A Bisection Algorithm for the Mixed μ Upper Bound and its Supremum

A new approach to computing the mixed μ upper bound (ν) is presented. The method exploits the fact that a positive de£nite matrix V (α) becomes singular when the scalar parameter α decreases to a critical value for a given frequency. A two-level optimization strategy is used with a bisection algorithm branching on the de£niteness of V in an outer loop, and a Semi-De£nite Programming (SDP) problem is formulated in an inner loop. Three different formulations are posed for the inner loop. The £rst uses a feasibility formulation (no objective function) with a constraint V 0, which tends to make V singular if possible. The second introduces an additional variable that makes the SDP feasible at all times. In the third formulation, the trace of V is minimized with the constraint V 0, which tends to minimize the rank of V and hence make V singular. The method is applied to a distillation column benchmark problem. Although it is computationally more expensive than existing methods when computing ν for a single frequency, it is a conceptually simple method that can be ef£cient when computing the supremum of ν with respect to frequency.