NONLINEAR COMPLEMENTARITY AS UNCONSTRAINED AND CONSTRAINED MINIMIZATION

The nonlinear complementarity problem is cast as an unconstrained mini mization problem that is obtained from an augmented Lagrangian formula tion. The dimensionality of the unconstrained problem is the same as t hat of the original problem, and the penalty parameter need only be gr eater than one. Another feature of the unconstrained problem is that i t has global minima of zero at precisely all the solution points of th e complementarity problem without any monotonicity assumption. If the mapping of the complementarity problem is differentiable, then so is t he objective of the unconstrained problem, and its gradient vanishes a t all solution points of the complementarity problem. Under assumption s of nondegeneracy and linear independence of gradients of active cons traints at a complementarity problem solution, the corresponding globa l unconstrained minimum point is locally unique. A Wolfe dual to a sta ndard constrained optimization problem associated with the nonlinear c omplementarity problem is also formulated under a monotonicity and dif ferentiability assumption. Most of the standard duality results are es tablished even though the underlying constrained optimization problem may be nonconvex. Preliminary numerical tests on two small nonmonotone problems from the published literature converged to degenerate or non degenerate solutions from all attempted starting points in 7 to 28 ste ps of a BFGS quasi-Newton method for unconstrained optimization.