Geometry of the Gass-Saaty parametric cost LP algorithm

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 region P. In particular, the avoidance of cycling does not require any restriction on the facial structure of P or on the algebraic relationships among the linear equalities and inequalities by means of which P is defined. © 1990 Springer-Verlag New York Inc.