A multidimensional systems approach to polynomial optimization

With any system of multivariate polynomial equations we can associate a system of multidimensional difference equations by interpreting the variables in the polynomial equations as shift operators working on a multidimensional time series. If the solution set of the system of multivariate polynomial equations is finite, including possible complex solutions, then the associated system of multidimensional difference equations has a finite dimensional solution set and one can describe the solution set as the outcomes of a finite dimensional state space system, a so-called state space realization of the system of multidimensional difference equations. Scalar solutions of the original system of polynomial equations coincide with the multi-poles of such a state-space realization, provided the state-space realization is minimal, i.e. has smallest possible state-space dimension. Each multi-pole is a multi-eigenvalue of the tuple of matrices representing the various shift operators. A multi-eigenvalue corresponds to a well-defined eigenspace. Here a special class of polynomials is considered for which the global minimization problem is solved by associating a multidimensional system to the first order conditions for minimization of the polynomial and solving for the first order conditions using the above method. The method leads to large eigenvalue problems. One way to solve them is by Jacobi-Davidson methods. For such methods one does not need to calculate the matrix that is involved explicitly, one only needs to specify the action of the linear operator on any given vector. Using the multidimensional system structure the action of the linear operator can be obtained by solving for certain elements of the multidimensional sequence that forms the solution of the system of difference equations given a sufficient number of initial values of the sequence.