Solving the Nonlinear Complementarity Problem by a Homotopy Method

Let F be a $C^2 $ map from n-dimensional Euclidean space into itself. It is proved that, under some mild conditions on F, the complementarily problem $z \geqq 0$, $F(z) \geqq 0$, $zF(z) = 0$ can be solved by a homotopy algorithm developed by Chow, Mallet-Paret, Yorke, and Watson. The algorithm is globally convergent with probability one, and uses Mangasarian’s nonlinear system equivalent to the complementarity problem. Convergence theorems for the algorithm simultaneously prove existence of a solution, although existence is already well known. Some computational results are included.