Nadia Zlateva

Second-order subdifferentials of C 1,1 functions and optimality conditions

We present second-order subdifferentials of Clarkes type of C1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ℝn, considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C1,1 data are obtained.

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Holderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquins primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

Integrability of subdifferentials of directionally Lipschitz functions

Using a quantitative version of the subdifferential characterization of directionally Lipschitz functions, we study the integrability of subdifferentials of such functions over arbitrary Banach space.

Integrability of subdifferentials of certain bivariate functions

Abstract We study integrability properties of bivariate functions defined on the product of Banach spaces.

Parameterized Minimax Problem: On Lipschitz-Like Dependence of the Solution with Respect to the Parameter

We study Lipschitz continuity with respect to the parameter of the set of solutions of a parameterized minimax problem on a product Banach space. We present a sufficient condition, ensuring that the map which to any value of the parameter assigns the set of solutions of the problem (possibly multi-valued, and unbounded) possesses Aubin property.

A new proof of the integrability of the subdifferential of a convex function on a Banach space

We provide a simple proof of the Moreau-Rockafellar theorem that a proper lower semicontinuous convex function on a Banach space is determined up to a constant by its subdifferential.

On nonconvex version of the inequality of Clarke and Ledyaev

The purpose of the paper is to extend the Clarke-Ledyaev multidirectional mean value inequality that estimates in terms of the subdifferential of a lower semicontinous function the difference between the infimum of the function on certain closed, bounded and convex set and its value on a certain point. The assumption of convexity is relaxed by showing that a similar inequality holds for any closed and bounded set and any point outside its closed convex hull from which the set “seems convex�?. The boundaries of convex set seem convex from each point outside its closed convex hull. The technique applied allows proving the Clarke-Ledyaev inequality without assuming that the function is bounded bellow. Also, in the convex case the infimum can be taken only over the boundary of the set. An inverse theorem shows that for arbitrary lower semicontinuous function no inequality of such type can be expected if the set does not seem convex from the point. If it does seem convex then no stronger inequality can be obtained in general.

Generic Gateaux differentiability via smooth perturbations

We prove that in a Banach space with an uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given. The usual applications of the variational principles in Banach spaces refer to differentiability of real valued functions. For example the papers of [B-P] and [D-G-Z] contain results about Gateaux differentiability on dense sets. An application of Ekeland’s variational principle to generic Frechet differentiability is given in the proff of famous Ekeland-Lebourg’s theorem (see [E-L]). In [Ge] an application of the smooth variational principle to generic Gateaux differentiability is presented. In this paper we prove some results about generic Gateaux differentiability of directionally differentiable functions. The tool for proving the main ∗The research was partially supported by the Bulgarian Ministry of Education and Science under contract number MM 506/1995.

Perturbation Method for a Non-convex Integral Functional

We consider the problem of minimization of an integral functional over Lipschitzian curves. In order to ensure the existence of a minimum for some perturbed function, a novel variational principle is developed.

Partially Ball Weakly Inf-Compact Saddle Functions

We study on a product Banach space the properties of a class of saddle functions called partially ball weakly inf-compact. For such a function we prove that the domain of the subdifferential is nonempty, that the operator naturally associated with the subdifferential is maximal monotone, and that the subdifferential of the function is integrable. For a function in a large subclass of that class we prove the density of the domain of the subdifferential in the domain of the function.