TY - GEN

T1 - Semidefinite programming relaxations applied to determining upper bounds of C-numerical ranges

AU - Tibken, Bernd

AU - Fan, Youping

AU - Glaser, Steffen J.

AU - Schulte-Herbrüggen, Thomas

PY - 2007

Y1 - 2007

N2 - In this contribution the global optimal upper bounds of the C-numerical range of an arbitrary square matrix A is investigated. In general the geometry of the C-numerical range is quite complicated and can be yet only partially understood. However, quadratically constrained quadratic programs (QQPs), as an important modelling tool, are used to describe this optimization problem, where the quadratic constraints are in this case the unitary matrix condition U†U = I und its seemingly redundant unitary matrix condition UU† = I. Generally the QQPs are NP-hard and numerically intractable. However the Semidefinite Programming (SDP) Relaxations to the QQPs, based upon the Positivstellensatz, can be solved in a numerically stable way and then offer sharp approximate solutions to these optimization problems. Numerical results for some physical benchmark examples are presented which indicate that the proposed method yields at least competitive upper bounds of the C-numerical ranges in comparison with other methods.

AB - In this contribution the global optimal upper bounds of the C-numerical range of an arbitrary square matrix A is investigated. In general the geometry of the C-numerical range is quite complicated and can be yet only partially understood. However, quadratically constrained quadratic programs (QQPs), as an important modelling tool, are used to describe this optimization problem, where the quadratic constraints are in this case the unitary matrix condition U†U = I und its seemingly redundant unitary matrix condition UU† = I. Generally the QQPs are NP-hard and numerically intractable. However the Semidefinite Programming (SDP) Relaxations to the QQPs, based upon the Positivstellensatz, can be solved in a numerically stable way and then offer sharp approximate solutions to these optimization problems. Numerical results for some physical benchmark examples are presented which indicate that the proposed method yields at least competitive upper bounds of the C-numerical ranges in comparison with other methods.

UR - http://www.scopus.com/inward/record.url?scp=43049155052&partnerID=8YFLogxK

U2 - 10.1109/CCA.2006.285999

DO - 10.1109/CCA.2006.285999

M3 - Conference contribution

AN - SCOPUS:43049155052

SN - 0780397959

SN - 9780780397958

T3 - Proceedings of the IEEE International Conference on Control Applications

SP - 2601

EP - 2606

BT - Proceedings of the 2006 IEEE International Conference on Control Applications

T2 - Joint 2006 IEEE Conference on Control Applications (CCA), Computer-Aided Control Systems Design Symposium (CACSD) and International Symposium on Intelligent Control (ISIC)

Y2 - 4 October 2006 through 6 October 2006

ER -