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Generalized Linear Programming Solves the Dual

Author

Listed:
  • T. L. Magnanti

    (Massachusetts Institute of Technology)

  • J. F. Shapiro

    (Massachusetts Institute of Technology)

  • M. H. Wagner

    (New York University)

Abstract

The generalized linear programming algorithm allows an arbitrary mathematical programming minimization problem to be analyzed as a sequence of linear programming approximations. Under fairly general assumptions, it is demonstrated that any limit point of the sequence of optimal linear programming dual prices produced by the algorithm is optimal in a concave maximization problem that is dual to the arbitrary primal problem. This result holds even if the generalized linear programming problem does not solve the primal problem. The result is a consequence of the equivalence that exists between the operations of convexification and dualization of a primal problem. The exact mathematical nature of this equivalence is given.

Suggested Citation

  • T. L. Magnanti & J. F. Shapiro & M. H. Wagner, 1976. "Generalized Linear Programming Solves the Dual," Management Science, INFORMS, vol. 22(11), pages 1195-1203, July.
    • Handle: RePEc:inm:ormnsc:v:22:y:1976:i:11:p:1195-1203
    DOI: 10.1287/mnsc.22.11.1195
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    Cited by:

    1. Thomas L. Magnanti, 2021. "Optimization: From Its Inception," Management Science, INFORMS, vol. 67(9), pages 5349-5363, September.
    2. Angelos Georghiou & Angelos Tsoukalas & Wolfram Wiesemann, 2019. "Robust Dual Dynamic Programming," Operations Research, INFORMS, vol. 67(3), pages 813-830, May.
    3. Larsson, Torbjörn & Patriksson, Michael & Rydergren, Clas, 2004. "A column generation procedure for the side constrained traffic equilibrium problem," Transportation Research Part B: Methodological, Elsevier, vol. 38(1), pages 17-38, January.
    4. Claude Lemaréchal, 2007. "The omnipresence of Lagrange," Annals of Operations Research, Springer, vol. 153(1), pages 9-27, September.
    5. Huisman, D. & Jans, R.F. & Peeters, M. & Wagelmans, A.P.M., 2003. "Combining Column Generation and Lagrangian Relaxation," ERIM Report Series Research in Management ERS-2003-092-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    6. Panagiotis Andrianesis & Dimitris Bertsimas & Michael C. Caramanis & William W. Hogan, 2020. "Computation of Convex Hull Prices in Electricity Markets with Non-Convexities using Dantzig-Wolfe Decomposition," Papers 2012.13331, arXiv.org, revised Oct 2021.
    7. Torbjörn Larsson & Michael Patriksson, 2006. "Global Optimality Conditions for Discrete and Nonconvex Optimization---With Applications to Lagrangian Heuristics and Column Generation," Operations Research, INFORMS, vol. 54(3), pages 436-453, June.

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