A global optimization approach for the BMI problem

The biaffine matrix inequality (BMI) is a potentially very flexible new framework for approaching complex robust control system synthesis problems with multiple plants, multiple objectives and controller order constraints. The BMI problem may be viewed as the nondifferentiable biconvex programming problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. The BMI problem is non-local-global in general, i.e. there may exist local minima which are not global minima. While local optimization techniques sometimes yield good results, global optimization procedures need to be considered for the complete solution of the BMI problem. In this paper, we present a global optimization algorithm for the BMI based on the branch and bound approach. A simple numerical example is included