The subdifferential formula for the sum of two convex functions defined on a locally convex space is proved under a general qualification condition. It is proved that all the similar results which are already known can be derivated from the formula.

Motivated by a classical result concerning the ε-subdifferential of the sum of two proper, convex and lower semicontinuous functions, we give in this paper a similar result for the enlargement of the sum of two maximal monotone operators defined on a Banach space. This is done by establishing a necessary and sufficient condition for a bivariate inf-convolution formula.

We prove the existence of viable solutions to the Cauchy problem x′′ ∈ F (x, x′), x(0) = x0, x′(0) = y0, where F is a set-valued map defined on a locally compact set M ⊂ R, contained in the Fréchet subdifferential of a φ-convex function of order two.

In this paper by using the scalarization method, we consider Stamppachia variational-like inequalities in terms of normal subdifferential for set-valued maps and study their relations with set-valued optimization problems. Furthermore, some characterizations of the solution sets of K-pseudoinvex extremum problems are given.

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

We introduce generalized definitions of Peano and Riemann directional derivatives in order to obtain second-order optimality conditions for vector optimization problems involving C 1,1 data. We show that these conditions are stronger than those in literature obtained by means of second-order Clarke subdifferential.

Given a lower semicontinuous function f : Rn → R ∪ {+∞}, we prove that the set of points of Rn where the lower Dini subdifferential has convex dimension k is countably (n − k)-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.

The applicant develops new tools of variational analysis; in particular, two new versions of the Ekeland variational principle and subdifferential variational principle for set-valued mappings, as well as new constructions of extremal set systems. They play a significant role to studying an important class of constrained optimization problems called Multiobjective Optimization Problems with Equ...

Generalised convexity is revisited from a geometrical point of view. A substitute to the subdifferential is proposed. Then generalised monotonicity is considered. A representation of generalised monotone maps allows to obtain a symmetry between maps and their inverses. Finally, maximality of generalised monotone maps is analysed.