Linear Programming

  • Bernhard Korte
  • Jens Vygen
Part of the Algorithms and Combinatorics book series (AC, volume 21)


In this chapter we review the most important facts about Linear Programming. Although this chapter is self-contained, it cannot be considered to be a comprehensive treatment of the field. The reader unfamiliar with Linear Programming is referred to the textbooks mentioned at the end of this chapter.


General Literature

  1. Bertsimas, D., and Tsitsiklis, J.N. [1997]: Introduction to Linear Optimization. Athena Scientific, Belmont 1997Google Scholar
  2. Chvátal, V. [1983]: Linear Programming. Freeman, New York 1983Google Scholar
  3. Matoušek, J., and Gärtner, B. [2007]: Understanding and Using Linear Programming. Springer, Berlin 2007Google Scholar
  4. Padberg, M. [1999]: Linear Optimization and Extensions. Second Edition. Springer, Berlin 1999Google Scholar
  5. Schrijver, A. [1986]: Theory of Linear and Integer Programming. Wiley, Chichester 1986Google Scholar

Cited References

  1. Avis, D., and Chvátal, V. [1978]: Notes on Bland’s pivoting rule. Mathematical Programming Study 8 (1978), 24–34Google Scholar
  2. Bland, R.G. [1977]: New finite pivoting rules for the simplex method. Mathematics of Operations Research 2 (1977), 103–107Google Scholar
  3. Borgwardt, K.-H. [1982]: The average number of pivot steps required by the simplex method is polynomial. Zeitschrift für Operations Research 26 (1982), 157–177Google Scholar
  4. Carathéodory, C. [1911]: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rendiconto del Circolo Matematico di Palermo 32 (1911), 193–217Google Scholar
  5. Dantzig, G.B. [1951]: Maximization of a linear function of variables subject to linear inequalities. In: Activity Analysis of Production and Allocation (T.C. Koopmans, ed.), Wiley, New York 1951, pp. 359–373Google Scholar
  6. Dantzig, G.B., Orden, A., and Wolfe, P. [1955]: The generalized simplex method for minimizing a linear form under linear inequality restraints. Pacific Journal of Mathematics 5 (1955), 183–195Google Scholar
  7. Farkas, G. [1894]: A Fourier-féle mechanikai elv alkalmazásai. Mathematikai és Természettudományi Értesitö 12 (1894), 457–472Google Scholar
  8. Gale, D., Kuhn, H.W., and Tucker, A.W. [1951]: Linear programming and the theory of games. In: Activity Analysis of Production and Allocation (T.C. Koopmans, ed.), Wiley, New York 1951, pp. 317–329Google Scholar
  9. Hoffman, A.J., and 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, eds.), Princeton University Press, Princeton 1956, pp. 223–246Google Scholar
  10. Kelner, J.A., and Spielman, D.A. [2006]: A randomized polynomial-time simplex algorithm for linear programming. Proceedings of the 38th Annual ACM Symposium on Theory of Computing (2006), 51–60Google Scholar
  11. Klee, V., and Minty, G.J. [1972]: How good is the simplex algorithm? In: Inequalities III (O. Shisha, ed.), Academic Press, New York 1972, pp. 159–175Google Scholar
  12. Kuhn, H.W. [1956]: Solvability and consistency for linear equations and inequalities. The American Mathematical Monthly 63 (1956), 217–232Google Scholar
  13. Minkowski, H. [1896]: Geometrie der Zahlen. Teubner, Leipzig 1896Google Scholar
  14. Motzkin, T.S. [1936]: Beiträge zur Theorie der linearen Ungleichungen (Dissertation). Azriel, Jerusalem 1936Google Scholar
  15. von Neumann, J. [1947]: Discussion of a maximum problem. Working paper. Published in: John von Neumann, Collected Works; Vol. VI (A.H. Taub, ed.), Pergamon Press, Oxford 1963, pp. 27–28Google Scholar
  16. Spielman, D.A., and Teng, S.-H. [2004]: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. Journal of the ACM 51 (2004), 385–463Google Scholar
  17. Steinitz, E. [1916]: Bedingt konvergente Reihen und konvexe Systeme. Journal für die reine und angewandte Mathematik 146 (1916), 1–52Google Scholar
  18. Weyl, H. [1935]: Elementare Theorie der konvexen Polyeder. Commentarii Mathematici Helvetici 7 (1935), 290–306Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  • Bernhard Korte
    • 1
  • Jens Vygen
    • 1
  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations