Advertisement

Reformulation-Linearization Techniques for Discrete Optimization Problems

  • Hanif D. Sherali
  • Warren P. Adams
Chapter

Abstract

Discrete and continuous nonconvex programming problems arise in a host of practical applications in the context of production, location-allocation, distribution, economics and game theory, process design, and engineering design situations. Several recent advances have been made in the development of branch-and-cut algorithms for discrete optimization problems and in polyhedral outer-approximation methods for continuous nonconvex programming problems. At the heart of these approaches is a sequence of linear programming problems that drive the solution process. The success of such algorithms is strongly tied in with the strength or tightness of the linear programming representations employed.

Keywords

Convex Hull Discrete Optimization Valid Inequality Quadratic Assignment Problem Discrete Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, W. P. and T. A. Johnson, An Exact Solution Strategy for the Quadratic Assignment Problem Using RLT-Based Bounds, Working paper, Department of Mathematical Sciences, Clemson University, Clemson, SC, 1996.Google Scholar
  2. [2]
    Adams, W. P. and T. A. Johnson, Improved Linear Programming-Based Lower Bounds for the Quadratic Assignment Problem, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Quadratic Assignment and Related Problems, eds. P. M. Pardalos and H. Wolkowicz, 16, 43–75, 1994.Google Scholar
  3. [3]
    Adams, W. P. and H. D. Sherali, A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems, Management Science, 32 (10), 1274–1290, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Adams, W. P. and H. D. Sherali, Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems, Operations Research, 38 (2), 217–226, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Adams, W. P. and H. D. Sherali, Mixed-Integer Bilinear Programming Problems, Mathematical Programming, 59 (3), 279–305, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Adams, W. P., A. Billionnet, and A. Sutter, Unconstrained 0–1 Optimization and Lagrangean Relaxation, Discrete Applied Mathematics, 29 (2–3), 131–142, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Adams, W. P., J. B. Lassiter, and H. D. Sherali Persistency in 0–1 Optimizationunder revision for Mathematics of Operations Research, Manuscript, 1993. Google Scholar
  8. [8]
    Balas, E., Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems, SIAM Journal on Algebraic and Discrete Methods, 6 (3), 466–486, 1985.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Balas, E. and J. B. Mazzola, Nonlinear 0–1 Programming: I. Linearization Techniques, Mathematical Programming, 30, 2–12, 1984a.Google Scholar
  10. [10]
    Balas, E. and J. B. Mazzola, Nonlinear 0–1 Programming: II. Dominance Relations and Algorithms, Mathematical Programming, 30, 22–45, 1984b.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Balas, E. and E. Zemel, Facets of the Knapsack Polytope from Minimal Covers, SIAM Journal of Applied Mathematics, 34, 119–148, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Balas, E., S. Ceria, and G. Cornuéjols, A Lift-and-Project Cutting Plane Algorithm for Mixed 0–1 Programs, Mathematical Programming, 58 (3), 295–324, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Boros, E., Y. Crama and P. L. Hammer, Upper Bounds for Quadratic 0–1 Maximization Problems, RUTWR Report RRR # 14–89, Rutgers University, New Brunswick, NJ 08903, 1989.Google Scholar
  14. [14]
    Camerini, P. M., L. Fratta, and F. Maffioli, On Improving Relaxation Methods by Modified Gradient Techniques, Mathematical Programming Study 3, North-Holland Publishing Co., New York, NY, 26–34, 1975.Google Scholar
  15. [15]
    CPLEXUsing the CPLEX Linear Optimizer CPLEX Optimization, Inc., Suite 279, 930 Tahoe Blvd., Bldg. 802, Incline Village, NV 89451, 1990.Google Scholar
  16. [16]
    Crowder, H., E. L. Johnson, and M. W. Padberg, Solving Large-Scale Zero-One Linear Programming Problems, Operations Research, 31, 803–834, 1983.CrossRefzbMATHGoogle Scholar
  17. [17]
    Desrochers, M. and G. Laporte, Improvements and Extensions to the Miller-Tucker-Zemlin Subtour Elimination Constraints, Operations Research Letters, 10 (1), 27–36, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    Fisher, M. L., The Lagrangian Relaxation Method for Solving Integer Programming Problems, Management Science, 27 (1), 1–18, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Garfinkel, R. S. and G. L. Nemhauser, A Survey of Integer Programming Emphasizing Computation and Relations Among Models, In Mathematical Programming: Proceedings of an Advanced Seminar, T. C. Hu and S. Robinson (eds.), Academic Press, New York, NY, 77–155, 1973.Google Scholar
  20. [20]
    Geoffrion, A. M., Lagrangian Relaxation for Integer Programming, Mathematical Programming Study 2, M. L. Balinski (ed.), North- Holland Publishing Co., Amsterdam, 82–114, 1974.Google Scholar
  21. [21]
    Geoffrion, A. M. and R. McBryde, Lagrangian Relaxation Applied to Facility Location Problems, AIIE Transactions, 10, 40–47, 1979.Google Scholar
  22. [22]
    Glover, F., Improved Linear Integer Programming Formulations of Nonlinear Integer Problems, Management Science, 22 (4), 455–460, 1975.CrossRefMathSciNetGoogle Scholar
  23. [23]
    Hoffman, K. L. and M. Padberg, Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut, ORSA Journal on Computing, 3 (2), 121–134, 1991.zbMATHGoogle Scholar
  24. [24]
    Jeroslow, R. G. and J. K. Lowe, Modeling with Integer Variables, Mathematical Programming Study 22, 167–184, 1984.zbMATHMathSciNetGoogle Scholar
  25. [25]
    Jeroslow, R. G. and J. K. Lowe, Experimental Results on New Techniques for Integer Programming Formulations, Journal of the Operational Research Society, 36, 393–403, 1985.zbMATHGoogle Scholar
  26. [26]
    Johnson, E. L., Modeling and Strong Linear Programs for Mixed Integer Programming, Algorithms and Model Formulations in Mathematical Programming, NATO ASI 51, (ed.)S. Wallace, Springer-Verlag, 3–43, 1989.Google Scholar
  27. [27]
    Johnson, E. L., M. M. Kostreva, and U. H. Suhl, Solving 0–1 Integer Programming Problems Arising From Large Scale Planning Models, Operations Research, 33 (4), 803–819, 1985.CrossRefzbMATHGoogle Scholar
  28. [28]
    Lovâsz, L. and A. Schrijver, Cones of Matrices and Set Functions, and 0–1 Optimization, SIAM J. Opt., 1, 166–190, 1991.zbMATHGoogle Scholar
  29. [29]
    Magnanti, T. L. and R. T. Wong, Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria, Operations Research, 29, 464–484, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    Martin, K. R., Generating Alternative Mixed-Integer Programming Models Using Variable Redefinition, Operations Research, 35, 820–831, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    Meyer, R. R., A Theoretical and Computational Comparison of ‘Equivalent’ Mixed-Integer Formulations, Naval Research Logistics Quarterly, 28, 115–131, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    Nemhauser, G. L. and L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, New York, 1988.zbMATHGoogle Scholar
  33. [33]
    Nemhauser, G. L. and L. A. Wolsey, A Recursive Procedure for Generating all Cuts for Mixed-Integer Programs, Mathematical Programming, 46, 379–390, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    Oley, L. A. and R. J. Sjouquist, Automatic Reformulation of Mixed and Pure Integer Models to Reduce Solution Time in Apex IV, Presented at the ORSA/TIMS Fall Meeting, San Diego, 1982.Google Scholar
  35. [35]
    Overton, M. and H. Wolkowicz, “Semidefinite Programming,” Mathematical Programming, 77 (2), 105–110, 1997zbMATHMathSciNetGoogle Scholar
  36. [36]
    Padberg, M. W., (1,k)-Configurations and Facets for Packing Problems, Mathematical Programming, 18, 94–99, 1980.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [37]
    Padberg, M. and G. Rinaldi, A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems, SIAM Review, 33, 60–100, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    Ramachandran, B. and J. F. Pekny, Dynamic Factorization Methods for Using Formulations Derived from Higher Order Lifting Techniques in the Solution of the Quadratic Assignment Problem, in State of the Art in Global Optimization, eds. C. A. Floudas and P. M. Pardalos, Kluwer Academic Publishers, 7, 75–92, 1996.CrossRefGoogle Scholar
  39. [39]
    Ramakrishnan, K. G., M. G. C. Resende, and P. M. Pardalos A Branch and Bound Algorithm for the Quadratic Assignment Problem Using a Lower Bound Based on Linear Programming, in State of the Art in Global Optimization, eds. C. A. Floudas and P. M. Pardalos, Kluwer Academic Publishers, 7, 57–74, 1996.CrossRefGoogle Scholar
  40. [40]
    Sherali, H. D., On the Derivation of Convex Envelopes for Multlinear Functions, Working Paper, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061–0118, 1996.Google Scholar
  41. [41]
    Sherali, H. D. and W. P. AdamsA Decomposition Algorithm for a Discrete Location-Allocation Problem Operations Research, 32(4), 878900, 1984.Google Scholar
  42. [42]
    Sherali, H. D. and W. P. Adams, A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems, SIAM Journal on Discrete Mathematics, 3 (3), 411–430, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [43]
    Sherali, H. D. and W. P. Adams, A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed- Integer Zero-One Programming Problems, Discrete Applied Mathematics, 52, 83–106, 1994. (Manuscript, 1989 ).CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    Sherali, H. D. and E. L. Brown, A Quadratic Partial Assignment and Packing Model and Algorithm for the Airline Gate Assignment Problem, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Quadratic Assignment and Related Problems, eds. P. M. Pardalos and H. Wolkowicz, 16, 343–364, 1994.Google Scholar
  45. [45]
    Sherali, H. D. and G. Choi, Recovery of Primal Solutions When Using Subgradient Optimization Methods to Solve Lagrangian Duals of Linear Programs, Operations Research Letters, 19 (3), 105–113, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    Sherali, H. D. and P. J. Driscoll, On Tightening the Relaxations of Miller-Tucker-Zemlin Formulations for Asymmetric Traveling Salesman Problems, Working Paper, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1996.Google Scholar
  47. [47]
    Sherali, H. D. and Y. Lee, Sequential and Simultaneous Liftings of Minimal Cover Inequalities for GUB Constrained Knapsack Polytopes, SIAM Journal on Discrete Mathematics, 8 (1), 133–153, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [48]
    Sherali, H. D. and Y. Lee, Tighter Representations for Set Partitioning Problems, Discrete Applied Mathematics, 68, 153–167, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    Sherali, H. D. and D. C. Myers, Dual Formulations and Subgradient Optimization Strategies for Linear Programming Relaxations of Mixed-Integer Programs, Discrete Applied Mathematics, 20 (S-16), 51–68, 1989.MathSciNetGoogle Scholar
  50. [50]
    Sherali, H. D. and O. Ulular, A Primal-Dual Conjugate Subgradient Algorithm for Specially Structured Linear and Convex Programming Problems, Applied Mathematics and Optimization, 20, 193–221, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    Sherali, H. D. and C. H. Tuncbilek, A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation- Linearization Technique, Journal of Global Optimization, 2, 101–112, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    Sherali, H. D. and C. H. Tuncbilek, A Reformulation-Convexification Approach for Solving Nonconvex Quadratic Programming Problems, Journal of Global Optimization, 7, 1–31, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  53. [53]
    Sherali, H. D. and C. H. Tuncbilek, New Reformulation-Linearization Technique Based Relaxations for Univariate and Multivariate Polynomial Programming Problems, Operation Research Letters, to appear, 1996.Google Scholar
  54. [54]
    Sherali, H. D., W. P. Adams, and P. Driscoll, Exploiting Special Structures in Constructing a Hierarchy of Relaxations for 0–1 Mixed Integer Problems, Operations Research, to appear, 1996.Google Scholar
  55. [55]
    Sherali, H. D., G. Choi, and C. H. Tuncbilek, A Variable Target Value Method, under revision for Mathematical Programming, 1995.Google Scholar
  56. [56]
    Sherali, H. D., R. Krishnamurthy, and F. A. Al-Khayyal, A Reformulation -Linearization Approach for the General Linear Complementarity Problem, Presented at the Joint National ORSA/TIMS Meeting, Phoenix, Arizona, 1993.Google Scholar
  57. [57]
    Sherali, H. D., R. S. Krishnamurthy, and F. A. Al-Khayyal, An Enhanced Intersection Cutting Plane Approach for Linear Complementarity Problems, Journal of Optimization Theory and Applications, to appear, 1995.Google Scholar
  58. [58]
    Sherali, H. D., Y. Lee, and W. P. Adams, A Simultaneous Lifting Strategy for Identifying New Classes of Facets for the Boolean Quadric Polytope, Operations Research Letters, 17 (1), 19–26, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [59]
    Van Roy, T. J. and L. A. Wolsey, Solving Mixed Integer Programs by Automatic Reformulation, Operations Research, 35, 45–57, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  60. [60]
    Van Roy, T. J. and L. A. Wolsey, Valid Inequalities for Mixed 0–1 Programs, CORE Discussion Paper No. 8316, Center for Operations Research and Econometrics, Universite Catholique de Louvain, Belgium, 1983.Google Scholar
  61. [61]
    Williams, H. P., Model Building in Mathematical Programming. John Wiley and Sons (Wiley Interscience), Second Edition, New York, NY, 1985.Google Scholar
  62. [62]
    Wolsey, L. A. Facets and Strong Valid Inequalities for Integer Programs, Operations Research, 24, 367–373, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  63. [63]
    Wolsey, L. A., Strong Formulations for Mixed Integer Programming: A Survey, Mathematical Programming, 45, 173–191, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [64]
    Wolsey, L. A., Valid Inequalities for 0–1 Knapsacks and MIPs with Generalized Upper Bound Constraints, Discrete Applied Mathematics, 29, 251–262, 1990.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Hanif D. Sherali
    • 1
  • Warren P. Adams
    • 2
  1. 1.Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Math SciencesClemson UniversityClemsonUSA

Personalised recommendations