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Decomposition and Nondifferentiable Optimization with the Projective Algorithm

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Author Info

  • 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)

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    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.

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    File URL: http://dx.doi.org/10.1287/mnsc.38.2.284
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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 38 (1992)
    Issue (Month): 2 (February)
    Pages: 284-302

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    Handle: RePEc:inm:ormnsc:v:38:y:1992:i:2:p:284-302

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    Related research

    Keywords: projective algorithm; interior point method; cutting plane; decomposition; nondifferentiable optimization;

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    Cited by:
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Benno Bueeler & Socrates Kypreos, . "Multiregional Markal-Macro: Introduction of CO Certificate Trade and Solution Concepts," Computing in Economics and Finance 1996 _011, Society for Computational Economics.
    8. 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.
    9. 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.
    10. 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.

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