link to my thesis

58 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
to ξ 2 times its preceding row. Obviously, in such a situation the value y 0,0 can not
be zero and it therefore can be used to produce a normalized common eigenvector
(1,ξ 1 ,ξ 2 1,ξ 2 ,ξ 1 ξ 2 ,ξ 2 1ξ 2 ,ξ 2 2,ξ 1 ξ 2 2,ξ 2 1ξ 2 2) T of the matrices A T x 1
and A T x 2
from which the
multi-eigenvalue (ξ 1 ,ξ 2 ) can be read off directly.
5.2 Usage of the nD-system in polynomial optimization
Modern methods for the solution of large eigenvalue problems have the attractive
feature that they do not operate on the matrix A T r directly. Instead they iteratively
perform the action of the linear operator at hand, for which it suffices to
implement a computer routine that computes this action for any given vector. The
nD-systems approach supports this and it offers a framework to compute the action
of A T r on a vector v, by initializing the initial state as w 0,...,0 := v and using
the n recursions (5.3) in combination with the relationship (5.10) to obtain the vector
r(σ 1 ,...,σ n )w 0,...,0 = A T r w 0,...,0 . Such an approach entirely avoids an explicit
construction of the matrix A r . Note that r(σ 1 ,...,σ n )w 0,...,0 consists of a linear
combination of state vectors w t1,...,t n
; each monomial term r α1,...,α n
x α1
1 ···xαn n that
occurs in r(x 1 ,...,x n ) corresponds to a weighted state vector r α1,...,α n
w α1,...,α n
. This
makes clear that for any choice of polynomial r the vector r(σ 1 ,...,σ n )w 0,...,0 can be
constructed from the same multidimensional time series y t1,...,t n
which is completely
determined by (and computable from) the n difference equations (5.3) and the initial
state w 0,...,0 .
There are two possible ways to retrieve the minimal value and the global minimizer(s)
of the polynomial p λ (x 1 ,...,x n ).
(i) If attention is focused on the computation of all the stationary points of the
criterion p λ first, then the actions of the matrices A T x i
, for all i = 1,...,n, play
a central role since their eigenvalues constitute the coordinates of these stationary
points. Given an initial state w 0,...,0 , which is composed of m n values of the time
series y t1,...,t n
(those for which t j