Equivalence of Nonlinear Complementarity Problems and Least Element Problems in Banach Lattices

A class of not necessarily linear operators A: V → V* is introduced, where the Banach space V and its dual V* carry dual vector-lattice orderings ≥. These operators, called Z-maps, generalize the n × n real matrices with nonnegative off-diagonal elements. If A is a strictly monotone Z-map with certain regularity and growth conditions, and if F denotes the set of all vectors v ∈ V for which v ≥ 0, and Av ≥ 0, then it is shown that the complementarity problem, to find v ∈ F such that = 0, and the least element problem, to find v ∈ F with v ≤ w for all w ∈ F, have the same unique solution. Some other problems equivalent to these, and some examples are discussed.