## Contributed Session Mon.2.H 2035

#### Monday, 13:15 - 14:45 h, Room: H 2035

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

### Eigenvalue and semi-infinite optimization

**Chair: Sara Maria Grundel**

**Monday, 13:15 - 13:40 h, Room: H 2035, Talk 1**

**Sara Maria Grundel**

Variational analysis of the spectral abscissa for defective and derogatory matrices

**Coauthor: Michael L. Overton**

**Abstract:**

The spectral abscissa and radius are respectively the largest of the real parts and the largest

of the moduli of the eigenvalues of a matrix. They are non-Lipschitz, nonconvex functions

on the space of complex *n× n* matrices. To motivate our work, we briefly discuss

a spectral radius optimization problem for parametrized matrices arising from

subdivision surfaces, a method to construct smooth surfaces from

polygonal meshes used in computer graphics and geometric modeling.

In this case, we find that the minimizing matrix is both defective and derogatory:

the Jordan form of the optimal matrix has four blocks corresponding to the eigenvalue zero,

with sizes *4, 3, 2* and *2*. Generally, spectral functions are not Clarke regular at such

points in matrix space, and hence their subdifferential analysis is complicated. By refining

results of Burke and Overton, we address the simplest such case, presenting a complete

characterization of the Mordukhovich subgradients of the spectral abscissa for a matrix with

an eigenvalue having two Jordan blocks of size *2* and *1*.

**Monday, 13:45 - 14:10 h, Room: H 2035, Talk 2**

**Tatiana Tchemisova**

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

**Abstract:**

In the paper, we consider a problem of convex Semi-Infinite Programming with multi-dimensional index set in the form of a multidimensional polyhedron. In study of these problems we apply the approach suggested in our recent paper [Kost-Tchem] and based on the notions of immobile indices and their immobility orders. For this problem, we formulate explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison with other known optimality conditions is provided.

**Monday, 14:15 - 14:40 h, Room: H 2035, Talk 3**

**Julia Eaton**

On the subdifferential regularity of functions of roots of polynomials

**Coauthor: James V. Burke**

**Abstract:**

Eigenvalue optimization problems arise in the control of continuous and discrete time dynamical systems. The spectral abscissa and spectral radius are examples of functions of eigenvalues, or spectral functions, connected to these problems. A related class of functions are polynomial root functions. In 2001, Burke and Overton showed that the abscissa mapping on polynomials is subdifferentially regular on the monic polynomials of degree *n*. We extend this result to the class of max polynomial root functions which includes both the polynomial abscissa and the polynomial radius mappings. The approach to the computation of the subgradient simplifies that given by Burke and Overton and provides new insight into the variational properties of these functions.