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Chapter 5
An nD-systems approach in
polynomial optimization
In Chapter 4 an extension of the Stetter-Möller matrix method is introduced for
computing the global minimum of a multivariate Minkowski dominated polynomial
p λ (x 1 ,...,x n ). The Stetter-Möller matrix method can be applied to the class of
Minkowski dominated polynomials because its system of first-order conditions is in
Gröbner basis form and has a finite number of zeros due to its construction. This
approach yields a matrix A T p λ (x 1,...,x n)
whose real eigenvalues are the function values
of the stationary points of the polynomial p λ (x 1 ,...,x n ). The global minimum
of p λ (x 1 ,...,x n ) can therefore be computed by the smallest real eigenvalue of this
matrix. The involved eigenvectors are structured and because of this structure the
values of the minimizers (x 1 ,...,x n ) can be read off from the eigenvectors (see the
previous chapter for details).
A serious bottleneck in this approach from a computational point of view is constituted
by solving the eigenproblem of the matrix A T r introduced in Chapter 4. As
a matter of fact, the N × N matrix A T r quickly grows large, since N = m n where m
equals 2d − 1 and 2d denotes the total degree of the polynomial p λ (x 1 ,...,x n ). On
the other hand, the matrix A T r may be highly sparse and structured. This holds in
particular for the choices r(x 1 ,...,x n )=x i .
Large and sparse eigenproblems are especially suited for a modern iterative eigenproblem
solver (e.g., based on a Jacobi–Davidson [39], [94], or an Arnoldi method
[77]). An advantage of such an iterative solver is that it has the ability to focus on
certain eigenvalues first. In our application, this would be the smallest real eigenvalue.
This allows to address the critical values of a polynomial p λ (x 1 ,...,x n ) directly, as
they appear among the (real) eigenvalues of the matrix A T p λ
, without determining
all the stationary points of p λ (x 1 ,...,x n ) first. Iterative eigenvalue solvers and the
modifications of their implementations to improve their efficiency in this special application
of polynomial optimization, are discussed in Chapter 6.
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