Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems

We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer–Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.