Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

The goal of the paper is to study the particular class of regularly ${\mathcal{H}}$-convex functions, when ${\mathcal{H}}$ is the set ${\mathcal{L}\widehat{C}}(X,{\mathbb{R}})$ of real-valued Lipschitz continuous classically concave functions defined on a real normed space $X$. For an extended-real-valued function $f:X \mapsto \overline{\mathbb{R}}$ to be ${\mathcal{L}\widehat{C}}$-convex it is necessary and sufficient that $f$ be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each ${\mathcal{L}\widehat{C}}$-convex function is regularly ${\mathcal{L}\widehat{C}}$-convex as well. We focus on ${\mathcal{L}\widehat{C}}$-subdifferentiability of functions at a given point. We prove that the set of points at which an ${\mathcal{L}\widehat{C}}$-convex function is ${\mathcal{L}\widehat{C}}$-subdifferentiable is dense in its effective domain. Using the subset ${\mathcal{L}\widehat{C}}_\theta$ of the set ${\mathcal{L}\widehat{C}}$ consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of ${\mathcal{L}\widehat{C}}_\theta$-subgradient and ${\mathcal{L}\widehat{C}}_\theta$-subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Symmetric notions of abstract ${\mathcal{L}\widecheck{C}}$-concavity and ${\mathcal{L}\widecheck{C}}$-superdifferentiability of functions where ${\mathcal{L}\widecheck{C}}:= {\mathcal{L}\widecheck{C}}(X,{\mathbb{R}})$ is the set of Lipschitz continuous convex functions are also considered. Some properties and simple calculus rules for ${\mathcal{L}\widehat{C}}_\theta$-subdifferentials as well as ${\mathcal{L}\widehat{C}}_\theta$-subdifferential conditions for global extremum points are established.Comment: 18 page

Necessary conditions for an extremum for 2-regular problems

The paper deals with the problem of minimizing a real-valued smooth function fcolon X to Bbb R over the set D={x in X,|,F(x) in Q}, where Fcolon X to Y is a smooth mapping, X and Y are Banach spaces, and Q is a closed convex set of Y. The authors say that the mapping Fcolon X to Y is 2-regular at a point overline{x} with respect to the set Q in a direction h in X if 0 in {rm int}(F(overline{x}) + {rm Im}F'(overline{x})+F"(overline{x})[h,,(F'(overline{x}))^{-1}(Q -F(overline{x}))] - Q). When h = 0 the 2-regularity coincides with the well-known Robinson regularity condition [S. M. Robinson, Math. Oper. Res. {bf 1} (1976), no.~2, 130--143; [msn] MR0430181 (55 #3188) [/msn]]. Moreover Robinson's regularity condition implies that F is 2-regular at overline{x} with respect to Q in any direction h in X (including h=0). However, the mapping F may be 2-regular at overline{x} in some nonvanishing directions h in X,,h ne 0, and not satisfy Robinson's regularity condition. The authors show that in the case when the 2-regularity condition holds one can obtain representations both for the contingent cone and for the set of second-order tangent vectors to the constraint set D at overline{x} in the terms of local approximations of the set Q and Fréchet derivatives of F. Using these representations the authors derive first- and second-order necessary conditions for local optimal solutions of the optimization problem being considered