Polyhedral Complementarity Problem With Quasimonotone Decreasing Mappings

Abstract

The fixed point problem of piecewise constant mappings in R n is investigated. This is a polyhedral complementarity problem, which is a generalization of the linear complementarity problem. Such mappings arose in the author’s research on the problem of economic equilibrium in exchange models, where mappings were considered on the price simplex. The author proposed an original approach of polyhedral complementarity, which made it possible to obtain simple algorithms for solving the problem. The present study is a generalization of linear complementarity methods to related problems of a more general nature and reveals a close relationship between linear complementarity and polyhedral complementarity. The investigated method is an analogue of the well-known Lemke method for linear complementarity problems. A class of mappings is described for which the process is monotone, as it is for the linear complementarity problems with positive principal minors of the constraint matrix (class P). It is shown that such a mapping has always unique fixed point.