Skip to main content
SpringerLink
Log in

Ganzzahlige Optimierung

  • Chapter
  • First Online:
Kombinatorische Optimierung

Part of the book series: Masterclass ((MASTERCLASS))

  • 6269 Accesses

Zusammenfassung

In diesem Kapitel betrachten wir lineare Programme mit ganzzahligen Nebenbedingungen. Unter bestimmten Bedingungen existieren immer optimale ganzzahlige Lösungen. Außerdem betrachten wir Schnittebenenverfahren und Lagrange-Relaxierung.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

Allgemeine Literatur:

  • Bertsimas, D., und Weismantel, R. [2005]: Optimization Over Integers. Dynamic Ideas, Belmont 2005

    Google Scholar 

  • Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., und Schrijver, A. [1998]: Combinatorial Optimization. Wiley, New York 1998, Kapitel 6

    Google Scholar 

  • Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G. und Wolsey, L., Hrsg. [2010]: 50 Years of Integer Programming 1958–2008. Springer, Berlin 2010

    Google Scholar 

  • Nemhauser, G.L., und Wolsey, L.A. [1988]: Integer and Combinatorial Optimization. Wiley, New York 1988

    Book  Google Scholar 

  • Schrijver, A. [1986]: Theory of Linear and Integer Programming. Wiley, Chichester 1986

    Google Scholar 

  • Wolsey, L.A. [1998]: Integer Programming. Wiley, New York 1998

    Google Scholar 

Zitierte Literatur:

  • Anstreicher, K.M., und Wolsey, L.A. [2009]: Two “well-known’’ properties of subgradient optimization. Mathematical Programming B 120 (2009), 213–220

    Google Scholar 

  • Boyd, E.A. [1997]: A fully polynomial epsilon approximation cutting plane algorithm for solving combinatorial linear programs containing a sufficiently large ball. Operations Research Letters 20 (1997), 59–63

    Article  MathSciNet  Google Scholar 

  • Bruns, W., Gubeladze, J., Henk, M., Martin, A., und Weismantel, R. [1999]: A counterexample to an integral analogue of Carathéodory’s theorem. Journal für die Reine und Angewandte Mathematik 510 (1999), 179–185

    Google Scholar 

  • Carr, R., und Vempala, S. [2004]: On the Held-Karp relaxation for the asymmetric and symmetric traveling salesman problems. Mathematical Programming A 100 (2004), 569–587

    Google Scholar 

  • Chvátal, V. [1973]: Edmonds’ polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4 (1973), 305–337

    Article  MathSciNet  Google Scholar 

  • Cook, W.J. [1983]: Operations that preserve total dual integrality. Operations Research Letters 2 (1983), 31–35

    Article  MathSciNet  Google Scholar 

  • Cook, W.J., Fonlupt, J., und Schrijver, A. [1986]: An integer analogue of Carathéodory’s theorem. Journal of Combinatorial Theory B 40 (1986), 63–70

    Google Scholar 

  • Cook, W.J., Gerards, A., Schrijver, A., und Tardos, É. [1986]: Sensitivity theorems in integer linear programming. Mathematical Programming 34 (1986), 251–264

    Article  MathSciNet  Google Scholar 

  • Cook, W.J., Kannan, R., und Schrijver, A. [1990]: Chvátal closures for mixed integer programming problems. Mathematical Programming 47 (1990), 155–174

    Article  MathSciNet  Google Scholar 

  • Cornuéjols, G., und Li, Y. [2016]: Deciding emptyness of the Gomory-Chvátal closure is NP-complete, even for a rational polyhedron containing no integer point. In: Integer Programming and Combinatorial Optimization; Proceedings of the 18th International IPCO Conference; LNCS 9682 (Q. Louveaux, M. Skutella, Hrsg.), Springer, Switzerland 2016, pp. 387–397

    MATH  Google Scholar 

  • Dadush, D., Dey, S.S., und Vielma, J.P. [2014]: On the Chvátal-Gomory closure of a compact convex set. Mathematical Programming A 145 (2014), 327–348

    Google Scholar 

  • Dantzig, G., Fulkerson, R., und Johnson, S. [1954]: Solution of a large-scale travelingsalesman problem. Operations Research 2 (1954), 393–410

    Google Scholar 

  • Edmonds, J., und Giles, R. [1977]: A min-max relation for submodular functions on graphs. In: Studies in Integer Programming; Annals of Discrete Mathematics 1 (P.L. Hammer, E.L. Johnson, B.H. Korte, G.L. Nemhauser, Hrsg.), North-Holland, Amsterdam 1977, 185–204

    Chapter  Google Scholar 

  • Eisenbrand, F. [1999]: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19 (1999), 297–300

    Article  MathSciNet  Google Scholar 

  • Eisenbrand, F., und Schulz, A.S. [2003]: Bounds on the Chvátal rank of polytopes in the 0/1-cube. Combinatorica 23 (2003), 245–261

    Google Scholar 

  • Fulkerson, D.R. [1971]: Blocking and anti-blocking pairs of polyhedra. Mathematical Programming 1 (1971), 168–194

    Article  MathSciNet  Google Scholar 

  • Geoffrion, A.M. [1974]: Lagrangean relaxation for integer programming. Mathematical Programming Study 2 (1974), 82–114

    Google Scholar 

  • Giles, F.R., und Pulleyblank, W.R. [1979]: Total dual integrality and integer polyhedra. Linear Algebra and Its Applications 25 (1979), 191–196

    Article  MathSciNet  Google Scholar 

  • Ghouila-Houri, A. [1962]: Caractérisation des matrices totalement unimodulaires. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris) 254 (1962), 1192–1194

    Google Scholar 

  • Goemans, M.X. [1995]: Worst-case comparison of valid inequalities for the TSP. Mathematical Programming 69 (1995), 335–349

    Article  MathSciNet  Google Scholar 

  • Goemans, M.X., und Skutella, M. [2004]: Cooperative facility location games. Journal of Algorithms 50 (2004), 194–214

    Article  MathSciNet  Google Scholar 

  • Goffin, J.L. [1977]: On convergence rates of subgradient optimization methods. Mathematical Programming 13 (1977), 329–347

    Article  MathSciNet  Google Scholar 

  • Gomory, R.E. [1958]: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64 (1958), 275–278

    Article  MathSciNet  Google Scholar 

  • Gomory, R.E. [1963]: An algorithm for integer solutions of linear programs. In: Recent Advances in Mathematical Programming (R.L. Graves, P. Wolfe, Hrsg.), McGraw-Hill, New York, 1963, pp. 269–302

    Google Scholar 

  • Graver, J.E. [1975]: On the foundations of linear and integer programming I. Mathematical Programming 9 (1975), 207–226

    Article  MathSciNet  Google Scholar 

  • Harvey, W. [1999]: Computing two-dimensional integer hulls. SIAM Journal on Computing 28 (1999), 2285–2299

    Article  MathSciNet  Google Scholar 

  • Hochbaum, D.S., und Levin, A. [2006]: Optimizing over consecutive 1’s and circular 1’s constraints. SIAM Journal on Optimization 17 (2006), 311–330

    Article  MathSciNet  Google Scholar 

  • Hoffman, A.J. [1974]: A generalization of max flow-min cut. Mathematical Programming 6 (1974), 352–359

    Article  MathSciNet  Google Scholar 

  • Hoffman, A.J., und Kruskal, J.B. [1956]: Integral boundary points of convex polyhedra. In: Linear Inequalities and Related Systems; Annals of Mathematical Study 38 (H.W. Kuhn, A.W. Tucker, Hrsg.), Princeton University Press, Princeton 1956, 223–246

    Google Scholar 

  • Lasserre, J.B. [2004]: The integer hull of a convex rational polytope. Discrete & Computational Geometry 32 (2004), 129–139

    Article  MathSciNet  Google Scholar 

  • Meyer, R.R. [1974]: On the existence of optimal solutions to integer and mixed-integer programming problems. Mathematical Programming 7 (1974), 223–235

    Article  MathSciNet  Google Scholar 

  • Pokutta, S., und Schulz, A.S. [2010]: On the rank of cutting-plane proof systems. In: Integer Programming and Combinatorial Optimization; Proceedings of the 14th International IPCO Conference; LNCS 6080 (F. Eisenbrand, F.B. Shepherd, Hrsg.), Springer, Berlin 2010, pp. 450–463

    Chapter  Google Scholar 

  • Polyak, B.T. [1967]: A general method for solving extremal problems. Doklady Akademii Nauk SSSR 174 (1967), 33–36 [auf Russisch]. Englische Übersetzung: Soviet Mathematics Doklady 8 (1967), 593–597

    Google Scholar 

  • Rothvoß, T., und Sanità, L. [2017]: 0/1 polytopes with quadratic Chvátal rank. Operations Research 65 (2017), 212–220

    Article  MathSciNet  Google Scholar 

  • Schrijver, A. [1980]: On cutting planes. In: Combinatorics 79; Part II; Annals of Discrete Mathematics 9 (M. Deza, I.G. Rosenberg, Hrsg.), North-Holland, Amsterdam 1980, pp. 291–296

    Chapter  Google Scholar 

  • Schrijver, A. [1981]: On total dual integrality. Linear Algebra and its Applications 38 (1981), 27–32

    Article  MathSciNet  Google Scholar 

  • Schrijver, A. [1983]: Packing and covering of crossing families of cuts. Journal of Combinatorial Theory B 35 (1983), 104–128

    Article  MathSciNet  Google Scholar 

  • Sebő, A. [1990]: Hilbert bases, Carathéodory’s theorem and combinatorial optimization. In: Integer Programming and Combinatorial Optimization (R. Kannan undW.R. Pulleyblank, Hrsg.), University of Waterloo Press, 1990

    Google Scholar 

  • Seymour, P.D. [1980]: Decomposition of regular matroids. Journal of Combinatorial Theory B 28 (1980), 305–359

    Article  MathSciNet  Google Scholar 

  • Veinott, A.F., Jr., und Dantzig, G.B. [1968]. Integral extreme points. SIAM Review 10 (1968), 371–372

    Article  MathSciNet  Google Scholar 

  • Wolsey, L.A. [1981]: The b-hull of an integer program. Discrete Applied Mathematics 3 (1981), 193–201

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Forschungsinstitut für Diskrete Mathematik, Universität Bonn, Bonn, Deutschland

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernhard Korte .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Korte, B., Vygen, J. (2018). Ganzzahlige Optimierung. In: Kombinatorische Optimierung. Masterclass. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57691-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation