The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry

Author:
J. C. Lagarias

Journal:
Trans. Amer. Math. Soc. **320** (1990), 193-225

MSC:
Primary 90C05

DOI:
https://doi.org/10.1090/S0002-9947-1990-1058199-0

MathSciNet review:
1058199

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Abstract: This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm. A nonlinear change of variables, *projective Legendre transform coordinates*, is introduced to study these trajectories. The projective Legendre transform mapping has a coordinate-free geometric interpretation in terms of the notion of "centering by a projective transformation." Let be a set of linear programming constraints on such that its polytope of feasible solutions is bounded and contains in its interior. The projective Legendre transform mapping is given by

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DOI:
https://doi.org/10.1090/S0002-9947-1990-1058199-0

Article copyright:
© Copyright 1990
American Mathematical Society