## Invited Session Mon.1.H 1012

#### Monday, 10:30 - 12:00 h, Room: H 1012

**Cluster 17: Nonsmooth optimization** [...]

### Iterative methods for variational analysis

**Chair: Alain Pietrus**

**Monday, 10:30 - 10:55 h, Room: H 1012, Talk 1**

**Celia Jean-Alexis**

The second order generalized derivative and generalized equations

**Coauthors: Michel Geoffroy, Alain Pietrus**

**Abstract:**

We consider a generalized equation of the form

*0 ∈ f(x)+G(x)* where *f: ***R**^{n} → **R**^{n} is a *C*^{1,1} function such that its Fréchet-derivative *f'* is subanalytic and *G: ***R**^{n} → 2^{R}^{n} is a set-valued map metrically regular. First of all, we present some iterative methods introduced for solving this equation and then we state our main result. In fact, we propose a method using the second order generalized derivative and we show existence and convergence of a sequence defined by this method.

**Monday, 11:00 - 11:25 h, Room: H 1012, Talk 2**

**Robert Baier**

Set-valued Newton's method for computing convex invariant sets

**Abstract:**

A new realization of Newton's method for "smooth'' set-valued fixed-point problems is presented. For a dynamical system *x*_{k+1} = g(x_{k}) a convex invariant set *X ⊂ ***R**^{n} has to be determined with *g(X) = X*.

This fixed-point problem is transformed to a zero-finding problem in the Banach space of directed sets for which Newton's method can be formulated. The cone of convex, compact subsets of **R**^{n} can be embedded into this Banach space such that usual set arithmetics are extended and a visualization of differences of embedded convex compact sets as usually nonconvex subsets of **R**^{n} is available.

Important assumptions are the existence of a set of convex subsets such that their image under *g* remains convex and the existence of a differentiable extension of *g* to directed sets. The visualization of an embedded fixed set

for the transformed problem is a convex invariant set for the original problem.

First examples illustrate that the convergence assumptions can be verified and local quadratic convergence even to unstable convex invariant sets is observed in contrary to fixed set iterations. Further extensions of this approach are

indicated.

**Monday, 11:30 - 11:55 h, Room: H 1012, Talk 3**

**Elza Farkhi**

The directed subdifferential and applications

**Coauthors: Robert Baier, Vera Roshchina**

**Abstract:**

The directed subdifferential of quasidifferentiable functions

is introduced as the difference of two convex subdifferentials

embedded in the Banach space of directed sets.

Preserving the most important properties of the quasidifferential,

such as exact calculus rules, the directed subdifferential lacks

major drawbacks of the quasidifferential: non-uniqueness and

growing in size of the two convex sets representing the quasidifferential

after applying calculus rules. Its visualization, the Rubinov subdifferential,

is a non-empty, generally non-convex set in *R*^{n}.

Calculus rules for the directed subdifferentials are derived.

Important properties as well as necessary and suffcient optimality

conditions for the directed subdifferential are obtained.

The Rubinov subdifferential is compared with other well-known subdifferentials.