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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
    DOI: 10.1287/mnsc.38.2.284
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