## Contributed Session Thu.1.H 2035

#### Thursday, 10:30 - 12:00 h, Room: H 2035

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

### Structure and stability of optimization problems

**Chair: Jan-J. Ruckmann**

**Thursday, 10:30 - 10:55 h, Room: H 2035, Talk 1**

**Jan-J. Ruckmann**

Max-type objective functions: A smoothing procedure and strongly stable stationary points

**Abstract:**

We consider the minimization of a max-type function

over a feasible set *M* and apply the concept of strongly stable

stationary points to this class of problems. We use a logarithmic

barrier function and construct a family *M*^{\}gamma of interior point approximations of

*M* where *M*^{\}gamma is described by a single smooth inequality

constraint. We show that there is a one-to-one

correspondence between the stationary points (and their corresponding stationary indices) of the original problem and those

with the feasible set *M*^{\}gamma.

**Thursday, 11:00 - 11:25 h, Room: H 2035, Talk 2**

**Helmut Gfrerer**

Second-order conditions for a class of nonsmooth programs

**Abstract:**

We study infinite dimensional optimization problems with constraints given in form of an inclusion *0 ∈ F(x)-S(x)*,

where *F* denotes a smooth mapping and *S* is a generalized polyhedral multifunction, e.g., the normal cone mapping of a convex polyhedral set. By using advanced techniques of variational analysis we obtain second-order characterizations, both necessary and sufficient, for directional metric subregularity of the constraint mapping. These results can be used to obtain second-order optimality conditions for the optimization problem.

**Thursday, 11:30 - 11:55 h, Room: H 2035, Talk 3**

**Peter Fusek**

On metric regularity of the Kojima function in nonlinear semidefinite programming

**Abstract:**

The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming.

Having in mind the topological structure of the positive semidefinite cone we identify a class of Lipschitz metrically regular functions which turn out to have coherently oriented B-subdifferentials. This class is broad enough to include the Kojima function corresponding to the nonlinear semidefinite programming problem.