Equi-Calmness And Epi-Derivatives That Are Pointwise Limits

18. A. D. Iooe, \Variational analysis of a composite function: a formula for the lower second order epi-derivative," J. 843{865. 21. R. A. Poliquin, \An extension of Attouch's Theorem and its application to second-order epi-diierentiation of convexly composite functions," Trans. Amer. Math. Soc. 18 neighborhood of g ? F(x) and is twice epi-diierentiable with second-order epi-derivative given as a pointwise limit along rays. This is exactly our situation. Indeed according to Cor. 3.3 g is Lipschitz relative to the intersection of its domain and a neighborhood of g ? F(x). The assumptions on the second-order epi-derivative follow from Theorem 3.9. The formula comes from 20] (which generalized the formula in 5]). In 21] it was shown that a strongly amenable function f is twice epi-diierentiable at x relative to v if and only if its subgradient mapping @f is proto-diierentiable at x relative to v. Proto-derivatives were introduced in 22] and can be used to approximate set-valued mappings. The fact that the subgradient mapping is proto-diierentiable has many interesting applications; see 23]{{26]. Corollary 3.12. Under the assumptions of Theorem 3.11, there is a neighborhood X of x such that for all x 2 X, the subgradient mapping @f is proto-diierentiable at x for any v 2 @f(x). References 1. M. Moussaoui and A. Seeger, \On monotonicity of rst and second-order diierential quotients", preprint Universit e d'Avignon 1995. 5. R. T. Rockafellar, \First-and second-order epi-diierentiability in nonlinear programming ," Trans. 6. R. T. Rockafellar, \Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives," Math. 17 This shows that f is twice epi-diierentiable at x relative to v in any direction u. Corollary 3.10 5, Theorem 3.1]. Let f be piecewise linear-quadratic and x 2 dom f. Then f is epi-diierentiable at x, and the epi-derivative is given as a pointwise limit, i.e. f 0 x (u) = lim t & 0 ' x;t f(u) for every u 2 IR n. If in addition f is convex, then f is twice epi-diierentiable at x relative to any v 2 @f(x) and the second-order epi-derivative is given as a pointwise limit, i.e. f 00 x; v (u) = lim t & 0 ' 2 x; v;t f(u) for every u 2 IR n. For the composition f(x) = g ? F(x) we say that the basic constraint qualiication holds at x 2 dom f if the only y 2 N dom …