Equivalence of the Complementarity Problem to a System of Nonlinear Equations

Source abstract

It is shown that the complementarily problem of finding a z in $R^n$ satisfying $zF( z ) = 0$, $F( z )\geqq 0$, $z\geqq 0$, where $F:R^n \to R^n$, is completely equivalent to solving the system of n nonlinear equations in n unknowns $$\theta \left( {\left| {F_i ( z ) - z_i } \right|} \right) - \theta \left( {F_i ( z )} \right) - \theta \left( {z_i } \right) = 0,\qquad i = 1, \cdots ,n,$$ where $F_i ( z )$ and $z_i$ denote the components of $F( z )$ and z, respectively, and $\theta$ is any strictly increasing function from R into R such that $\theta ( 0 ) = 0$. If in addition, F is differentiable on $R^n$, $\theta$ is differentiable on R and $\theta '( 0 ) = 0$, 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.