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

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 ther methods.