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Dynamic Matrix Factorization Methods for Using Formulations Derived From Higher Order Lifting Techniques in the Solution of the Quadratic Assignment Problem

  • Bala Ramachandran
  • J. F. Pekny
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 7)

Abstract

This paper concerns the use of linear programming based methods for the exact solution of the Quadratic Assignment Problem (QAP). The primary obstacles facing such an approach are the large size of the formulation resulting from the linearization of the quadratic objective function and the poor quality of the lower bounds. Special purpose linear programming methods using dynamic matrix factorization provide a promising avenue for solving these large scale linear programs (LP). This enables a large portion of the LP basis to be represented implicitly and generated from the remaining explicit part. Computational results demonstrating the strength of this approach are also presented. For this approach to be effective in the solution of QAPs, dynamic matrix factorization should be combined with formulations that yield superior lower bounds. Lifting techniques have been theoretically proven to improve bound strength at the cost of a dramatic increase in formulation size. A formulation including third order interactions is derived using this methodology. However, degeneracy poses a significant problem in the solution of these linear programs. Incorporation of third order interaction costs in the objective function is proposed as a possible way to mitigate problems due to stalling. Computational results indicate that this formulation yields much stronger lower bounds than the currently best known lower bounds. Unifying these various observations, it is suggested that the development of specialized dual simplex algorithms using dynamic matrix factorization can provide a promising approach to overcome these barriers.

Keywords

Quadratic Assignment Problems Linear Programming Dynamic Matrix Factorization Lifting Techniques 

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References

  1. 1.
    Adams, W.P. and Johnson, T.A. (1994) “Improved Linear Programming Based Lower Bounds for the Quadratic Assignment Problem”, DIMACS Strict in Discrete Mathematics and Theoretical Computer Science, 16, 43–75.MathSciNetGoogle Scholar
  2. 2.
    Balas, E., Ceria, S. and Cornuejols, G. (1993) “A Lift-and-Project Cutting Plane Algorithm for Mixed 0-1 Programs”, Math. Prog 58, 295–324.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bazarra, M.S. and Sherali, M.D. (1980) “Bender’s Partitioning Scheme Applied to a New Formulation of the Quadratic Assignment Problem” Naval Res. Log. Quart., 27, 29–41.CrossRefGoogle Scholar
  4. 4.
    Bazarra, M.S. and Sherali, H.D. (1982) “On the Use of Exact and Heuristic Cutting Plane Methods for the Quadratic Assignment Problem”, J. Oper. Res. Soc., 33, 991–1003.MathSciNetGoogle Scholar
  5. 5.
    Bixby, R.E., Gregory, J.W., Lustig, I.J., Mawten, R.E. and Shanno, D.F. (1992) “Very Large- Scale Linear Programming: A Case Study in Combining Interior Point and Simplex Methods”, Oper. Res., 40, 885–897.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bokhari, S.H. (1987) Assignment Problems in Distributed and Parallel Computing, Kluwer Academic Publishers, Boston.Google Scholar
  7. 7.
    Brown, G.G. and Thomen, D. (1980) “Automatic Identification of Generalized Upper Bounds in Large-scale Optimization Models”, Mgmt. Sci., 26, 1166–1184.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brown, G.G. and Olson, M.P. (1994) “Dynamic Factorization in Large-scale Optimization”, Math. Prog., 64, 17–51.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bui, T.N. and Moon, B.R. (1994) “A Genetic Algorithm for a Special Class of the Quadratic Assignment Problem” DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, 16, 99–116.MathSciNetGoogle Scholar
  10. 10.
    Burkard, R.E. and Bonniger, T. (1983) “A Heuristic for Quadratic Boolean Programs with Applications to Quadratic Assignment Problems”, Eur. J. Oper. Ret., 13, 374–386.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Burkard, R.E. (1984) “A Thermodynamically Motivated Simulation Procedure for Combinatorial Optimization Problems”, Eur. J. Oper. Res., 17, 169–174.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chen, S. and Saigal, R. (1977) “A Primal Algorithm for solving a Capacitated Network Flow Problem with Additional Linear Constraints”, Networks, 7, 59–79.zbMATHCrossRefGoogle Scholar
  13. 13.
    Chvatal, V . (1983) Linear Programming, W.H. FYeeman and Co., New York.zbMATHGoogle Scholar
  14. 14.
    Dantzig, G.B. and Van Slyke, R.M. (1967) “Generalised Upper Bounding Techniques”, J. Comp. Sys. Sci., 1, 213–226.zbMATHCrossRefGoogle Scholar
  15. 15.
    Frieze, A.M. and Yadegar, J. (1983) “On the Quadratic Assignment Problem”, Disc. Appl Math., 5, 89–98.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gal, T., Kruse, H. and Zornig, P. (1988) “Survey of Solved and Open Problems in the Degeneracy Phenomenon”, Math. Prog., 42, 125–133.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Geoffrion, A.M. and Graves, G.W. (1976) “Scheduling Parallel Production Lines with Changeover Costs: Practical Applications of a Quadratic Assignment/LP Approach, Oper. Res., 24, 595–610zbMATHCrossRefGoogle Scholar
  18. 18.
    Gill, P.E., Murray, W., Saunders, M.A. and Wright, M.H. (1989) “A Practical Anti-cycling Procedure for Linearly Constrained Optimization”, Math. Prog., 45, 437–474.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gilmore, P.C. (1962) “Optimal and suboptimal algorithms for the Quadratic Assignment Problem”, J. SI AM., 10, 305–313.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Graves, G.W. and McBride, R.D. (1976) “The Factorization Approach to Large-scale Linear Programming”, Math. Prog., 10, 91–110.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hartman, J.K. and Lasdon, L.S. (1972) “A Generalized Upper Bounding Algorithm for Multicommodity Network Flow Problems”, Networks, 1, 333–354.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Skorin-Kapov, J. (1990) “Tabu Search Applied to the Quadratic Assignment Problem”, ORSA J. Comput., 2, 33.zbMATHGoogle Scholar
  23. 23.
    Kaufman, L. and Broeckx, F. (1978) “An Algorithm for the Quadratic Assignment Problem using Bender’s Decomposition” Eur. J. Op. Res., 2, 204–211.Google Scholar
  24. 24.
    Kettani, O. and Oral, M. (1993) “Reformulating Quadratic Assignment Problems for Efficient Optimization”, IIE Trans., 25, 6, 97–107.CrossRefGoogle Scholar
  25. 25.
    Klingman, D. and Russell, R. (1975) “On Solving Constrained Transportation Problems”, Oper. Res., 23, 91–107.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Koopmans, T.C. and Beckmann, M.J. (1957) “Assignment Problems and the Location of Economic Activities”, Econometrica, 25, 53–76.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lawler, E.L. (1963) “The Quadratic Assignment Problem”, Mmgt. Sci., 9, 586–599.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Li, Y., Pardalos, P.M. and Resende M.G.C. (1994) “A Greedy Randomized Adaptive Search Procedure for the Quadratic Assignment Problem”, DIM A CS Series in Discrete Mathematics and Theoretical Computer Science, 16, 237–261.MathSciNetGoogle Scholar
  29. 29.
    Lovasz, L. and Schrijver, A. (1991) “Cones of Matrices and Set-Functions and 0-1 Optimization”, SIAM J. Opt., 1, 166–190.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Mautor, T. and Roucairol, C. (1994) “Difficulties of Exact Methods for Solving the Quadratic Assignment Problem”, DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, 16, 263–274.MathSciNetGoogle Scholar
  31. 31.
    McBride, R.D. (1985) “Solving Embedded Generalized Network Problems”, Eur. J. Oper. Res., 21, 82–92.zbMATHCrossRefGoogle Scholar
  32. 32.
    Mirchandani, P.B., and Obata, T. (1979) “Locational Decisions with Interactions Between Facilities: The Quadratic Assignment Problem — A Review”, Working Paper PS-79-1, Rensselaer Polytechnic Institute, TVoy, New York.Google Scholar
  33. 33.
    Pardalos, P.M. and Crouse, J.V. (1989) “A Parallel Algorithm for the Quadratic Assignment Problem”, Proc. Supercomputing ’89, 351–360.Google Scholar
  34. 34.
    Pardalos, P.M., Rendl, F. and Wolkowicz, H. (1994) “The Quadratic Assignment Problem: A Survey and Recent Developments”, DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, 16, 1–42.MathSciNetGoogle Scholar
  35. 35.
    Pollatschek, M.A., Gershoni, H. and Radday, Y.T. (1976) “Optimization of the Typewriter Keyboard by Computer Simulation”, Angewandte Informatik, 10, 438–439.Google Scholar
  36. 36.
    Powell, S. (1975) “A Development of the Product Form Algorithm for the Simplex Method using Reduced Transformation Vectors”, Math. Prog. Study, 4, 93–107.Google Scholar
  37. 37.
    QAPLIB — A Quadratic Assignment Problem Library February 1994. (Available by ftp at ftp.tu-graz.ac.at).Google Scholar
  38. 38.
    Ramachandran, B. and Pekny, J.F. (1994) “An Approach Based on Combinatorial Optimization Guarantees For the Determination of a Global Minimum on a Lattice”, Submitted to Biopolymers.Google Scholar
  39. 39.
    Ramachandran, B. and Pekny, J.F. (1995) “Lower Bounds for Nonlinear Assignment Problems using Many Body Interactions”, Submitted to Discrete Applied Mathematics.Google Scholar
  40. 40.
    Resende, M.G.C., Ramakrishnan, K.G. and Dresner, Z. (1994) “Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming”, Technical Report, Mathematical Sciences Research Center, AT & T Bell Laboratories, Murray Hill, New Jersey.Google Scholar
  41. 41.
    Roucairol, C. (1987) “A Parallel Branch and Bound Algorithm for the Quadratic Assignment Problem”, Dis. Appl. Math., 18, 211–225.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Schrage, L. (1975) “Implicit Representation of Variable Upper Bounds in Linear Programming”, Math. Prog. Study, 4, 118–132.MathSciNetGoogle Scholar
  43. 43.
    Sherali, H. and Adams, W. (1990) “A Hierarchy of Relaxations Between the Continuous and the Convex Hull Representations for zero-one problems”, SIAM J. Disc. Math., 3, 411–430.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Steinberg, L. (1961) “The Backboard Wiring Problem: A Placement Algorithm”, SIAM Review, 3, 37–50.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Suhl, U.H. and Suhl, L.M. (1990) “Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases”, ORSA J. Comp., 2, 325–335.zbMATHGoogle Scholar
  46. 46.
    Wilhelm, M.R., and Ward, T.L. (1987) “Solving Quadratic Assignment Problems by Simulated Annealing”, IIE Transactions, 19, 107–119.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Bala Ramachandran
    • 1
  • J. F. Pekny
    • 1
  1. 1.Department of Chemical EngineeringPurdue UniversityWest LafayetteUSA

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