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

**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.

Talk 2 of the invited session Wed.3.H 2035

**"Nonsmooth analysis with applications in engineering"** [...]

Cluster 24

**"Variational analysis"** [...]