Optimization Problems over Positive Pseudopolynomial Matrices

The Nesterov characterizations of positive pseudopolynomials on the real line, the imaginary axis, and the unit circle are extended to the matrix case. With the help of these characterizations, a class of optimization problems over the space of positive pseudopolynomial matrices is considered. These problems can be solved in an efficient manner due to the inherent block Toeplitz or block Hankel structure induced by the characterization in question. The efficient implementation of the resulting algorithms is discussed in detail. In particular, the real line setting of the problem leads naturally to ill-conditioned numerical systems. However, adopting a Chebyshev basis instead of the natural basis for describing the polynomial matrix space yields a restatement of the problem and of its solution approach with much better numerical properties.