Multidisk problems in H Optimization: A method for analyzing numerical algorithms

Abstract

The theory of block Toeplitz operators and block Hankel operators is exploited to analyze numerical algorithms for solving optimization problems of the form y* = inf sup ||Γ(e iθ ,f(e iθ ))|| m×m , where r(e iθ ,z) is a smooth positive semidefinite matrix valued function of e iθ z = (z 1 ,...,z n ) and z = (z 1 ,...,z N ), and A N is a prescribed set of N-tuples f = (f 1 ,...,f N ) of functions that are analytic in the open unit disk D of the complex plane C. The algorithms under consideration are based on writing the equations that an optimum must satisfy (in terms of primal and dual variables f, y, Ψ) as T(Ψ,f,y) = 0 and then invoking a Newton algorithm (or something similar) to solve these equations. The convergence of Newton's method depends critically upon whether or not the differential T' is invertible. For the class of problems under consideration, this is a very difficult issue to resolve. However, it is relatively easy to determine when T' is a Fredholm operator with Fredholm index equal to zero, as will be shown in this paper. Fortunately, it turns out that this weaker condition seems to characterize effective numerical algorithms and is reasonably easy to check. Explicit tests for the differential T' to be a Fredholm operator of index zero are presented and compared with numerical experiments on a few randomly chosen two and three disk problems. The experimental results lend credence to our contention that whether or not the differential T' is a Fredholm operator of index zero determines the numerical behavior for almost all multidisk problems.