Constrained optimization and analytic spectral factorization
Mikhail Iakoubovski · 1999
Let [special characters omitted] be function of [special characters omitted], 0 [special characters omitted], and [special characters omitted], that takes m x m nonnegative matrix values and such that [special characters omitted] for all [special characters omitted]) and all [special characters omitted]. Denote by [special characters omitted] the space of N-vector valued functions continuous on the closed unit disc in [special characters omitted] and analytic inside the disc whose Taylor series expansion about 0 has real coefficients. We consider the following problem: [special characters omitted]where [special characters omitted] is a subset of [special characters omitted] that may depend on given continuous N-vector valued functions [special characters omitted] whose entries have real Fourier coefficients, on given continuous scalar valued functions [special characters omitted] with real Fourier coefficients, and on given real constants cl. The set [special characters omitted] we consider takes one of the following forms:[special characters omitted]We establish optimality conditions for each of these three cases that must be satisfied by a solution of the (CONSTRAINED-OPT) problem. These optimality conditions can be used for finding good candidates to solutions of the (CONSTRAINED-OPT), and we discuss algorithms for finding such candidates for [special characters omitted] as in (1) and (2). Numerical examples are presented. We also consider the problem of finding a spectral factorization of a given, possibly low rank, positive semidefinite matrix valued function on the unit circle of the complex plane. We derive operator equations that must be satisfied by solutions to the spectral factorization problem for either the low or the full rank cases. The equations we derive can be solved numerically using Newton's method. We prove that sufficient conditions for local quadratic convergence of the Newton's method are satisfied. Numerical examples are given. Finally, we develop an implementation of a high level algorithm of Peller-Young for finding superoptimal approximants to a given matrix valued continuous function. Our implementation uses spectral factorization as well as other tools that we developed for this purpose. Also a numerical example is presented.