Multivariate Polynomial Optimization in Complex Variables Is a (Rectangular) Multiparameter Eigenvalue Problem

We extend the relation between univariate polynomial optimization in one complex variable and the polynomial eigenvalue problem to the multivariate case. The first-order necessary conditions for optimality of the multivariate polynomial optimization problem, which are computed using Wirtinger derivatives, constitute a system of multivariate polynomial equations in the complex variables and their complex conjugates. Wirtinger calculus provides an elegant way to differentiate real-valued (cost) functions in complex variables. An elimination of the complex conjugate variables, via the Macaulay matrix, results in a (rectangular) multiparameter eigenvalue problem, (some of) the eigentuples of which correspond to the stationary points of the original real-valued cost function. We illustrate our novel globally optimal optimization approach with several (didactical) examples.