Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

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.

This is a preview of subscription content, access via your institution.

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.

    Google Scholar 

  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.

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  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.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

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

    Article  Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

  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.

    Article  Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

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