The Generalized Order Linear Complementarity Problem

The generalized order linear complementarity problem (in the setting of a finite dimensional vector lattice) is the problem of finding a solution to the piecewise-linear system \[ x \wedge ( M_1 x + q_1 ) \wedge ( M_2 x + q_2 ) \wedge \cdots \wedge ( M_k x + q_k ) = 0, \] where $M_i $’s are linear transformations and $q_i $’s are vectors. This problem is equivalent to the generalized linear complementarity problem considered by Cottle and Dantzig [J. Combin. Theory, 8 (1970), pp. 79–90.]. Using degree theory, a comprehensive analysis of existence, uniqueness, and stability aspects of this problem is presented.