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.

 

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