A Semismooth Active-set Algorithm for Degenerate Nonlinear Complementarity Problems

We propose a semismooth active-set Newton algorithm for solving the nonlinear complementarity problems with degenerate solutions. This method introduces the active-set technique to identify the degenerate set. At each iteration, the search direction is obtained by two reduced linear systems. Instead of employing gradient steps as adjustments to guarantee the sufficient reduction of the merit function, the algorithm employs a Newton-type direction, which is more efficient than gradient direction, in the adjustment step. This method has globally convergence. When near the solution, the degenerate set will be identified correctly, and only one reduced linear system is solved at each iteration. Under some mild assumptions, locally superlinear convergence is obtained as well. Numerical experiments on MATLAB shows the efficiency of the method.