Equivalence of the complementarity problem to a system of nonlinear equations. (English) Zbl 0339.90051

Mangasarian, O. L.

SIAM J. Appl. Math. 31, 89-92 (1976).

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

Reviewer: O. L. Mangasarian

MSC:

90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C30	Nonlinear programming