## Invited Session Fri.1.H 2051

#### Friday, 10:30 - 12:00 h, Room: H 2051

**Cluster 24: Variational analysis** [...]

### Generalized differentiation and applications

**Chair: Vera Roshchina and Robert Baier**

**Friday, 10:30 - 10:55 h, Room: H 2051, Talk 1**

**Diethard Ernst Pallaschke**

Quasidifferentiable calculus and minimal pairs of compact convex sets

**Coauthor: Ryszard Urbanski**

**Abstract:**

The quasidifferential calculus developed by V.,F. Demyanov and A.,M. Rubinov provides a

complete analogon to the classical calculus of differentiation for a wide class of non-smooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiations whose gradients are elements of a suitable

Minkowski-Rådström-Hörmander space. Since the elements of the Minkowski-Rådström-Hörmander space are not uniquely determined, we mainly focused our attention to smallest possible representations of quasidifferentials, i.e. to minimal representations.

**Friday, 11:00 - 11:25 h, Room: H 2051, Talk 2**

**Adil Bagirov**

Subgradient methods in nonconvex nonsmooth optimization

**Coauthors: Alia Al Nuaimat, Napsu Karmitsa, Nargiz Sultanova**

**Abstract:**

The subgradient method is known to be the simplest method

in nonsmooth optimization. This method requires only one

subgradient and function evaluation at each iteration and it does

not use a line search procedure. The simplicity of the subgradient method makes it very attractive. This method was studied for only convex problems. In this talk we will present new versions of the subgradient method for solving nonsmooth nonconvex optimization problems. These methods are easy to implement. The efficiency of the proposed algorithms will be demonstrated by applying them to the well known nonsmooth optimization test problems.

**Friday, 11:30 - 11:55 h, Room: H 2051, Talk 3**

**Vladimir Goncharov**

Well-posedness of minimal time problem with constant convex dynamics via differential properties of the value function

**Coauthors: Giovanni Colombo, Boris Mordukhovich**

**Abstract:**

We consider a general minimal time problem with a constant convex dynamics in a (reflexive) Banach space, which can be seen as a mathematical programming problem. First, we obtain a general formula for the minimal time projection onto a closed set in terms of the duality mapping associated with the dynamics. Based on this formula we deduce then necessary and sufficient conditions of existence and uniqueness of a minimizer in terms of either dynamics rotundity (equivalently, smoothness of the dual set) or differential properties of the target. In both cases the (Fréchet) differentiability of the value function is extremely relevant. Some counter-examples are presented.