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Decomposition and Nondifferentiable Optimization with the Projective Algorithm

Author

Listed:
  • J. L. Goffin

    (GERAD, Faculty of Management, McGill University, Montreal, Quebec, Canada H3A 1G5)

  • A. Haurie

    (GERAD, Ecole des Hautes Etudes Commerciales de Montreal, Montreal, Quebec, Canada and Departement d'Economie Commerciale et Industrielle, Université de Genève, Geneva, Switzerland)

  • J. P. Vial

    (Departement d'Economie Commerciale et Industrielle, Université de Genève, Geneva, Switzerland)

Abstract

This paper deals with an application of a variant of Karmarkar's projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem. This problem is closely related to the Dantzig-Wolfe decomposition technique used in large-scale convex programming. The proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems. Associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm. When a point close to the minimum of the potential function is reached, a corresponding point in the dual space is constructed, which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem. An admissible cut of the polytope, corresponding to a new supporting hyperplane of the epigraph of the function to minimize, is then generated at this approximate analytic center. In the primal space this new cut translates into a new column for the associated linear programming problem. The algorithm has performed well on a set of convex nondifferentiable programming problems.

Suggested Citation

  • J. L. Goffin & A. Haurie & J. P. Vial, 1992. "Decomposition and Nondifferentiable Optimization with the Projective Algorithm," Management Science, INFORMS, vol. 38(2), pages 284-302, February.
  • Handle: RePEc:inm:ormnsc:v:38:y:1992:i:2:p:284-302
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    File URL: http://dx.doi.org/10.1287/mnsc.38.2.284
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    References listed on IDEAS

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    1. Leech, Dennis, 1985. "Ownership Concentration and the Theory of the Firm : A Simple-Game-Theoretic Approach to Applied US Corporations in the 1930's," The Warwick Economics Research Paper Series (TWERPS) 262, University of Warwick, Department of Economics.
    2. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
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    Cited by:

    1. Klose, Andreas & Gortz, Simon, 2007. "A branch-and-price algorithm for the capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 179(3), pages 1109-1125, June.
    2. Csaba Fábián & Olga Papp & Krisztián Eretnek, 2013. "Implementing the simplex method as a cutting-plane method, with a view to regularization," Computational Optimization and Applications, Springer, vol. 56(2), pages 343-368, October.
    3. Nagih, Anass & Soumis, Francois, 2006. "Nodal aggregation of resource constraints in a shortest path problem," European Journal of Operational Research, Elsevier, vol. 172(2), pages 500-514, July.
    4. Samir Elhedhli & Jean-Louis Goffin, 2005. "Efficient Production-Distribution System Design," Management Science, INFORMS, pages 1151-1164.
    5. Gondzio, J. & Sarkissian, R. & Vial, J.-P., 1997. "Using an interior point method for the master problem in a decomposition approach," European Journal of Operational Research, Elsevier, vol. 101(3), pages 577-587, September.
    6. Júlíus Atlason & Marina A. Epelman & Shane G. Henderson, 2008. "Optimizing Call Center Staffing Using Simulation and Analytic Center Cutting-Plane Methods," Management Science, INFORMS, pages 295-309.
    7. Klose, Andreas, 2000. "A Lagrangean relax-and-cut approach for the two-stage capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 126(2), pages 408-421, October.
    8. Rustem, Berc & Becker, Robin G. & Marty, Wolfgang, 2000. "Robust min-max portfolio strategies for rival forecast and risk scenarios," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1591-1621, October.
    9. Bueler, Benno, 1997. "Solving an equilibrium model for trade of CO2 emission permits," European Journal of Operational Research, Elsevier, vol. 102(2), pages 393-403, October.
    10. Fischer, I. & Gruber, G. & Rendl, F. & Sotirov, R., 2006. "Computational experience with a bundle approach for semidenfinite cutting plane relaxations of max-cut and equipartition," Other publications TiSEM 03dfd8c3-9216-4c75-8921-3, Tilburg University, School of Economics and Management.
    11. Attila Bernáth & Tamás Király & Erika Kovács & Gergely Mádi-Nagy & Gyula Pap & Júlia Pap & Jácint Szabó & László Végh, 2013. "Algorithms for multiplayer multicommodity flow problems," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(4), pages 699-712, December.
    12. Kurt Jörnsten & Andreas Klose, 2016. "An improved Lagrangian relaxation and dual ascent approach to facility location problems," Computational Management Science, Springer, vol. 13(3), pages 317-348, July.
    13. Gondzio, Jacek & González-Brevis, Pablo & Munari, Pedro, 2013. "New developments in the primal–dual column generation technique," European Journal of Operational Research, Elsevier, vol. 224(1), pages 41-51.
    14. Gondzio, J. & du Merle, O. & Sarkissian, R. & Vial, J. -P., 1996. "ACCPM -- A library for convex optimization based on an analytic center cutting plane method," European Journal of Operational Research, Elsevier, vol. 94(1), pages 206-211, October.
    15. Dulce Rosas & Jordi Castro & Lídia Montero, 2009. "Using ACCPM in a simplicial decomposition algorithm for the traffic assignment problem," Computational Optimization and Applications, Springer, vol. 44(2), pages 289-313, November.
    16. Klose, Andreas & Drexl, Andreas, 2001. "Lower bounds for the capacitated facility location problem based on column generation," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 544, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    17. Laurent Drouet & Alain Haurie & Francesco Moresino & Jean-Philippe Vial & Marc Vielle & Laurent Viguier, 2008. "An oracle based method to compute a coupled equilibrium in a model of international climate policy," Computational Management Science, Springer, vol. 5(1), pages 119-140, February.
    18. Andreas Klose & Andreas Drexl, 2005. "Lower Bounds for the Capacitated Facility Location Problem Based on Column Generation," Management Science, INFORMS, pages 1689-1705.
    19. Klose, Andreas & Drexl, Andreas, 2001. "Combinatorial optimisation problems of the assignment type and a partitioning approach," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 545, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    20. Daniel Aloise & Pierre Hansen & Caroline Rocha & Éverton Santi, 2014. "Column generation bounds for numerical microaggregation," Journal of Global Optimization, Springer, vol. 60(2), pages 165-182, October.
    21. Elhedhli, Samir & Naoum-Sawaya, Joe, 2015. "Improved branching disjunctions for branch-and-bound: An analytic center approach," European Journal of Operational Research, Elsevier, vol. 247(1), pages 37-45.
    22. A. Ouorou & P. Mahey & J.-Ph. Vial, 2000. "A Survey of Algorithms for Convex Multicommodity Flow Problems," Management Science, INFORMS, pages 126-147.
    23. Benno Bueeler & Socrates Kypreos, "undated". "Multiregional Markal-Macro: Introduction of CO Certificate Trade and Solution Concepts," Computing in Economics and Finance 1996 _011, Society for Computational Economics.
    24. Haurie, A., 1995. "Time scale decomposition in production planning for unreliable flexible manufacturing systems," European Journal of Operational Research, Elsevier, vol. 82(2), pages 339-358, April.

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