A SVD approach to multivariate polynomial optimization problems

We present a novel method to compute all stationary points of optimization problems, of which the objective function and equality constraints are expressed as multivariate polynomials, in the linear algebra setting. It is shown how Stetter-Möller matrix methods can be obtained through a parameterization of the objective function, subsequently manipulated using Macaulay matrices. An algorithm is provided to extend this framework to circumvent the necessity of a Gröbner basis. The generalized eigenvalue problem is obtained through a sequence of unitary transformations and rank tests operating directly on the polynomial coefficients (data-driven). The proposed method is illustrated by means of a structured total least squares (STLS) example.