Optimization and Engineering

, Volume 3, Issue 3, pp 281–303 | Cite as

Product Disaggregation in Global Optimization and Relaxations of Rational Programs

  • Mohit Tawarmalani
  • Shabbir Ahmed
  • Nikolaos V. Sahinidis
Article

Abstract

We consider the product of a single continuous variable and the sum of a number of continuous variables. We show that “product disaggregation” (distributing the product over the sum) leads to tighter linear programming relaxations, much like variable disaggregation does in mixed-integer linear programming. We also derive closed-form expressions characterizing the exact region over which these relaxations improve when the bounds of participating variables are reduced.

In a concrete application of product disaggregation, we develop and analyze linear programming relaxations of rational programs. In the process of doing so, we prove that the task of bounding general linear fractional functions of 0–1 variables is \(\mathcal{N}\mathcal{P}\)-hard. Finally, we present computational experience to demonstrate that product disaggregation is a useful reformulation technique for global optimization problems.

global optimization relaxation gap convex extensions range reduction 

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References

  1. S. C. Aggarwal, “An alternative method of integer solutions to linear fractional functionals by a branch and bound technique,” Z. Angew. Math. Mech. vol. 57, pp. 52–53, 1977.Google Scholar
  2. F. A. Al-Khayyal and J. E. Falk, “Jointly constrained biconvex programming,” Mathematics of Operations Research vol. 8, pp. 273–286, 1983.Google Scholar
  3. S. R. Arora, K. Swarup, and M. C. Puri, “The set covering problem with linear fractional functional,” Indian Journal of Pure and Applied Mathematics vol. 8, pp. 578–588, 1977.Google Scholar
  4. E. Balas, Lecture Notes on Integer Programming. GSIA, Carnegie-Melon University, January 1988.Google Scholar
  5. E. Balas and J. Mazzola, “Nonlinear 0–1 programming: I. Linearization techniques,” Mathematical Programming vol. 30, pp. 1–21, 1984.Google Scholar
  6. L. T. Biegler and I. B. Tjoa, “Catalyst mixing for packed bed reactor,” in CACHE Design Case Study Volume 6: Chemical Engineering Optimization Models with GAMS, I. E. Grossmann, ed., CACHE Corporation: Austin, TX, 1991.Google Scholar
  7. A. Charnes and W. W. Cooper, “Programming with linear fractional functionals,” Naval Research Logistics Quarterly vol. 9, pp. 181–186, 1962.Google Scholar
  8. CPLEX, CPLEX 7.0 User's Manual. ILOG CPLEX Division, Incline Village, NV, 2000.Google Scholar
  9. Y. Crama, “Concave extensions for non-linear 0–1 maximization problems,” Mathematical Programming vol. 61, pp. 53–60, 1993.Google Scholar
  10. A. Ghosh, S. McLafferty, and S. Craig, “Multifacility retail networks,” in Facility Location: A Survey of Applications and Methods, Z. Drezner, ed., Springer-Verlag: New York, pp. 301–330, 1995.Google Scholar
  11. P. C. Gilmore and R. E. Gomory, “A Linear programming approach to the cutting stock problem–Part II,” Operations Research vol. 11, pp. 52–53, 1963.Google Scholar
  12. F. Glover, “Improved linear integer programming formulations of nonlinear integer problems,” Management Science vol. 2, pp. 455–460, 1975.Google Scholar
  13. F. Glover and E. Woolsey, “Converting a 0–1 polynomial programming problem to a 0–1 linear program,” Operations Research vol. 22, pp. 180–182, 1974.Google Scholar
  14. D. Granot and F. Granot, “On solving fractional (0–1) programs by implicit enumeration,” INFOR vol. 14, pp. 241–249, 1976.Google Scholar
  15. D. Granot and F. Granot, “On integer and mixed integer fractional programming problems,” Annals of Discrete Mathematics vol. 1, pp. 221–231, 1977.Google Scholar
  16. M. Grunspan and M. E. Thomas, “Hyperbolic integer programming,” Naval Research Logistics Quarterly vol. 20, pp. 341–356, 1973.Google Scholar
  17. D. J. Gunn and W. J. Thomas, “Mass transport and chemical reaction in multifunctional catalyst systems,” Chemical Engineering Science vol. 20, pp. 89–100, 1965.Google Scholar
  18. P. L. Hammer and S. Rudeanu, Boolean Methods in Operations Research and Related Areas, Springer: NewYork, 1968.Google Scholar
  19. P. Hansen, B. Jaumard, and V. Mathon, “Constrained nonlinear 0–1 programming,” ORSA Journal of Computing vol. 5, pp. 87–119, 1993.Google Scholar
  20. P. Hansen, M. V. Poggi de Aragao, and C. C. Ribeiro, “Hyperbolic 0–1 programming and query optimization in information retrieval,” Mathematical Programming vol. 52, pp. 255–263, 1991.Google Scholar
  21. S. Hashizume, M. Fukushima, N. Katoh, and T. Ibaraki, “Approximation algorithms for combinatorial fractional programming problems,” Mathematical Programming vol. 37, pp. 255–267, 1987.Google Scholar
  22. W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes vol. 187 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag: New York, 1981.Google Scholar
  23. J. R. Isbelle and W. H. Marlow, “Attrition games,” Naval Research Logistics Quarterly vol. 3, pp. 71–93, 1956.Google Scholar
  24. J. Krarup and O. Bilde, “Plant location, set covering and economic lot size: An O(mn) algorithm for structured problems,” in International Series of Numerical Mathematics vol. 36, L. Collatz et al., eds., Birkhäuser Verlag: Basel, pp. 155–180, 1977.Google Scholar
  25. E. L. Lawler, “Sequencing jobs to minimize total weighted completion time subject to precedence constraints,” Annals of Discrete Mathematics vol. 2, pp. 75–90, 1978.Google Scholar
  26. H. Li, “A global approach for general 0–1 fractional programming,” European Journal of Operational Research vol. 73, pp. 590–596, 1994.Google Scholar
  27. J. S. Logsdon and L. T. Biegler, “Accurate solution of differential-algebraic optimization problems,” Industrial & Engineering Chemistry Research vol. 28, pp. 1628–1639, 1989.Google Scholar
  28. R. K. Martin, “Generating alternative mixed-integer programming models using variable redefinition,” Operations Research vol. 35, pp. 820–831, 1987.Google Scholar
  29. G. P. McCormick, “Computability of global solutions to factorable nonconvex programs: Part I–convex underestimating problems,” Mathematical Programming vol. 10, pp. 147–175, 1976.Google Scholar
  30. N. Megiddo, “Combinatorial optimization with rational objective functions,” Mathematics of Operations Research vol. 4, pp. 414–424, 1979.Google Scholar
  31. B. A. Murtagh and M. A. Saunders, “MINOS 5.5 user's guide,” Technical report, Technical Report SOL 83–20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, CA, 1995.Google Scholar
  32. G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons: New York, 1988.Google Scholar
  33. J. Picard and M. Queyranne, “A network flow solution to some nonlinear 0–1 programming problems with applications to graph theory,” Networks vol. 12, pp. 141–159, 1982.Google Scholar
  34. I. Quesada and I. E. Grossmann, “A global optimization algorithm for linear fractional and bilinear programs,” Journal of Global Optimization vol. 6, pp. 39–76, 1995.Google Scholar
  35. A. Quist, Personal Communication, April 2000a.Google Scholar
  36. A. J. Quist, “Application of mathematical optimization techniques to nuclear reactor reload pattern design,” Ph.D. Thesis, Technische Universiteit, Delft, 2000b.Google Scholar
  37. M. R. Rao, “Cluster analysis and mathematical programming,” Journal of the American Statistical Association vol. 66, pp. 622–626, 1971.Google Scholar
  38. R. L. Rardin and U. Choe, “Tighter relaxations of fixed charge network flow problems,” Technical Report J-79–18, Industrial and Systems Engineering Report Series, Georgia Institute of Technology, Atlanta, GA, 1979.Google Scholar
  39. A. D. Rikun, “A convex envelope formula for multilinear functions,” Journal of Global Optimization vol. 10, pp. 425–437, 1997.Google Scholar
  40. P. Robillard, “(0, 1) hyperbolic programming problems,” Naval Research Logistics Quarterly vol. 18, pp. 47–57, 1971.Google Scholar
  41. R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series. Princeton University Press, 1970.Google Scholar
  42. A. L. Saipe, “Solving a (0, 1) hyperbolic program by branch and bound,” Naval Research Logistics Quarterly vol. 22, pp. 497–515, 1975.Google Scholar
  43. S. Schaible, “Fractional programming with sums of ratios,” in Proceedings of the Workshop held in Milan on March 28, 1995, E. Castagnoli and J. Giorgi, eds., Scalar and Vector Optimization in Economic and Financial Problems, pp. 163–175, 1995.Google Scholar
  44. H. D. Sherali and W. P. Adams, “Ahierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems,” SIAM Journal of Discrete Mathematics vol. 3, pp. 411–430, 1990.Google Scholar
  45. H. D. Sherali and W. P. Adams, “A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems,” Discrete Applied Mathematics vol. 52, no. 1, pp. 83–106, 1994.Google Scholar
  46. I. M. Stancu-Minasian, Fractional Programming, Kluwer Academic Publishers: Netherlands, 1997.Google Scholar
  47. J. B. Sydney, “Decomposition algorithm for single-machine sequencing with precedence relations and deferral costs,” Operations Research vol. 23, pp. 283–298, 1975.Google Scholar
  48. M. Tawarmalani, S. Ahmed, and N. V. Sahinidis, “Global optimization of 0–1 hyperbolic programs,” Journal of Global Optimization vol. 24, pp. 385–417, 2002.Google Scholar
  49. M. Tawarmalani and N. V. Sahinidis, “Convexification and global optimization in continuous and mixed-integer nonlinear programming: Theory, algorithms, software and applications,” in Nonconvex Optimization and Its Applications Series, vol. 65, Kluwer Academic Publishers: Dordrecht, 2002a.Google Scholar
  50. M. Tawarmalani and N. V. Sahinidis, “Convex extensions and convex envelopes of l.s.c. functions,” Mathematical Programming, DOI 10.1007/s10107–002–0308–z, September 5, 2002b.Google Scholar
  51. H. P. Williams, “Experiments in the formulation of integer programming problems,” Mathematical Programming Study vol. 2, pp. 180–197, 1974.Google Scholar
  52. T. Wu, “A note on a global approach for general 0–1 fractional programming,” European Journal of Operational Research vol. 101, pp. 220–223, 1997.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mohit Tawarmalani
    • 1
  • Shabbir Ahmed
    • 2
  • Nikolaos V. Sahinidis
    • 3
  1. 1.Krannert School of ManagementPurdue UniversityWest Lafayette
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlanta
  3. 3.Department of Chemical and Biomolecular EngineeringUniversity of Illinois at Urbana-ChampaignUrbana

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