Polynomial Optimization Problems are Eigenvalue Problems

Abstract

Many problems encountered in systems theory and system identification require the solution of polynomial optimization problems, which have a polynomial objective function and polynomial constraints. Applying the method of Lagrange multipliers yields a set of multivariate polynomial equations. Solving a set of multivariate polynomials is an old, yet very relevant problem. It is little known that behind the scene, linear algebra and realization theory play a crucial role in understanding this problem.We show that determining the number of roots is essentially a linear algebra question, from which we derive the inspiration to develop a root-finding algorithm based on realization theory, using eigenvalue problems. Moreover, since one is only interested in the root that minimizes the objective function, power iterations can be used to obtain the minimizing root directly.We highlight applications in systems theory and system identification, such as analyzing the convergence behaviour of prediction error methods and solving structured total least squares problems.