A New Homotopy Method for Nonlinear Complementarity Problems

̈ find x ∈ R s. t. x i ≥ 0, Fi(x)≥ 0, x iFi(x) = 0, for i = 1, · · · ,n, where F(x) : R → R is assumed to be continuously differentiable. There have been extensive studies for complementarity problems, see for example [5] and a recent review paper [7]. The NCP was introduced by Cottle [4] in his Ph.D. thesis in 1964. At the beginning, it was recognized that the NCP is a special case of a variation inequality problem [11]. The nonlinear complementarity problem and the linear complementarity problem, where F(x) is an affine map, have been studied extensively, see, e.g., [2,8,10]. There exist many methods for solving the NCP, such as semi-smooth equation method [15], smoothing or non-smoothing Newton method [17,19], neural network method [13, 14] etc. Homotopy method, as a kind of fixed-point method, had been developed in 1970s: Scarf [18] introduced the notion of primitive sets and described the first algorithm to approximate a fixed point of a continuous mapping; some powerful theoretical tools were available [3]; Eaves [6] introduced piecewise-linear maps into the computational fixedpoint literature. The classical reference by Todd [20] and the text by Garcia and Zangwill ∗Corresponding author.