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Discrete Analysis and Operations Research pp 59–79Cite as

Regular Partitions and Cuts in Integer Programming

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 355))

Abstract

In [1]-[22] an approach to problems of integer programming (IP) based on the use of regular partitions of their relaxation sets is proposed and developed. Using this approach, new classes of cuts are described and investigated. The upper and lower bounds for the number of iterations (cuts) are found for fractional cutting-plane algorithms (including some Gomory algorithms) as well as upper bounds for the number of iterations for several lexicographical enumeration and branch-and-bound algorithms. The algorithms are suggested for solving and analyzing IP problems using the structure of set partitions into ¿-classes; the results of numerical experiments are presented. The present paper reviews the results obtained on this subject. Purely integer problems are mainly considered. In addition, a number of new results related to Boolean programming (BP) problems are given. The survey [11] and a part of the book [23], both focussed mainly on L-partitions, are devoted to this subject as well.

This research was partially supported by the Russian Foundation for Fundamental Research (Grant 93–01–00557).

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References

  1. O. A. Zablotskaya (1984) Comparison of completely regular truncation algorithms (in Russian), Upravlyaemye Sistemy 25, 68–74.

    MathSciNet  MATH  Google Scholar 

  2. O. A. Zablotskaya and A. A. Kolokolov (1983) Completely regular truncations in Boolean programming (in Russian), Upravlyaemye Sistemy 23, 55–63.

    MathSciNet  MATH  Google Scholar 

  3. G.G. Zabudskiǐ(1990) On an integer formulation of a problem on the arrangement of objects on a line (in Russian), Upravlyaemye Sistemy 30, 35–45.

    Google Scholar 

  4. G. M. Zaikina and A. A. Kolokolov (1989) Experimental researches in integer programming by using of L-partition (in Russian), in: Discrete Optimization and Analysis of Complicated Systems, Novosibirsk, pp. 26–56.

    Google Scholar 

  5. G. M. Zaikina, A. A. Kolokolov, and T. V. Levanova (1992) Experimental comparison of some integer programming methods (in Russian), in: Methods of Solution and Analysis of Discrete Optimization Problems, Omsk, pp. 25–41.

    Google Scholar 

  6. A. A. Kolokolov (1980) On the lexicographical structure of some convex polyhedral sets (in Russian), in: Fifth All-Union Conference on Problems of Theoretical Cybernetics. Abstracts of Reports, Novosibirsk, pp. 77–79.

    Google Scholar 

  7. A. A. Kolokolov (1981) Regular cuttings in the solution of problems of integer optimization (in Russian), Upravlyaemye Sistemy 21, 18–25.

    MathSciNet  MATH  Google Scholar 

  8. A. A. Kolokolov (1983) Lower bound for the number of iterations for a class of truncation algorithms (in Russian), Upravlyaemye Sistemy 23, 64–69.

    MathSciNet  MATH  Google Scholar 

  9. A. A. Kolokolov (1984) Discrete Optimization Methods (in Russian), Omsk. University, Omsk.

    Google Scholar 

  10. A. A. Kolokolov (1984) On fastest algorithm in one class of regular cut processes (in Russian), in: Abstracts of the Third All-Union Conference (Tashkent), Akad. Nauk SSSR, Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, Part 2, p. 70.

    Google Scholar 

  11. A. A. Kolokolov (1986) Truncation algorithms and some set partition (in Russian), in: Discrete Optimization and Numerical Methods for Solving Applied Problems, Akad. Nauk SSSR, Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, pp. 50–67.

    Google Scholar 

  12. A. A. Kolokolov (1987) Convex sets with alternating L-partition (in Russian), in: Modelling and Optimization of Complicated Structure Systems, Omsk, pp. 144–150.

    Google Scholar 

  13. A. A. Kolokolov (1988) Researches of problems and methods of integer programming on the base of L-partition (in Russian), in: Wiss. Koll. Ilmenau: TH, pp. 53–55.

    Google Scholar 

  14. A. A. Kolokolov (1989) Method of evaluating partitions in integer programming (in Russian), in: Theory and Programmed Realization of Discrete Optimization Methods, Naukova Dumka, Kiev, pp. 44–47.

    Google Scholar 

  15. A. A. Kolokolov (1992) Regular partitions at integer programming (in Russian), in: Methods of Solution and Analysis of Discrete Optimization Problems, Omsk, pp. 67–93.

    Google Scholar 

  16. A. A. Kolokolov (1993) Regular partitions and branch-and-bounds method (in Russian), in: Abstracts of the Conference on Mathematical Programming and Applications (Ekaterinburg, 1993), Ekaterinburg, No. 4, p. 61 (Inform., Association of Mathematical Programming Bulletin).

    Google Scholar 

  17. A. A. Kolokolov and E. V. Tsepkova (1984) Upper bound for the number of regular truncations for a family of problems on a cone (in Russian), Upravlyaemye Sistemy 25, 75–79.

    MathSciNet  MATH  Google Scholar 

  18. A. A. Kolokolov and E. V. Tsepkova (1991) Investigation of the knapsack problem on the basis of L-partition (in Russian), Kibernetika (Kiev), No. 2, 38–43.

    MathSciNet  Google Scholar 

  19. L. A. Saǐko (1989) Investigation of a class of partitions in integer programming (in Russian), Upravlyaemye Sistemy 29, 72–82.

    Google Scholar 

  20. R. Yu. Simanchëv (1990) Completely regular cuttings for problems of integer optimization (in Russian), Upravlyaemye Sistemy 30, 61–71.

    MATH  Google Scholar 

  21. A. A. Kolokolov (1992) Upper bound of L-covering power in the boolean programming problems (in Russian), in: Third International Seminar on Global Optimization. Abstracts, Irkutsk, p. 108.

    Google Scholar 

  22. A. A. Kolokolov (1993) On the L-structure of integer linear programming problems, in: Abstracts of the 16th IFIP Conference on System Modelling and Optimization, IFIP, Compiegne, France, pp. 183–184.

    Google Scholar 

  23. I. V. Sergienko (1988) Mathematical Models and Solving Methods of Discrete Optimization Problems (in Russian), Naukova Dumka, Kiev.

    Google Scholar 

  24. A. A. Kolokolov (1975) On cutting planes number of first Gomory algorithm (in Russian), in: Problems of Analysis of Discrete Information, Part 1, Novosibirsk, pp. 84–96.

    Google Scholar 

  25. A. A. Kolokolov (1976) Upper bounds on cutting planes number for cyclical Gomory algorithm (in Russian), in: Methods of Information Modelling and Processing, Nauka, Novosibirsk, pp. 106–116.

    Google Scholar 

  26. A. Schrijver (1986) Theory of Linear and Integer Programming, John Wiley & Sons, Chichester.

    Google Scholar 

  27. V. P. Bulatov (1977) Imbedding Methods in Optimization Problems (in Russian), Nauka, Novosibirsk.

    Google Scholar 

  28. A. A. Corbut and Yu. Yu. Finkel’shteǐn (1976) Discrete Programming (in Russian), Nauka, Moscow.

    Google Scholar 

  29. T. C. Hu (1970) Integer Programming and Network Flows, Addison-Wesley Publ. Co., Reading.

    Google Scholar 

  30. Yu. Yu. Chervak (1979) Search for a lexicographically maximal discretely defined point of a convex set (in Russian), in: Mathematical Methods for Solution of Economical Problems, No. 8, Nauka, Moscow, pp. 69–75.

    Google Scholar 

  31. V. N. Shevchenko (1979) Convex polyhedral cones, comparison systems and valid cuts in integer programming (in Russian), in: Combinatorial-Algebraic Methods in Applied Mathematics, Gor’kov. University, Gor’kiǐ, pp. 109–119.

    Google Scholar 

  32. R. E. Gomory (1958) Outline of an algorithm for integer solution to linear programs Bull. Amer. Math. Soc. 64, No. 5, 275–278.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Grötschel, L. Lovász, and A. Schrijver (1988) Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics, Part 2, Springer-Verlag, Berlin etc.

    Google Scholar 

  34. R. Jeroslow (1978) Cutting-plane theory: algebraic methods, Discrete Math., 23, No. 2, 121–150.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Piehler (1975) Einige Bemerkungen zum Schnittrang in der rein-ganzzahligen linearen Optimierung, Math. Operationsforsch. Statist. 6, No. 4, 523–533.

    MathSciNet  Google Scholar 

  36. S. Walukiewicz (1991) Integer programming, PWN-Polish Scientific Publ., Warszawa; Kluwer Acad. Publ., Dordrecht, etc.

    MATH  Google Scholar 

  37. V. A. Emelichev, M. M. Kovalev, and M. K. Kravtsov (1981) Polytopes, Graphs and Optimization (in Russian), Nauka, Moscow.

    Google Scholar 

  38. A. A. Kolokolov (1973) On length of lexicographically monotonous sequences (in Russian), in: Optimization of Territorial and Branch Systems, Economical Problems Solving Methods, Part 3, Novosibirsk, pp. 93–99.

    Google Scholar 

  39. F. J. Nourie and E. R. Venta (1981) An upper bound on the number of cuts needed in Gomory’s method of integer forms Oper. Res. Lett. 1, No. 4, 129–133.01D0

    Article  MathSciNet  MATH  Google Scholar 

  40. A. E. Bakhtin, A. A. Kolokolov, and Z. V. Korobkova (1978) Discrete Problems of Production-Transportation Type (in Russian), Nauka, Novosibirsk.

    Google Scholar 

  41. V. L. Beresnev, È. Kh. Gimadi, and V. T. Dement’ev (1978) Extremal Standardization Problems (in Russian), Nauka, Novosibirsk.

    Google Scholar 

  42. C. A. Trauth and R. E. D. Woosley (1969) Integer programming: a study in computation efficiency, Management Sci. 15, No. 9, 481–493

    Article  Google Scholar 

  43. C. C. Petersen (1967) Computational experience with variants of the Balas algorithm applied to the selection of R&D projects, Management Sci. 13, No. 9, 736–750.

    Article  Google Scholar 

  44. L. Kivistick (1986) On acceleration of all-integer Gomory algorithm (in Russian), Eesti NSV Tead. Akad. Toimetised Füüs.-Mat. 34, No. 1, 11–19.

    Google Scholar 

  45. M. G. Chang and F. Shepardson (1982) An integer programming test problem generator, Lecture Notes in Economics and Mathematical Systems 199, 146–160.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Informational Technology and Applied Mathematics, ul. Andrianova, 28, Omsk, 644077, Russia

    A. A. Kolokolov

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  1. A. A. Kolokolov
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Editors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Siberia, Russia

    Alekseǐ D. Korshunov

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© 1996 Kluwer Academic Publishers

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Kolokolov, A.A. (1996). Regular Partitions and Cuts in Integer Programming. In: Korshunov, A.D. (eds) Discrete Analysis and Operations Research. Mathematics and Its Applications, vol 355. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1606-7_6

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  • DOI: https://doi.org/10.1007/978-94-009-1606-7_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7217-5

  • Online ISBN: 978-94-009-1606-7

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