Title: Algorithm for Optimization and Interpolation based on Hyponormality,
Abstract: On one hand, consider the problem of finding global solutions to a polynomial
optimization problem and, on the other hand, consider the problem of
interpolating a set of points with a complex exponential function. This paper
proposes a single algorithm to address both problems. It draws on the notion of
hyponormality in operator theory. Concerning optimization, it seems to be the
first algorithm that is capable of extracting global solutions from a
polynomial optimization problem where the variables and data are complex
numbers. It also applies to real polynomial optimization, a special case of
complex polynomial optimization, and thus extends the work of Henrion and
Lasserre implemented in GloptiPoly. Concerning interpolation, the algorithm
provides an alternative to Prony's method based on the Autonne-Takagi
factorization and it avoids solving a Vandermonde system. The algorithm and its
proof are based exclusively on linear algebra. They are devoid of notions from
algebraic geometry, contrary to existing methods for interpolation. The
algorithm is tested on a series of examples, each illustrating a different
facet of the approach. One of the examples demonstrates that hyponormality can
be enforced numerically to strenghten a convex relaxation and to force its
solution to have rank one.