Piecewise-linear pathways to the optimal solution set in linear programming
Pınar, M. Ç.
Journal of Optimization Theory and Applications
Kluwer Academic Publishers - Plenum Publishers
619 - 634
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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 l1-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.
KeywordsCharacterization of optimal solutions
Piece - wise - linear path - following algorithms
Quadratic penalty functions