 137–172.
Dorigo, M. and L.M. Gambardella (1997a), “Ant colony system: A cooperative learning approach to the traveling salesman problem”, IEEE Transactions on Evolutionary Computation, 1 (1), 53–66.
Dorigo, M. and L.M. Garbardella (1997b), “Ant colonies for the traveling salesman problem”, Biosystems 43, 73–81.
Dorigo, M., Di Caro, G. and Gambardella, L.M. (1999), “Ant algorithms for discrete optimization, Artificial Life,5(2), 137–172.
Dorigo, M., Maniezzo, V. and A. Colorni (1996), “The ant system: Optimization by a colony of cooperating agents”, IEEE Trans. on Systems, Man, and Cybernetics—Part B, 26, 29–41.
Gutjahr, W.J. (2000), “A graph-based ant system and its convergence”, Future Generations Computing, 16, 873–888.
Gutjahr, W.J. (2000), “A generalized convergence result for the graph-based ant system methaheuristic”, Manuscript, University of Vienna.
Dyer, M., Frieze, A. and R. Kannan (1991), “A random polynomial-time algorithm for approximation the volume of convex bodies”, Journal on ACM, 38, 1–17.
Fox, B.L. and G.W. Heine (1996), “Probabilistic search with overrides”, Annals of Applied Probability, 6, 1087–1094.
Geyer, C.J. and E.A. Thompson (1995), “Annealing Markov chain Monte-Carlo with applications to ancestral inference”, Journal of the American Statistical Association, 90, 909–920.
Gilks, W.R., Richardson, S. and D.J. Spiegelhalter (1996), Markov Chain Monte Carlo in Practice,Chapman and Hall.
Glover, F. and M. Laguna (1992), “Tabu search”, a chapter in Modern Heuristic Techniques for Combinatorial Optimization.
Goldberg, D. (1989), Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, MA.
Gong, W.B., Ho, Y.C. and W. Zhai (1992), “Stochastic comparison algorithm for discrete optimization with estimation”, Proceedings of the 31st IEEE Conference on Decision and Control, 795–800.
Gutjahr, W.J. and Pflug, G.Ch. (1996), “Simulated annealing for noisy cost functions”, Journal of Global Optimization 8, 1–13.
Hastings, W.K. (1970), “Monte Carlo sampling methods using Markov chains and their applications”, Biometrika, 57, 92–109.
Horst, R., Pardalos, P.M. and N.V. Thoai (1996), Introduction to Global Optimization,Kluwer Academic Publishers.
Jerrum, M.R. and A.J. Sinclair (1989), “Approximating the permanent”, SIAM Journal on Computing, 18, 1149–1178.
Jerrum, M.R. and A.J. Sinclair (1993), “Polynomial-time approximation algorithms for the Ising model”, SIAM Journal on Computing, 22, 1087–1116.
Johnson, N.L. and S. Kotz (1977), Urn Models and Their Applications: An Approach to Modern Discrete Probability Theory John Wiley and Sons.
Karp, R.M. and M. Luby (1983), “Monte-Carlo algorithms for enumeration and reliability problems”, Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, 56–64.
Kapur J.N. and H.K. Kesavan (1992), Entropy Optimization Principles with Applications,Academic Press.
Kirkpatrick, S., Gelatt, C.D. and M.P. Vecchi (1983), “Optimization by simulated annealing”, Science, 220, 671–680.
Lieber, D. (1998), “Rare-events estimation via cross-entropy and importance sampling”, Ph.D. Thesis, William Davidson Faculty of Industrial Engineering and Management, Technion, Haifa, Israel.
Lieber, D., Rubinstein, R.Y. and D. Elmakis (1997), “Quick estimation of rare-events in stochastic networks”, IEEE Transactions on Reliability, 46 (2), 254–265.
Lovasz, L. (1995), “Randomized algorithms in combinatorial optimization”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 25, 153–179.
Marinari, E. and G. Parisi (1992), “Simulated tempering: A new Monte Carlo scheme”, Europhysics Letters, 19, 451–458.
Mead, R. and J.A. Nedler (1965), “A simplex method for function minimization”, Computer Journal, 7, 308–313.
Metropolis, M., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and E. Teller (1953), “Equations of state calculations by fast computing machines”, J. of Chemical Physics, 21, 1087–1092.
Norkin, W.I., Pflug, G.C. and A. Ruszczynski (1996), “A branchand-bound method for stochastic global optimization”. Working paper, International Institute for Applied System Analysis, WP-96–065, Laxenburg, Austria.
Osman, I.H. and G. Laporte (1996), “Metaheuristics: A bibliography”, Annals of Operations Research, 63, 513–523.
Pinter, J.D. (1996), Global Optimization in Action,Kluwer Academic Publishers.
Parker, R.G. and R.L. Rardin (1988), Discrete Optimization, Academic Press, San Diego.
Potamianos, G. and J. Goutsias (1997), “Stochastic approximation algorithms for partition function estimation of Gibbs random fields”, IEEE Transactions on Information Theory,43(6), 19481965.
Rayward-Smith, V.J., Osman, I.H., Reeves, C.R. and G.D. Smith (1996), Modern Heuristic Search Methods, Wiley, Chichester.
Reeves, C.R. (1996), “Modern heuristic techniques”, in: Modern Heuristic Search Methods (eds. V.J. Rayward-Smith, I.H. Osman, C.R. Reeves and G.D. Smith ), Wiley, Chichester.
Romeijn, H.E. and R.L. Smith (1994), “Simulated annealing for constrained global optimization”, Journal of Global Optimization, 5, 101–126.
Rubinstein, R.Y. (1999), “The cross-entropy method for combinatorial and continuous optimization”, Methodology and Computing in Applied Probability, 2, 127–190.
Rubinstein, R.Y. (2000), “Combinatorial optimization via cross-entropy ”, S. Gass and C. Harris (editors), Kluwer, Encyclopedia of Management Sciences.
R.Y. Rubinstein (2000), “The cross-entropy method and rare-events for maximal cut and bipartition problems”. Manuscript, Technion, Haifa, Israel.
Rubinstein, R.Y. and B. Melamed (1998), Efficient Simulation and Modeling,John Wiley and Sons.
Rubinstein, R.Y. and A. Shapiro (1993), Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization via the Score Function Method John Wiley and Sons.
Schoonderwoerd, R., Holland, O., Bruten, J. and Rothkrantz, L. (1997). Ant-based Load Balancing in Telecommunications Networks, Adaptive Behavior, 5 (2), 169–207.
Shi, L. and S. Olafsson (2000), “Nested partitioning method for global optimization”, Operations Research, 48 (3).
Shi, L., Olafsson, S. and N. Sun (1999), “New parallel randomized algorithm for traveling salesman problem”, Computers and Operations Research, 26, 371–394.
Stützle, T. and M. Dorigo (1999), “ACO algorithms for the quadratic assignment problem”, in D. Corne, M. Dorigo and F. Glover (eds.), New Ideas in Optimization, McGraw-Hill, 33–50.
Wagner, I.A., Lindenbaum, M. and A.M. Bruckstein (1998), “Efficient graph search by a smell-oriented vertex process”, Annals of Mathematics and Artificial Intelligence, 24, 211–223.
Wagner, I.A., Lindenbaum, M. and A.M. Bruckstein (1999), “Distributed covering by ant-robots using evaporating traces”, IEEE Transactions on Robotics and Automation, 15 (5), 918–933.
Wagner, I.A., Lindenbaum, M. and A.M. Bruckstein (2000), “ANTS: Agents on Networks, Trees and Subgraphs” (preprint).
Walker, A.J. (1977), “An efficient method for generating discrete random variables with general distributions”, Assoc. Comput. Mach. Trans. Math. Software 3, 253–256.
Yanovski, V.M., Wagner, I.A. and A.M. Bruckstein (2000), “Edge Ant Walk for Patrolling Networks”.
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Rubinstein, R.Y. (2001). Combinatorial Optimization, Cross-Entropy, Ants and Rare Events. In: Uryasev, S., Pardalos, P.M. (eds) Stochastic Optimization: Algorithms and Applications. Applied Optimization, vol 54. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6594-6_14
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DOI: https://doi.org/10.1007/978-1-4757-6594-6_14
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