A bisection algorithm for the mixed /spl mu/ upper bound and its supremum

A new approach to computing the mixed /spl mu/ upper bound (v) is presented. The method exploits the fact that a positive definite matrix V(/spl alpha/) becomes singular when the scalar parameter a decreases to a critical value for a given frequency. A two-level optimization strategy is used with a bisection algorithm branching on the definiteness of V in an outer loop, and a semi-definite programming (SDP) problem is formulated in an inner loop. Three different formulations are posed for the inner loop. The first uses a feasibility formulation (no objective function) with a constraint V/spl ges/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 /spl ges/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 v for a single frequency, it is a conceptually simple method that can be efficient when computing the supremum of v with respect to frequency.