An implicit-function theorem for C0, 1-equations and parametric C1, 1-optimization

Abstract

The implicit-function theorem deals with the solutions of the equation F( x, t) = a for locally Lipschitz functions F from R n + m into R n . 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 C 1, 1-optimization problems, we study the critical point map, the associated critical values, and derive first and second order formulas, respectively.