THE BASIC THEOREM OF COMPLEMENTARITY REVISITED

The basic theorm of (linear) complementarity was stated in a 1971 pape r [6] by B.C. Eaves who credited C.E. Lemke for giving a constructive proof based on his almost complementary Pivot algorithm. This theorem asserts that associated with an arbitrary linear complementarity probl em, a certain augmented problem always possesses a solution. Many well -known existence results pertaining to the linear complementarity prob lem are consequences of this fundamental theorem. In this paper, we ex plore some further implications of the basic theorem of complementarit y and derive new existence results for the linear complementarity prob lem. Based on these results, conditions for the existence of a solutio n to a linear complementarity problem with a fully-semimonotone matrix are examined. The class of the linear complementarity problems with a G-matrix is also investigated.