Tibken, Bernd (Univ. of Wuppertal), Fan, Youping (Univ. of Wuppertal), Schulte-Herbrueggen, Thomas (Tech. Univ. of Munich), Glaser, Steffen J. (Tech. Univ. of Munich)

Optimization Approach for Quadratically Constrained Quadratic Programs with Application to Determining C-Numerical Ranges

Scheduled for presentation during the Regular Session "Optimization" (TuP12), Tuesday, July 25, 2006, 15:20−15:45, Room 104b

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 25, 2024

Abstract

In this paper the maximization of the real part of the C-numerical range of an arbitrary complex square matrix A is studied. The geometry of the C-numerical range can be quite complicated and only partially understood. Through quadratically constrained quadratic optimization program (QQP) this problem can be formulated well, where the quadratic constraints are represented by the unitary matrix condition U⁺ U = I and the seemingly redundant unitary matrix condition U U⁺ = I, here ⁺ denotes the Hermitian conjugate of a matrix.

In general the QQPs are NP-hard und numerically intractable. However the Lagrangian relaxation can offer good approximate solutions to these hard problems. It is shown that in some special cases the Lagrangian dual relaxation provides even a zero duality gap. The results are compared with those obtained by other methods for physical benchmark examples.