SMOOTH REPRESENTATIONS OF EPI-LIPSCHITZIAN SUBSETS OF R-N

Closed epi-Lipschitzian subsets M of R-n are characterized as sets def ined by a Lipschitzian inequality constraint {x is an element of R-n\f (M)(x) less than or equal to 0} for some function f(M) : R-n --> R whi ch is Lipschitzian on R-n, ''smooth'' (C-infinity) on the complementar y of partial derivative M (the boundary of M), which satisfies, for ev ery x is an element of partial derivative M, both the ''nondegeneracy' ' condition: 0 is not an element of partial derivative f(M)(x) (Clarke 's subdifferential), and the ''normal representation'' condition: that N-M(x) (Clarke's normal cone) is the cone spanned by partial derivati ve f(M)(x). This geometrical characterization is also equivalent to a more analytic formulation only by using the function Delta(M) = d(M) - d(Rn)\M (where d(M) is the distance function to M). Applications of t his result are given here and in [2], [3], and [4], to (i) the smooth (normal) approximation of epi-lipschitzian sets; (ii) Green's formula in the nonsmooth case; and (iii) the study of ''variational inequaliti es'' (or ''generalized equations'') in the nonconvex and nonsmooth cas e.