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Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

Author

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  • M. Ç. Pinar

    (Bilkent University)

Abstract

This paper takes a fresh look at the application of quadratic penalty functions to linear programming. Recently, Madsen et al. (Ref. 1) described a continuation algorithm for linear programming based on smoothing a dual l 1-formulation of a linear program with unit bounds. The present paper is prompted by the observation that this is equivalent to applying a quadratic penalty function to the dual of a linear program in standard canonical form, in the sense that both approaches generate continuous, piecewise-linear paths leading to the optimal solution set. These paths lead to new characterizations of optimal solutions in linear programming. An important product of this analysis is a finite penalty algorithm for linear programming closely related to the least-norm algorithm of Mangasarian (Ref. 2) and to the continuation algorithm of Madsen et al. (Ref. 1). The algorithm is implemented, and promising numerical results are given.

Suggested Citation

  • M. Ç. Pinar, 1997. "Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 619-634, June.
    • Handle: RePEc:spr:joptap:v:93:y:1997:i:3:d:10.1023_a:1022651331550
    DOI: 10.1023/A:1022651331550
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    Cited by:

    1. O. L. Mangasarian, 2004. "A Newton Method for Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 1-18, April.

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