Equivalence of the Complementarity Problem to a System of Nonlinear Equations

It is shown that the complementarily problem of finding a z in Rⁿ satisfying zF (z) = 0, F (z) 0, z 0, where F: Rⁿ Rⁿ, is completely equivalent to solving the system of n nonlinear equations in n unknowns \ (| {Fᵢ (z) - zᵢ |}) - (Fᵢ (z) ) - (zᵢ) = 0, i = 1, , n, \ where Fᵢ (z) and zᵢ denote the components of F (z) and z, respectively, and is any strictly increasing function from R into R such that (0) = 0. If in addition, F is differentiable on Rⁿ, is differentiable on R and ' (0) = 0, then the above equations are globally differentiable, and at any solution z which satisfies the nondegeneracy condition F (z) + z > 0, the system of equations has a nonsingular Jacobian if F has a nonsingular Jacobian with nonsingular principal minors.