Nonlinear complementarity as unconstrained and constrained minimization

The nonlinear complementarity problem is cast as an unconstrained minimization problem that is obtained from an augmented Lagrangean formulation. The dimensionality of the unconstrained problems is the same as that of the original problem, and the penalty parameter need only be greater than one. Another feature of the unconstrained problem is that it has global minima of zero at precisely all the solution points of the complementarity problem without any monotonicity assumption. If the mapping of the complementarity problem is differentiable, then so is the objective of the unconstrained problem, and its gradient vanishes at all solution points of the complementarity problem. Under assumptions of nondegeneracy and linear independence of gradients of active constraints at a complementarity problem solution, the corresponding global unconstrained minimum point is locally unique. A Wolfe dual to a standard constrained optimization problem associated with the nonlinear complementarity problem is also formulated under a monotonicity and differentiabilty assumption. Most of the standard duality results are established even though the underlying constrained optimization problem may be nonconvex. Preliminary numerical tests on two small nomonotone problems from the published literature converged to degenerate or nodegenerate solutions from all attempted starting points in 7 to 28 steps of a BFGS quasi-Newton method for unconstrained optimization.