AN IMPLICIT-FUNCTION THEOREM FOR C0,1-EQUATIONS AND PARAMETRIC C1,1-OPTIMIZATION

The implicit-function theorem deals with the solutions of the equation F(x, t) = a for locally Lipschitz functions F from Rn + m into Rn. The existence of a locally well-defined and Lipschitzian solution function x = G(a, t) will be completely characterized in terms of certain multivalued directional derivatives of F which determine the corresponding derivatives of G in a simple way. Our directional derivatives are nothing but L. Thibault's (Ann. Mat. Pura Appl. (4) 125, 1980, 157-192) limit sets which have been introduced to extend Clarke's calculus to functions in abstract spaces. For parametric C1, 1-optimization problems, we study the critical point map, the associated critical values, and derive first and second order formulas, respectively. © 1991.