## Invited Session Fri.2.H 0107

#### Friday, 13:15 - 14:45 h, Room: H 0107

**Cluster 16: Nonlinear programming** [...]

### Stability and solution methods

**Chair: Diethard Klatte**

**Friday, 13:15 - 13:40 h, Room: H 0107, Talk 1**

**Diethard Klatte**

Metric regularity versus strong regularity for critical points of nonlinear programs

**Coauthor: Bernd Kummer**

**Abstract:**

In this talk, we study perturbed nonlinear optimization problems in a setting which includes standard nonlinear programs as well as cone constrained programs. We discuss conditions for metric regularity of the critical point system, or, equivalently, for the Aubin property of the critical point map. Our focus is on conditions under which the critical point map has the Aubin property if and only if it is locally single-valued and Lipschitz, or, equivalently, metric regularity and strong regularity coincide. In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map.

**Friday, 13:45 - 14:10 h, Room: H 0107, Talk 2**

**Stephan Bütikofer**

Influence of inexact solutions in a lower level problem on the convergence behavior of a nonsmooth newton method

**Coauthor: Diethard Klatte**

**Abstract:**

In recent works of the authors a nonsmooth Newton was developed in an abstract framework and applied to certain finite dimensional optimization problems with *C*^{1,1} data. The *C*^{1,1} structure stems from the presence of an optimal value function of a lower level problem in the objective or the constraints. Such problems arise, for example, in semi-infinite programs under a reduction approach without strict complementarity and in generalized Nash equilibrium models. Using results from parametric optimization and variational analysis, the authors worked out in detail the concrete Newton schemes for these applications and discussed wide numerical results for (generalized) semi-infinite problems. This Newton scheme requests the exact computation of stationary points of a lower level problem, which is problematic from a numerical point of view. In this talk we discuss the influence of inexact stationary points on the feasibility and the convergence properties of the Newton scheme. We will make use of a perturbed Newton scheme and give concrete estimates of the convergence radius resp. rate for the perturbed scheme.

**Friday, 14:15 - 14:40 h, Room: H 0107, Talk 3**

**Bernd Kummer**

Newton schemes for functions and multifunctions

**Abstract:**

To solve an inclusion *0* in *H(x)* we consider iterations of the type *0* in *F(x*_{k+1},x_{k}) in Euclidean spaces. To ensure convergence, the approximations between *H* and *F* play the crucial role and will be discussed. We present situations where the auxiliary problems require to study eps-subdifferentials or proximal point steps. Additionally, we consider the case when *H* is a continuous or only locally Lipschitz function *f*, and *F* represents inclusions *0* in *f(x*_{k})+ Gf(x_{k})(x_{k+1}-x_{k}) with some generalized derivative *Gf*. Then the iterations describe certain standard and non-standard nonsmooth Newton methods with more or less strong convergence conditions.