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Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

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Abstract

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.

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References

  1. Alidaee, B., G. Kochenberger, and A. Ahmadian. (1994). “0—1 Quadratic Programming Approach for the Optimal Solution of Two Scheduling Problems.” International Journal of Systems Science 25, 401-408.

  2. Barahona, F., M. Grötschel, M. Jünger, and G. Reinelt. (1988). “An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design.” Operations Research 36, 493-513.

  3. Barahona, F., M. Jünger, and G. Reinelt. (1989). “Experiments in Quadratic 0—1 Programming.” Mathematical Programming 44, 127-137.

  4. Beasley, J.E. (1996). “Obtaining Test Problems Via Internet.” Journal of Global Optimization 8, 429-433.

  5. Beasley, J.E. (1998). “Heuristic Algorithms for the Unconstrained Binary Quadratic Programming Problem.” Working paper, The Management School, Imperial College, London, England.

  6. Billionnet, A., and A. Sutter. (1994). “Minimization of a Quadratic Pseudo-Boolean Function.” European Journal of Operational Research 78, 106-115.

  7. Boros, E., and P.L. Hammer. (1991). “The Max-Cut Problem and Quadratic 0—1 Optimization: Polyhedral Aspects, Relaxations and Bounds.” Annals of Operations Research 33, 151-180.

  8. Carter, M.W. (1984). “The Indefinite Zero—One Quadratic Problem.” Discrete Applied Mathematics 7, 23- 44.

  9. Çela, E. (1998). The Quadratic Assignment Problem: Theory and Algorithms. Dordrecht: Kluwer Academic.

  10. Charon, I., and O. Hudry. (1993). “The Noising Method: A New Method for Combinatorial Optimization.” Operations Research Letters 14, 133-137.

  11. Codenotti, B., G. Manzini, L. Margara, and G. Resta. (1996). “Perturbation: An Efficient Technique for the Solution of Very Large Instances of the Euclidean TSP.” INFORMS Journal on Computing 8, 125-133.

  12. Dearing, P.M., P.L. Hammer, and B. Simeone. (1988). “Boolean and Graph-Theoretic Formulations of the Simple Plant Location Problem.” RUTCOR Research Report 3-88, Rutgers University, New Brunswick, USA.

  13. De Simone, C., M. Diehl, M. Jünger, P. Mutzel, G. Reinelt, and G. Rinaldi. (1995). “Exact Ground States of Ising Spin Glasses: New Experimental ResultsWith a Branch and Cut Algorithm.” Journal of Statistical Physics 80, 487-496.

  14. Feo, T.A., and M.G.C. Resende. (1995). “Greedy Randomized Adaptive Search Procedures.” Journal of Global Optimization 6, 109-133.

  15. Fleurent, C., and F. Glover. (1999). “Improved Constructive Multistart Strategies for the Quadratic Assignment Problem Using Adaptive Memory.” INFORMS Journal on Computing 11, 198-204.

  16. Gallo, G., P.L. Hammer, and B. Simeone. (1980). “Quadratic Knapsack Problems.”Mathematical Programming 12, 132-149.

  17. Glover, F. (1977). “Heuristics for Integer Programming Using Surrogate Constraints.” Decision Sciences 8, 156-166.

  18. Glover, F. (1986). “Future Paths for Integer Programming and Links to Artificial Intelligence.” Computers and Operations Research 13, 533-549.

  19. Glover, F., B. Alidaee, C. Rego, and G. Kochenberger. (2002). “One-Pass Heuristics for Large-Scale Unconstrained Binary Quadratic Problems.” European Journal of Operational Research 137, 272-287.

  20. Glover, F., G.A. Kochenberger, and B. Alidaee. (1998). “Adaptive Memory Tabu Search for Binary Quadratic Programs.” Management Science 44, 336-345.

  21. Glover, F., and M. Laguna. (1997). Tabu Search.Hingham, MA: Kluwer Academic.

  22. Hammer, P.L. (1968). “Plant Location — A Pseudo-Boolean Approach.” Israel Journal of Technology 6, 330-332.

  23. Hammer, P., and S. Rudeanu. (1968). Boolean Methods in Operations Research.New York: Springer.

  24. Hansen, P. (1979). “Methods of Nonlinear 0—1 Programming.” Annals of Discrete Mathematics 5, 53-70.

  25. Helmberg, C., and F. Rendl. (1998). “Solving Quadratic (0,1)-Problems by Semidefinite Programs and Cutting Planes.” Mathematical Programming 82, 291-315.

  26. Jünger, M., A. Martin, G. Reinelt, and R. Weismantel. (1994). “Quadratic 0/1 Optimization and a Decomposition Approach for the Placement of Electronic Circuits.” Mathematical Programming 63, 257-279.

  27. Krarup, J., and P.M. Pruzan. (1978). “Computer-Aided Layout Design.” Mathematical Programming Study 9, 75-94.

  28. Laughunn, D.J. (1970). “Quadratic Binary Programming.” Operations Research 14, 454-461.

  29. Li, Y., P.M. Pardalos, and M.G.C. Resende. (1994). “A Greedy Randomized Adaptive Search Procedure for the Quadratic Assignment Problem.” In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 16. Providence, RI: Amer. Math. Soc., pp. 237-261.

  30. Lodi, A., K. Allemand, and T.M. Liebling. (1999). “An Evolutionary Heuristic for Quadratic 0—1 Programming.” European Journal of Operational Research 119, 662-670.

  31. McBride, R.D., and J.S. Yormark. (1980). “An Implicit Enumeration Algorithm for Quadratic Integer Programming.” Management Science 26, 282-296.

  32. Merz, P., and B. Freisleben. (1999). “Genetic Algorithms for Binary Quadratic Programming.” In W. Banzhaf, J. Daida, A.E. Eiben, M.H. Garzon, V. Honavar, M. Jakiela, and R.E. Smith (eds.), Proceedings of the Genetic and Evolutionary Computation Conference, Vol. 1. Orlando, FL: Morgan Kaufmann, pp. 417-424.

  33. Palubeckis, G. (1992). “Heuristics with a Worst-Case Bound for Unconstrained Quadratic 0—1 Programming.” Informatica 3, 225-240.

  34. Palubeckis, G. (1995). “A Heuristic-Based Branch and Bound Algorithm for Unconstrained Quadratic Zero—One Programming.” Computing 54, 283-301.

  35. Pardalos, P.M. (1991). “Construction of Test Problems in Quadratic Bivalent Programming.” ACM Transactions on Mathematical Software 17, 74-87.

  36. Pardalos, P.M., and S. Jha. (1991). “Graph Separation Techniques for Quadratic Zero—One Programming.” Computers and Mathematics with Applications 21, 107-113.

  37. Pardalos, P.M., and G.P. Rodgers. (1990). “Computational Aspects of a Branch and Bound Algorithm for Quadratic Zero—One Programming.” Computing 45, 131-144.

  38. Pardalos, P.M., and G.P. Rodgers. (1992). “A Branch and Bound Algorithm for the Maximum Clique Problem.” Computers and Operations Research 19, 363-375.

  39. Pardalos, P.M., and J. Xue. (1994). “The Maximum Clique Problem.” Journal of Global Optimization 4, 301-328.

  40. Picard, J.C., and H.D. Ratliff. (1975). “Minimum Cuts and Related Problems.” Networks 5, 357-370.

  41. Poljak, S., and Z. Tuza. (1995). “Maximum Cuts and Large Bipartite Subgraphs.” In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 20. Providence, RI: Amer. Math. Soc., pp. 181-244.

  42. Shih, M., and E.S. Kuh. (1993). “Quadratic Boolean Programming for Performance-Driven System Par282 PALUBECKIS titioning.” In Proceedings of the 30th ACM/IEEE Design Automation Conference, Dallas, TX. ACM Press, pp. 761-765.

  43. Skorin-Kapov, J. (1990). “Tabu Search Applied to the Quadratic Assignment Problem.” ORSA Journal on Computing 2, 33-45.

  44. Storer, R.H., S.D. Wu, and R. Vaccari. (1992). “New Search Spaces for Sequencing Problems with Application to Job Shop Scheduling.” Management Science 38, 1495-1509.

  45. Warszawski, A. (1974). “Pseudo-Boolean Solutions to Multidimensional Location Problems.” Operations Research 22, 1081-1085.

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Palubeckis, G. Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem. Ann Oper Res 131, 259–282 (2004). https://doi.org/10.1023/B:ANOR.0000039522.58036.68

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  • binary quadratic optimization
  • tabu search
  • multistart strategies
  • heuristics