## Contributed Session Thu.1.H 1012

#### Thursday, 10:30 - 12:00 h, Room: H 1012

**Cluster 17: Nonsmooth optimization** [...]

### Nonsmooth optimization theory

**Chair: Waltraud Huyer**

**Thursday, 10:30 - 10:55 h, Room: H 1012, Talk 1**

**Anna-Laura H. WickstrĂ¶m**

Generalized derivatives of the projection onto the cone of positive semidefinite matrices

**Abstract:**

We are interested in sensitivity and stability analysis of solution sets of nonlinear optimization problems under set or cone constraints. A main motivation behind our work is the analysis of semidefinite programs (SDPs). We wish to explore the sensitivity analysis of SDPs with help of generalized derivatives.

In order to study critical points and solutions of SDPs we construct a Kojima kind of locally Lipschitz functions of the Karush-Kuhn-Tucker conditions for *C*^{2}-optimization problems over the space of symmetric matrices.

We will study generalized derivatives of this Kojima-function in order to show regularity of our problem.

The Kojima-function is the product of a continuously differentiable and a nonsmooth function. The latter contains the projection function onto the cone of positive semidefinite matrices. We shall look at the construction of it's Thibault derivatives (strict graphical derivatives). Moreover, we examine the relations between Thibault derivatives and Clarke's generalized Jacobians of these projections.

**Thursday, 11:00 - 11:25 h, Room: H 1012, Talk 2**

**Alain B. Zemkoho**

Optimization problems with value function objectives

**Abstract:**

The family of optimization problems with value function objectives includes the minmax programming problem and the (pessimistic and original optimistic) bilevel optimization problem. In this talk, we would like to discuss necessary optimality conditions for this class of problems while assuming that the functions involved are nonsmooth and the constraints are the very general operator constraints.

**Thursday, 11:30 - 11:55 h, Room: H 1012, Talk 3**

**Waltraud Huyer**

Minimizing functions containing absolute values of variables

**Coauthor: Arnold Neumaier**

**Abstract:**

We propose an algorithm for the minimization of the function *f(Ax-b,|x|)* on a box **x** ⊆ **R**^{n} with nonempty interior, where *|x|* denotes the componentwise absolute value, *f* is a *C*^{2} function, *b ∈ ***R**^{m} and *A ∈ ***R**^{m× n}. Moreover, we assume that gradients and Hessian matrices are available and that the Hessian matrices of *f* can be represented in the form *G = D + R*^{T}ER, where *D* and *E* are diagonal matrices.

The algorithm MINABS proceeds from a starting point by making coordinate searches, minimizing local quadratic models and checking the optimality conditions. For the minimization of the quadratic models, an algorithm for minimizing a quadratic function of the form

q(x) = γ + c^{T}x + \frac12(Bx-c)^{T}F(Bx-c),

*F* a diagonal matrix, is developed.

Finally, MINABS is applied to problems of the above kind in order to demonstrate the applicability of the algorithm.