## Invited Session Thu.2.H 2051

#### Thursday, 13:15 - 14:45 h, Room: H 2051

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

### Variational analysis of optimal value functions and set-valued mappings with applications

**Chair: Mau Nam Nguyen**

**Thursday, 13:15 - 13:40 h, Room: H 2051, Talk 1**

**Messaoud Bounkhel**

Regularity concepts of perturbed distance functions at points outside of the set in Banach spaces

**Coauthor: Chong Li**

**Abstract:**

In this talk I will present some new results on the

(Fréchet, proximal, Clarke, Mordukhovich)

subdifferential of the perturbed distance function

*d*_{S}^{J}(·) determined by a closed subset *S* and a

Lipschitz function *J(·)*. Using these results, I will estabilsh some important relationships between the regularity of the set and the perturbed distance function

at points outside of *S* in arbitrary Banach space.

**Thursday, 13:45 - 14:10 h, Room: H 2051, Talk 2**

**Sangho Kum**

A geometric mean of parameterized arithmetic and harmonic means of convex functions

**Abstract:**

Recently Bauschke et al. (2008) introduced a new notion of proximal average, and studied this subject systemically from various viewpoints. The proximal average can be an attractive and powerful alternative to the classical arithmetic and

epigraphical averages in the context of convex analysis and optimization problems. The present work aims at providing a further development of the proximal average. For that purpose, exploiting the geometric mean of convex functions by Atteia and Rassouli (2001), we develop a new algorithmic self-dual operator for convex functions termed "the geometric mean of parameterized arithmetic and harmonic means of convex functions'', and investigate its essential properties.

**Thursday, 14:15 - 14:40 h, Room: H 2051, Talk 3**

**Nguyen Dong Yen**

Coderivatives of a Karush-Kuhn-Tucker point set map and applications

**Coauthor: Gue Myung Lee**

**Abstract:**

*The trust-region subproblem* corresponding to the triple

*{A,b,α}*, where *A ∈ ***R**^{n× n} is a symmetric

matrix, *b ∈ ***R**^{n} a given vector, and *α>0* a real number,

is the optimization problem

\begin{equation*}%\label{TRS}

min\Big{f(x):=1⁄2x^{T}Ax+b^{T}x, :,

||x||^{2} ≤ α^{2}\Big}.\eqno{(P)}

\end{equation*} One often encounters with *(P)* in

the development of trust-region methods for nonlinear programs.

Since the feasible region of *(P)* is a convex compact set with an

infinite number of extreme points, the structure of its solution set

(resp. of its Karush-Kuhn-Tucker point set) is quite different from

that of quadratic programs with linear constraints. By using some

tools from Variational Analysis, this paper investigates the

stability of *(P)* with respect to the perturbations of all the

three components of its data set *{A,b,α}*.