## Invited Session Wed.3.H 2035

#### Wednesday, 15:15 - 16:45 h, Room: H 2035

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

### Nonsmooth analysis with applications in engineering

**Chair: Radek Cibulka**

**Wednesday, 15:15 - 15:40 h, Room: H 2035, Talk 1**

**Alfredo Noel Iusem**

Th effect of calmness on the solution set of nonlinear equaltions

**Coauthor: Roger Behling**

**Abstract:**

We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, called also upper Lipschitz continuity in the literature, can be described as a local error bound, and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions, which imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. As a consequence, we conclude that, locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications.

**Wednesday, 15:45 - 16:10 h, Room: H 2035, Talk 2**

**Amos Uderzo**

On some calmness conditions for nonsmooth constraint systems

**Abstract:**

In various contexts of mathematical programming, constraints

appearing in optimization problems, which depend on parameters,

can be formalized as follows

*f(p,x) ∈ C,*

where *f:P× X → Y* and *C ⊂ Y* are given

problem data, and *p* plays the role of a parameter.

Useful insights on the problem

behaviour (stability and sensitivity) can be achieved

by a proper analysis of the corresponding feasible set

mapping, i.e. *S:P → 2*^{X}

*S(p)={x ∈ X: f(p,x) ∈ C}.*

In this vein, whenever *P* and *X* have a metric space structure,

a property of *S* playing a crucial role, both from the

theoretical and the computational viewpoint, is calmness.

Mapping *S* is said to be calm at *(p*_{0},x_{0}) if

*x*_{0} ∈ S(p_{0}) and there exist *r,, \zeta>0* and

*l ≥ 0* such that

S(p)∩ B(x_{0},r) ⊆ B(S(p_{0}),l d(p,p_{0})), ∀ p ∈ B(p_{0},\zeta),

where *B(A,r)={x ∈ X: ∈ f*_{a ∈ A}d(x,a) ≤ r}.

This talk is devoted to the analysis of conditions for the calmness

of *S*. Such task is carried out by referring to recent developments

of variational analysis. Emphasis is given to the case in

which mapping *f* defining *S* is nonsmooth.

**Wednesday, 16:15 - 16:40 h, Room: H 2035, Talk 3**

**Radek Cibulka**

Quantitative stability of a generalized equation: Application to non-regular electrical circuits

**Coauthors: Samir Adly, Jiří V. Outrata**

**Abstract:**

Given matrices *B ∈ ***R**^{n× m}, *C ∈ ***R**^{m × n}, and mappings *f: ***R**^{n} → **R**^{n} , *F: ***R**^{m} rightarrows **R**^{m} with *m ≤ n*, consider the problem of finding for a vector *p ∈ ***R**^{n} the solution *z ∈ ***R**^{n} to the inclusion

\begin{equation} \label{eqIN}

p ∈ f(z) + B F (Cz).

\end{equation}

Denote by *Φ* the set-valued mapping from **R**^{n} into itself defined by

*Φ(z) = f(z) + BF(C z)* whenever *z ∈ ***R**^{n}. Our aim is to investigate stability properties such as Aubin continuity, calmness and isolated calmness of the solution mapping *Ψ:= Φ*^{-1}. Under appropriate assumptions, the verifiable conditions ensuring these properties are given in terms of the input data *f*, *F*, *B* and *C*. We illustrate our consideration on a particular examples arising from electronics.