An existence theorem for the complementarity problem

Abstract

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ. The complementarity problem (q|M) overC is that of finding a solution to the system

$$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$

It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ.