Abstract
In previous discussions of the Gass-Saaty algorithm, the possibility of cycling is avoided by making strong nondegeneracy assumptions or by incorporating a lexicographic decision rule. By analyzing the geometric ideas on which the algorithm is based, it is shown here that even without any “lexicography,” cycling is impossible unless the two objective functions are related in a very special way to each other or to the constraints defining the feasible regionP. In particular, the avoidance of cycling does not require any restriction on the facial structure ofP or on the algebraic relationships among the linear equalities and inequalities by means of whichP is defined.
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Work on this paper by the first author was supported in part directly by the National Science Foundation, and in part by the Mathematical Sciences Research Institute and the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation. Work by the second author was supported in part by the Alexander von Humboldt Stiftung.
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Klee, V., Kleinschmidt, P. Geometry of the Gass-Saaty parametric cost LP algorithm. Discrete Comput Geom 5, 13–26 (1990). https://doi.org/10.1007/BF02187776
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DOI: https://doi.org/10.1007/BF02187776