Primal-Dual Pairs

  • Manfred Padberg
Part of the Algorithms and Combinatorics book series (AC, volume 12)


For every linear programming problem we have a “dual” linear programming problem. Whereas in the original or primal linear program the variables are associated with the columns of the constraint matrix, in the dual linear program the variables are associated with the rows of the constraint matrix. To bring out the symmetry of the construction of a primal-dual pair of linear programs we consider first the general case.


Feasible Solution Dual Basis Dual Solution Simplex Algorithm Choice Rule 
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. Cahn, A.S. [ 1948 ] “The warehouse problem”, Bull. Amer.Math.Soc. 54 10–73.Google Scholar
  2. Dantzig, G.B. [ 1982 ] “Reminiscences about the origins of linear programming”, Operations Research Letters 1 43–48.MathSciNetCrossRefGoogle Scholar
  3. Gale, D., Kuhn, H.W. and A.W. Tucker [ 1951 ] “Linear programming and the theory of games”, in Koopmans (ed) Activity Analysis of Production and Allocation, Wiley, New York.Google Scholar
  4. Goldman, A.J. and A.W. Tucker [ 1956 ] “Theory of linear programming”, in Kuhn and Tucker (eds), Linear Inequalities and Related Systems, Princeton U. Press, Princeton, 53–97.Google Scholar
  5. Weingartner, H.M. [ 1967 ] Mathematical Programming and the Analysis of Capital Budgeting Problems, Markham Publ. Co., Chicago.Google Scholar
  6. Farkas, J. [ 1902 ] “Theorie der einfachen Ungleichungen”, Journal für die reine und angewandte Mathematik 124 1–27.Google Scholar
  7. Goldman, A.J. [ 1956 ] “Resolution and separation theorems for polyhedral convex sets”, in Kuhn and Tucker (eds) Linear Inequalities and Related Systems, Princeton U. Press, Princeton, 41–51.Google Scholar
  8. Gordan, P. [ 1873 ] “Über die Auflösung linearer Gleichungen mit reellen Coefficienten”, Mathematische Annalen VI 23–28.MathSciNetCrossRefGoogle Scholar
  9. Kirchberger, P. [ 1903 ] “Über Tschebyschefsche Annäherungsmethoden”, Mathematische Annalen 57 509–540.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Kuhn, H.W. [ 1956 ] “Solvability and consistency for systems of linear equations and inequalities”, The American Mathematical Monthly 63 217–232.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Mangasarian, O.L. [ 1969 ] Nonlinear Programming, McGraw-Hill, New York.zbMATHGoogle Scholar
  12. Motzkin, Th.S. [1933] Beiträge zur Theorie der linearen Ungleichungen,DoctoralGoogle Scholar
  13. Thesis, University of Basel. English translation in Cantor et al (eds) Theodore S. Motzkin: Selected Papers, Birkhäuser, Boston, 1983 1–81.Google Scholar
  14. Stiemke, E. [ 1915 ] “Über positive Lösungen homogener linearer Gleichungen”, Mathematische Annalen 76 340–342.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Tucker, A.W. [ 1956 ] “Dual systems of homogeneous linear equations”, in Kuhn and Tucker (eds) Linear Inequalities and Related Systems, Princeton U. Press, Princeton, 3–18.Google Scholar
  16. Beale, E.M.L. [ 1954 ] “An alternative method for linear programming”, Proceedings of the Cambridge Philosophical Society 50 513–523.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Bland, R. [ 1977 ] “New finite pivot rules for the simplex method”, Mathematics of Operations Research 2 103–107.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Fourier, J. [ 1822 ] Théorie analytique de la chaleur, republished in Grattan-Guiness, I. [1972] Joseph Fourier 1768–1830, MIT Press, Cambridge.Google Scholar
  19. Gay, D.M. [ 1987 ] “A variant of Karmarkar’s linear programming algorithm for problems in standard form”, Mathematical Programming 37 81–90.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Grigoriadis, M.D. [ 1971 ] “A dual generalized upper bounding technique”, Management Science 17 269–284.zbMATHCrossRefGoogle Scholar
  21. Lemke, C.E. [ 1954 ) “The dual method of solving the linear programming problem”, Naval Research Logistics Quarterly 1 36–47.MathSciNetCrossRefGoogle Scholar
  22. Wagner, H.M. [ 1958 ] “The dual simplex algorithm for bounded variables”, Naval Research Logistics Quarterly 5 257–261.MathSciNetCrossRefGoogle Scholar
  23. Freund, R.M. [ 1985 ] “Postoptimal analysis of a linear program under simultaneous changes in the matrix coefficients”, Mathematical Programming Study 24 1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Gass, S.I. and T.L. Saaty [ 1955 ] “The computational algorithm for the parametric objective function”, Naval Logistics Research Quarterly 2 39–45.MathSciNetCrossRefGoogle Scholar
  25. Gass, S.I. and T.L. Saaty [ 1955 ] “Parametric objective function. Part II: Generalization”, Operations Research 3 395–401.MathSciNetCrossRefGoogle Scholar
  26. Saaty, T.L. and S.I. Gass [ 1954 ] “The parametric objective function. Part I”, Operations Research 2 316–319.MathSciNetCrossRefGoogle Scholar
  27. Bixby, R.E. [1989] Personal communication.Google Scholar
  28. Crowder, H. and M. Padberg [ 1980 ] “Solving large-scale symmetric traveling salesman problems to optimality”, Management Science 26 393–410.MathSciNetCrossRefGoogle Scholar
  29. Crowder, H., E.L. Johnson and M. Padberg [ 1983 ] “Solving large-scale zero-one linear programming problems”, Operations Research 31 803–834.zbMATHCrossRefGoogle Scholar
  30. Forrest, J.H. [1989] Personal communication.Google Scholar
  31. Grötschel, M., M. Jünger and G. Reinelt [ 1984 ] “A cutting plane algorithm for the linear ordering problem”, Operations Research 32 1195–1220.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Grötschel, M. and O. Holland [ 1991 ] “Solution of large-scale symmetric travelling salesman problems”, Mathematical Programming 51 141–202.MathSciNetzbMATHCrossRefGoogle Scholar
  33. Hoffman, K.L. and M. Padberg [ 1985 ] “LP-based combinatorial problem solving”, Annals of Operations Research 4 145–194.MathSciNetCrossRefGoogle Scholar
  34. Hoffman, K.L. and M. Padberg [ 1991 ] “Improving LP-representations of zero-one linear programs for branch-and-cut”, ORSA Journal on Computing 3 121–134.zbMATHCrossRefGoogle Scholar
  35. Hoffman, K.L. and M. Padberg [ 1993 ] “Solving airline crew scheduling problems by branch-and-cut”, Management Science 39 657–682.zbMATHCrossRefGoogle Scholar
  36. Padberg, M. and S. Hong [ 1980 ] “On the symmetric travelling salesman problem: a computational study”, Mathematical Programming Study 12 78–107.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Padberg, M. and G. Rinaldi [ 1987 ] “Optimization of a 532-city symmetric traveling salesman problem by branch-and-cut”, Operations Research Letters 6 1–7.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Padberg, M. and G. Rinaldi [ 1991 ] “A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems”, SIAM Review 33 60–100.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Manfred Padberg
    • 1
  1. 1.Leonard N. Stern School of Business, Statistics and Operations Research DepartmentNew York UniversityNew York CityUSA

Personalised recommendations