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Reformulation-Linearization Techniques for Discrete Optimization Problems

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Handbook of Combinatorial Optimization

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.

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References

  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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  5. Adams, W. P. and H. D. Sherali, Mixed-Integer Bilinear Programming Problems, Mathematical Programming, 59 (3), 279–305, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  9. Balas, E. and J. B. Mazzola, Nonlinear 0–1 Programming: I. Linearization Techniques, Mathematical Programming, 30, 2–12, 1984a.

    Google Scholar 

  10. Balas, E. and J. B. Mazzola, Nonlinear 0–1 Programming: II. Dominance Relations and Algorithms, Mathematical Programming, 30, 22–45, 1984b.

    Article  MATH  MathSciNet  Google Scholar 

  11. Balas, E. and E. Zemel, Facets of the Knapsack Polytope from Minimal Covers, SIAM Journal of Applied Mathematics, 34, 119–148, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. CPLEXUsing the CPLEX Linear Optimizer CPLEX Optimization, Inc., Suite 279, 930 Tahoe Blvd., Bldg. 802, Incline Village, NV 89451, 1990.

    Google Scholar 

  16. Crowder, H., E. L. Johnson, and M. W. Padberg, Solving Large-Scale Zero-One Linear Programming Problems, Operations Research, 31, 803–834, 1983.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  18. Fisher, M. L., The Lagrangian Relaxation Method for Solving Integer Programming Problems, Management Science, 27 (1), 1–18, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Geoffrion, A. M. and R. McBryde, Lagrangian Relaxation Applied to Facility Location Problems, AIIE Transactions, 10, 40–47, 1979.

    Google Scholar 

  22. Glover, F., Improved Linear Integer Programming Formulations of Nonlinear Integer Problems, Management Science, 22 (4), 455–460, 1975.

    Article  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  24. Jeroslow, R. G. and J. K. Lowe, Modeling with Integer Variables, Mathematical Programming Study 22, 167–184, 1984.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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. 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.

    Article  MATH  Google Scholar 

  28. Lovâsz, L. and A. Schrijver, Cones of Matrices and Set Functions, and 0–1 Optimization, SIAM J. Opt., 1, 166–190, 1991.

    MATH  Google Scholar 

  29. Magnanti, T. L. and R. T. Wong, Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria, Operations Research, 29, 464–484, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  30. Martin, K. R., Generating Alternative Mixed-Integer Programming Models Using Variable Redefinition, Operations Research, 35, 820–831, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  31. Meyer, R. R., A Theoretical and Computational Comparison of ‘Equivalent’ Mixed-Integer Formulations, Naval Research Logistics Quarterly, 28, 115–131, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  32. Nemhauser, G. L. and L. A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, New York, 1988.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. Overton, M. and H. Wolkowicz, “Semidefinite Programming,” Mathematical Programming, 77 (2), 105–110, 1997

    MATH  MathSciNet  Google Scholar 

  36. Padberg, M. W., (1,k)-Configurations and Facets for Packing Problems, Mathematical Programming, 18, 94–99, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. Sherali, H. D. and W. P. AdamsA Decomposition Algorithm for a Discrete Location-Allocation Problem Operations Research, 32(4), 878900, 1984.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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 ).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  48. Sherali, H. D. and Y. Lee, Tighter Representations for Set Partitioning Problems, Discrete Applied Mathematics, 68, 153–167, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Sherali, H. D., G. Choi, and C. H. Tuncbilek, A Variable Target Value Method, under revision for Mathematical Programming, 1995.

    Google Scholar 

  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. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  59. Van Roy, T. J. and L. A. Wolsey, Solving Mixed Integer Programs by Automatic Reformulation, Operations Research, 35, 45–57, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  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. Williams, H. P., Model Building in Mathematical Programming. John Wiley and Sons (Wiley Interscience), Second Edition, New York, NY, 1985.

    Google Scholar 

  62. Wolsey, L. A. Facets and Strong Valid Inequalities for Integer Programs, Operations Research, 24, 367–373, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  63. Wolsey, L. A., Strong Formulations for Mixed Integer Programming: A Survey, Mathematical Programming, 45, 173–191, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  64. Wolsey, L. A., Valid Inequalities for 0–1 Knapsacks and MIPs with Generalized Upper Bound Constraints, Discrete Applied Mathematics, 29, 251–262, 1990.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0118, USA

    Hanif D. Sherali

  2. Department of Math Sciences, Clemson University, Clemson, SC, 29634-1907, USA

    Warren P. Adams

Authors
  1. Hanif D. Sherali
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  2. Warren P. Adams
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Editor information

Editors and Affiliations

  1. University of Minnesota, Minneapolis, USA

    Ding-Zhu Du

  2. University of Florida, Gainesville, USA

    Panos M. Pardalos

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© 1998 Kluwer Academic Publishers

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Sherali, H.D., Adams, W.P. (1998). Reformulation-Linearization Techniques for Discrete Optimization Problems. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_7

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  • DOI: https://doi.org/10.1007/978-1-4613-0303-9_7

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7987-4

  • Online ISBN: 978-1-4613-0303-9

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