Second-order necessary conditions for optimality in nonsmooth nonlinear programming

On Second-Order Directional Derivatives in Nonsmooth Optimization

“…In our approach it is true [15] that if D 2 f(x;u,v) exists and is finite then the first-order derivative f'(x;u) also exists. This property is not shared by the approach of [20], [24] (see Example 3.6 in [24]). A sequence (i^) is said to converge to x in the direction u, denoted by (it) -> u x, if (x/,) converges to x, x/c ^ x for every k, and the sequence (|| u \\ ,,**"*,.)…”

“…in Penot [20] and Studniarski [24]; these authors use D+f and D-f in the above definitions to replace £>_/ and D + f respectively. In our approach it is true [15] that if D 2 f(x;u,v) exists and is finite then the first-order derivative f'(x;u) also exists.…”

Second-order optimality conditions for inequality constrained problems with locally Lipschitz data

“…The papers, in which more general classes of functions are the object of investigations are rather limited. In [18,28] are obtained conditions for problems with locally Lipschitz data: in [18] in terms of G-functions, and in [28] necessary conditions under some constraint qualifications. Our aim is to obtain as weaker conditions, as possible.…”

First- and second-order directional differentiability of locally Lipschitzian functions

“…Demyanov's theory of quasidifferentials [3] concentrates on this special case. Similarly, in the study of second-order optimality conditions, it is of interest to know when second-order directional derivatives like f (x, v; y) = lim t↓0 2 f x + tv + t 2 y/2 − f (x) − tf (x; v) t 2 exist [5].…”