We consider convex optimization problems with the constraint that the variables form a finite autocorrelation sequence, or equivalently, that the corresponding power spectral density is nonnegative. This constraint is often approximated by sampling the power spectral density, which results in a set of linear inequalities. It can also be cast as a linear matrix inequality via the positive-real lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interior-point methods for semidefinite programming. However, these methods require O(n^6) floating point operations per iteration, if a general-purpose implementation is used. We introduce a much more efficient method with a complexity of O(n^3) flops per iteration.
To appear in the Proceedings of the 34th IEEE Asilomar Conference on Signals, Systems and Computer, Pacific Grove, California, October 29 through November 1, 2000.