## Invited Session Fri.3.H 0110

#### Friday, 15:15 - 16:45 h, Room: H 0110

**Cluster 9: Global optimization** [...]

### Structural aspects of global optimization

**Chair: Oliver Stein**

**Friday, 15:15 - 15:40 h, Room: H 0110, Talk 1**

**Georg Still**

Minimization of nonconvex quadratic functions on special feasible sets

**Abstract:**

We are interested in global minimization of general quadratic functions

on a feasible set *F*. It is well-known that depending on the specific

set F the problem is possibly tractable or hard. We are especially interested in

the minimization on the unit simplex *F*. This problem is just the feasibility problem for copositive programming.

The latter recently attracted much attention as it appeared that

many hard integer problems can be represented exactly by copositive programs.

In our talk we firstly discuss some interesting properties

of quadratic functions such as the number of components of the level sets and the number of (global) minimizers.

We then consider copositive programming and give some

recent results on the structure of this problem.

**Friday, 15:45 - 16:10 h, Room: H 0110, Talk 2**

**Tomas Bajbar**

Nonsmooth versions of Sard's theorem

**Abstract:**

We present a comparison between some versions of Sard's Theorem which have been proven recently for special function classes with different definitions of critical points. The motivation for

calling a given point a critical point of a function varies. Considering the class of *C*^{k} functions, the

class of min-type functions or min-max functions, the motivation for the definition of critical point is

the topological structure of the inverse image. Considering the class of set-valued definable

mappings, the motivation for the definition of critical points is the property of metric regularity. We

compare topological critical points and critical points defined via metric regularity in the class of min-type

and min-max functions. We illustrate the whole problematic by some examples.

**Friday, 16:15 - 16:40 h, Room: H 0110, Talk 3**

**Dominik Dorsch**

Local models in equilibrium optimization

**Coauthors: Hubertus Th. Jongen, Vladimir Shikhman**

**Abstract:**

We study equilibrium optimization from a structural point of view. For that, we consider equilibrium optimization problems up to the smooth coordinate transformations locally at their solutions. The latter equivalence relation induces classes of equilibrium optimization problems. We focus on the stable classes corresponding to a dense set of data functions. We prove that these classes are unique and call them "basic classes''. Their representatives in the simplest form are called local models. For particular realizations of equilibrium optimization problems basic classes and their local models are elaborated. The latter include bilevel optimization, general semi-infinite programming and Nash optimization.