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The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods

Richard E. Stone and Craig A. Tovey
SIAM Review
Vol. 33, No. 2 (Jun., 1991), pp. 220-237
Stable URL: http://www.jstor.org/stable/2031142
Page Count: 18
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The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods
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Abstract

The projective scaling algorithm in Karmarkar [Combinatorica, 4 (1984), pp. 373-395] and Dantzig's simplex method (see [Linear Programming and Extensions, Princeton University Press, 1963]) are usually thought of as fundamentally different approaches to linear programming. When viewed in Dantzig's column space geometry, however, both algorithms turn out to be iteratively reweighted least squares methods. The projective scaling algorithm (and the affine scaling method in Dikin [Soviet Math. Dokl., 8 (1967), pp. 674-675]) can then be derived as a natural generalization of the simplex method. This derivation shows how the dual variables arise in the interior methods. The insight is essentially geometric; suitable figures are provided.

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