Discrete Optimization

  • Alexander S. Belenky
Part of the Applied Optimization book series (APOP, volume 20)


If M is given by a system of inequalities with the additional stipulation that all its variables are integers, i.e.,
the problem of minimizing f(x) on M is called a discrete optimization problem or a discrete programming problem. One of the classes of the given type problems is formed by Boolean programming problems in which all variables may assume only two values: 0 and 1. The simplest discrete optimization problems are integer linear programming problems in which M is given by a system of linear inequalities of the type


Feasible Solution Integer Programming Dual Problem Linear Programming Problem Initial 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kivistik, L. On speeding up the first Gomory algorithm. Izvestiya Akademii Nauk ESSR, Fizika i matematika. 1988; 37, No. 1: 85–88 [in Russian].MathSciNetzbMATHGoogle Scholar
  2. [2]
    Geoffrion, A. M., and Marsten, R. E. Integer programming algorithms: a framework and state of the art survey. Management Science. 1972; 18, No. 9: 465–491.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Finkelshtein, Yu. Yu. Priblizhennye Metody i Prikladnye Zadachi Diskretnogo Programmirovania (Approximation Methods and Applied Problems of Discrete Programming). Moscow: Nauka, 1976 [in Russian].Google Scholar
  4. [4]
    Skaletskii, V. V. Modified branch-and-bound method for solving a series of problems. Automation and Remote Control. 1980; No. 4: 493–499.MathSciNetGoogle Scholar
  5. [5]
    Bellman, R. E. Dynamic Programming and Modern Control Theory. New York: Academic Press, 1965.Google Scholar
  6. [6]
    Venttsel, E. S. Operations Research. Wright-Patterson, Air Force Base, Ohio: Foreign Technology Division, Air Force System Command, 1978.Google Scholar
  7. [7]
    Mikhalevich, V. S., and Kuksa, A. I. Metody Posledovatel’noi Optimizatzii (Methods of Successive Optimization). Moscow: Nauka, 1983 [in Russian].Google Scholar
  8. [8]
    Sergienko, I. V. Matematicheskie Modeli i Metody Reschenia Zadach Diskretnoi Optimizatzii (Mathematical Models and Solution Methods of Discrete Optimization Problems). Kiev: Naukova dumka, 1985 [in Russian].Google Scholar
  9. [9]
    Holm, S., and Tind, J. A unified approach for price directive decomposition procedures in integer programming. Discrete Applied Mathematics. 1988; 20, No. 3: 205–219.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Shlyk, V. A. On the group theory approach in integer programming. Izvestija Akademii Nauk SSSR, Tekhnicheskaya kibernetika. 1988; No. 1: 94–105 [in Russian).Google Scholar
  11. [11]
    Barvinok, A. I., and Vershik, A. M. Methods of representations theory in combinatorial optimization problem. Soviet Journal of Computer and Systems Sciences (Formerly Engineering Cybernetics). 1989; 27, No. 5: 1–8.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Johnson, E. L. Integer Programming. Facets, Subadditivity and Duality for Group and Semi-Group Problems. Philadelphia: SIAM, 1980.Google Scholar
  13. [13]
    Hu, T. C. Integer Programming and Network Flows. Reading, Mass.: AdissonWesley Pub. Co., 1969.zbMATHGoogle Scholar
  14. [14]
    Grotschel, M., Lovasz, L., and Schrijver, A. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica. 1981; 1, No. 2: 169–197.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Korbut, A. A., and Finkelshtein, Yu. Yu. Diskretnoe Programmirovanie (Discrete Programming). Moscow: Nauka, 1969 [in Russian].Google Scholar
  16. [16]
    Polyak, B. T. Introduction to Optimization. New York: Optimization Software, Publication Division, 1987.Google Scholar
  17. [17]
    Azaryan, L. L., Lebedev, S. S., and Mestetskii, L. M. Solving transportation-type integer problems using generalized Lagrange multipliers. Economika i matematicheskie metody. 1977; XIII, No. 4: 723–731 [in Russian].Google Scholar
  18. [18]
    Shapiro, J. F. Generalized Lagrange multipliers in integer programming. Operations Research. 1971; 19, No. 1: 68–76.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Matematicheskii Apparat Ekonomicheskogo Modelirovania (Mathematical Tools of Economic Modelling). Moscow: Nauka, 1983 [in Russian].Google Scholar
  20. [20]
    Sheinman, O. K. Duality in some discrete minimization problems. Russian Mathematical Surveys. 1978; 32, No. 2: 251–252.CrossRefGoogle Scholar
  21. [21]
    Alekseyev, O. G. Kompleksnoe Primenenie Metodov Diskretnoi Optimizatzii (Complex Use of Discrete Optimization Methods). Moscow: Nauka, 1987, [in Russian].Google Scholar
  22. [22]
    Christofides, N. Graph Theory: an Algorithmic Approach. New York: Academic Press, 1975.zbMATHGoogle Scholar
  23. [23]
    Smith, B. M. IMPACS—a bus crew scheduling system using integer programming. Mathematical Programming. 1988; 42, No. 1: 181–187.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Davydov, G. V., and Davydova, I. M. Duality and non-tree search in discrete optimization. Izvestija Akademii Nauk SSSR, Tekhnicheskaya kibernetika. Moscow, 1988; No. 1: 86–93 [in Russian].Google Scholar
  25. [25]
    Korbut, F. F., Sygal, I. Kh., and Finkel’stein, Yu. Yu. Hybrid methods in discrete optimization. Izvestija Akademii Nauk SSSR, Tekhnicheskaya kibernetika. 1988; No. 1: 65–77 [in Russian].Google Scholar
  26. [26]
    Spielberg, K., and Suhl, U.H. “Solving large-scale integer optimization problems.” In Progress in Scientific Computing. Large Scale Scientific Computing. Edited by P.Deuflhard and B. Engquist. 7, Boston: Birkhauser, 1987.Google Scholar
  27. [27]
    Sigal, I. Kh. Kombinirovannye Algoritmy Reschenia Zadachi Kommivojazhera. Preprint. (Combined Algorithms of Solving the Travelling Salesman Problem. Preprint). Moscow: Izd. VTz Akademii Nauk SSSR ( Computer Center, USSR Academy of Sciences ), 1985 [in Russian].Google Scholar
  28. [28]
    Korbut, A. A., and Finkel’shteyn, Yu. Yu. Methods of discrete programming. Engineering Cybernetics. 1983; 21, No. 1: 124–134.Google Scholar
  29. [29]
    Land, A., and Powell, S. “Computer codes for problems of integer programming.” In Annals of Discrete Mathematics. Discrete Optimization II. Amsterdam: North-Holland, 1979; 5: 221–269.Google Scholar
  30. [30]
    Borodin, V. V., Lovetskii, S. E., Melamed, I. I., and Plotinskii, Yu. M. Zadachi Marschrutizatzii. Vychislitel’nyi Aspekt. Preprint. (Routing Problems. Computational Aspect. Preprint). Moscow: Izd. Institut Problem Upravlenia ( Institute of Control Sciences, USSR Academy of Sciences ), 1981 [in Russian].Google Scholar
  31. [31]
    Frumkin, M. A. Slozhnost’ Diskretnykh Zadach. Preprint (Complexity of Discrete Problems. Preprint.). Moscow: Izd. TsEMI AN SSSR ( Central Economic-Mathematical Institute, USSR Academy of Sciences ), 1981 [in Russian].Google Scholar
  32. [32]
    Yudin, D. B., and Yudin, A. D. Chislo i Mysl’. Matematiki Izmerjaut Slozhnost’. (Number and Thought. Mathematicians Measure Complexity). No. 8. Moscow: Znanie, 1985; [in Russian].Google Scholar
  33. [33]
    Kamburowski, J. On the computational complexity of the shortest route and maximum flow problems in stochastic networks. Foundations of Control Engineering. 1986; 11, No. 4: 167–175.Google Scholar
  34. [34]
    Afrati, F., Cosmadakis, S., Papadimitriou, C. H., Papageorgiou, G., and Papakostantinou, N. The complexity of the travelling repairman problem. RAIRO Informatique Theorique. 1986; 20, No. 1: 79–87.MathSciNetzbMATHGoogle Scholar
  35. [35]
    Pel’tsverger, B. V., and Khavronin, O. V. “Decomposition approach and discrete problems solution complexity.” In Decompozitzia i Coordinatzia v Kornpleksnykh Sistemakh. Vsesojuznaya Konferentzia. Tezisy Dokladov. Chelyabinsk, Mart 1986. Chast’ 1. (Decomposition and Coordination in Complex Systems. All- Union Scientific Conference. Abstracts of Papers. Chelyabinsk, March 1986, part I). Chelyabinsk: Izd. Chelyabinskii Politechnicheskii Institut (Chelyabinsk Polytechnic Institute), 1986, 101–102 [in Russian].Google Scholar
  36. [36]
    Fridman, A. A., Frumkin, M. A., Khmelevskii, Yu. I., and Levner, E. V. Issledovanie Algoritmov Reschenia Diskretnykh i Kombinatornykh Zadach, Teoria Svodimosti Zadach, Videlenie Universal’nykh Zadach (Investigation of the Effectiveness of Algorithms for Discrete Combinatorial Problems, Reducibility Theory, Universal Problems). Moscow: Izd. TsEMI AN SSSR ( Central Economic-Mathematical Institute, USSR Academy of Sciences ), 1976 [in Russian].Google Scholar
  37. [37]
    Yudin, D. B., and Yudin, A. D. Ekstremal’nye Modeli v Ekonomike (Extreme Models in Economics). Moscow: Ekonomika, 1979 [in Russian].Google Scholar
  38. [38]
    Trubin, V. A., and Sharifov, F. A. An efficient method of solving one class of allocation problems. Doklady Akademii Nauk AzSSR. 1986; 42, No. 11: 7–11 [in Russian].MathSciNetzbMATHGoogle Scholar
  39. [39]
    Grunspan, M., and Thomas, M. E. Hyperbolic integer programming. Naval Research Logistics Quarterly. 1973; 20, No. 2: 341–356.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Grunspan, M. Fractional Programming:a Survey. Technical report N50. Project THEMIS. Systems Research Center, Industrial and Systems Engineering Department, University of Florida, 1971.Google Scholar
  41. [41]
    Yemelichev, V. A., and Tyong Bui Kat. Decomposition approach for solving quasi-block discrete optimization problems on the basis of the plans sequence construction method. Kibernetika. 1988; No. 1: 116–118 [in Russian].Google Scholar
  42. [42]
    Tsurkov, V. I. “Block integer programming.” In Chislennye Metody i Optimizatzia. Materiali IV Simposiuma (Numerical Methods and Optimization. Papers of the 4th Symposium). Tallinn: Izd. Valgus, 1988; 194–196 [in Russian].Google Scholar
  43. [43]
    Lasdon, L. S. Optimization Theory for Large Systems. New York: Macmillan, 1970.zbMATHGoogle Scholar
  44. [44]
    Zykov, A. A. Osnovy Teorii Grafov (Fundamentals of the Theory of Graphs). Moscow: Nauka, 1987 [in Russian].Google Scholar
  45. [45]
    Karibskaya, Z. V., and Ostrovskii, V. A. “Automated subsystem of on-line planning and control of automobile transport in CAS ”RAPO“.” In Razrabotka Optimal’nykh Modul’nykh Sistem Obrabotki Dannykh (Developing Optimal Module Systems of Data Processing). Moscow: Izd. Institut Problem Upravleniya (Institute of Control Sciences, USSR Academy of Sciences ), 1987; 67–72 [in Russian].Google Scholar
  46. [46]
    Swamy, M., and Thulasirman, K. Graphs, Networks, and Algorithms. New York: John Wiley & Sons Publ. Co., 1981.Google Scholar
  47. [47]
    Lipatov, E. P. Teoria Graphov i ee Primenenia (Theory of Graphs and its Application). Moscow: Znanie, 1986 [in Russian].Google Scholar
  48. [48]
    Gallo, G., and Pallottino, S. Shortest path algorithms. Annals of Operations Research. 1988; 13, No. 1–4: 3–79.MathSciNetGoogle Scholar
  49. [49]
    de Queiros Viera Martins, E. An algorithm for ranking paths that may contain cycles. European Journal of Operational Research. 1984; 18, No. 1: 123–130.MathSciNetCrossRefGoogle Scholar
  50. [50]
    Laporte, G., Murcure, H., and Nobert, Y. Optimal tour planning with specified nodes. RAIRO Recherche Operationnelle/Operations Research. 1984; 18, No. 3: 203–210.zbMATHGoogle Scholar
  51. [51]
    Pederzoli, G., and Sancho, N. G. F. A shortest path routing problem with resource allocation. Journal of Mathematical Analysis and Applications. 1987; 124, No. 1: 33–42.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    Luby, M., and Radge, P. A bidirectional shortest-path algorithm with good average-case behavior. Algorithmica. 1989; 4, No. 4: 551–567.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    Hall, R. W. The fastest path through a network with random time-dependent travel times. Transportation Science. 1986; 20, No. 3: 182–188.CrossRefGoogle Scholar
  54. [54]
    Masao, F. On the dual approach to the traffic assignment problem. Transportation Research. 1984; 18B No. 3: 235–245.Google Scholar
  55. [55]
    Derigs, U., and Schrader, R. A short note on optimal assignable sets and allocation problems. Bolletino Unione Mathematica Italiana. 1984; 3, No. 1: 97–101.MathSciNetzbMATHGoogle Scholar
  56. [56]
    Carraresi, P., and Gallo, G. Optimization models in mass transit resources management. Ricerca Operativa. 1986; 16 No. 38 numero spec. 121–150.Google Scholar
  57. [57]
    Derigs, U. Solving non-bipartite matching problems via shortest path techniques. Annals of Operations Research. 1988; 13, No. 1–4: 225–261.MathSciNetCrossRefGoogle Scholar
  58. [58]
    Burkov, V. N., Lovetskii, S. E., and Gorgidze, I. A. Prikladnye Zadachi Teorii Graphov (Applied Problems of the Theory of Graphs). Tbilisi: Izd. VTs AN GSSR ( Computer Center, GSSR Academy of Sciences ), 1974 [in Russian].Google Scholar
  59. [59]
    Yemelichev, V. A., Perepelitsa, V. A., and Shungarov, H. D. “A study of one multi-criterion problem on graphs.” In Diskretnaja Optimizatzia i Komputery. III Vsesojuznaja Schkola. Tashtagol. Tezisy Dokladov (Discrete Optimization and Computers. III All- Union School, Tashtagol. Abstracts of Papers). Moscow: Izd. TsEMI AN SSSR ( Central Economic-Mathematical Institute, USSR Academy of Sciences ), 1987; 28–29 [in Russian].Google Scholar
  60. [60]
    Grabowski, J., and Skubalska, E. Optymalizacija struktury sieci transportowej przy kryterium minimalizacji kosztow przeplyuue. Archiwum Automatykii i Telemechaniki. 1985; 30, No. 1: 3–21.zbMATHGoogle Scholar
  61. [61]
    Basangova, E. O. “On one transportation problem on partially oriented graphs.” In Algebra i Discretnaja Matematika (Algebra and Discrete Mathematics). Elista: Izd. Kalmitzkii GU (Kalmik State University), 1985; 61–70 [in Russian].Google Scholar
  62. [62]
    Gomes, L. F. A. M. On modelling equilibrium traffic flow assignment with elastic demand as a stochastic nonlinear vector optimization problem. Foundations of Control Engineering. 1986; 11, No. 4: 157–166.zbMATHGoogle Scholar
  63. [63]
    Janecki, R., and Roznowski, M. Optymalizacja liczby stacji i punktow ladunkowych w arunkach koncentracji prac ladunkowych. In Problemy Ekonomiczne Transport y. Bulletin Informatzii. 1986; No. 2: 52–66.Google Scholar
  64. [64]
    Perykasza, E., and Janecki, R. Ein Modell fur die optimale Verteilung der Ladestellen im Eisenbahnnetz. Wissenschaftliche Zeitschrift der Hochshule fur Verkehrswesen. Friederich List. Dresden, 1987; No. 1: 87–95.Google Scholar
  65. [65]
    Balinski, M. L. The Hirsch conjecture for dual transportation polyhedra. Mathematics of Operations Research. 1984; 9, No. 4: 629–633.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [66]
    Suzuki, A., and Iri, M. A heuristic method for the Euclidean Steiner problem as a geographical optimization problem. Asia-Pacific Journal of Operational Research. 1986; 3, No. 2: 109–122.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Alexander S. Belenky

There are no affiliations available

Personalised recommendations