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 \[ θ ( {| {F_i ( z ) - z_i } |} ) - θ ( {F_i ( z )} ) - θ ( {z_i } ) = 0, i = 1, ⋯ ,n, \] where Fᵢ ( z ) and zᵢ denote the components of $F( z )$ and z, respectively, and θ is any strictly increasing function from R into R such that θ ( 0 ) = 0. If in addition, F is differentiable on Rⁿ, θ is differentiable on R and θ '( 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.