Abstract
A technique is described for resolving degeneracy in the simplex method for linear programming. It is shown that this technique enables a guarantee of termination to be given, not only for exact arithmetic but also for inexact arithmetic when roundoff errors are present. The method is recursive, and each level of recursion is obtained by localizing and dualizing the problem at a lower level. The existence of a cost function at any level is exploited to provide the guarantee. A number of interesting theoretical properties of the method are given. Data structures are described which enable the method to be implemented with very little overhead beyond what is normally required for the simplex method. There is no difficulty in using any of the currently popular techniques for representing and updating matrix factors. The technique is extended to handle degeneracy in l1 linear programming problems, including linear l1 approximation, and it is shown that all the above properties can be preserved. Some comparative numerical experiments are described.
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