IMPLICIT FUNCTIONS AND SENSITIVITY OF STATIONARY-POINTS

We consider the space L(D) consisting of Lipschitz continuous mappings from D to the Euclidean n-space ℝn, D being an open bounded subset of ℝn. Let F belong to L(D) and suppose that {Mathematical expression} solves the equation F(x) = 0. In case that the generalized Jacobian of F at {Mathematical expression} is nonsingular (in the sense of Clarke, 1983), we show that for G near F (with respect to a natural norm) the system G(x) = 0 has a unique solution, say x(G), in a neighborhood of {Mathematical expression} Moreover, the mapping which sends G to x(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied. © 1990 The Mathematical Programming Society, Inc.