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46 CHAPTER 4. GLOBAL OPTIMIZATION OF MULTIVARIATE POLYNOMIALS
which is readily expanded to n dimensions. Note that any other total degree monomial
ordering could be chosen instead.
By applying the Stetter-Möller matrix method, an n-tuple of commuting N ×N =
m n × m n matrices (A T x 1
,A T x 2
,...,A T x n
) can be constructed. These matrices yield a
matrix solution of the system of polynomial equations (4.2). Any common eigenvector
v of these matrices A T x 1
,A T x 2
,...,A T x n
leads to a scalar solution, constituted by the
n-tuple of eigenvalues of the matrices A T x 1
,A T x 2
,...,A T x n
corresponding to v.
A straightforward method to compute the global minimum of a dominated real
polynomial p λ (x 1 ,...,x n ) over R n , is to first compute all the real stationary points
of p λ (x 1 ,...,x n ) by solving the system of first-order conditions d (i) (x 1 ,..., x n )=0,
(i = 1,...,n) using the real eigenvalues of the matrices A T x 1
,A T x 2
,...,A T x n
. The
stationary points (x 1 ,...,x n ) can then be plugged into the polynomial p λ (x 1 ,...,
x n ) and selecting the real stationary point at which the minimal value is attained
yields the global optimum of the polynomial p λ (x 1 ,...,x n ).
However, there is another more efficient way of selecting the global optimum of
the polynomial under consideration, by slightly extending the Stetter-Möller matrix
method as discussed in the next section.
4.2 The Stetter-Möller matrix method for global optimization
In Section 3.3, the linear multiplication operators A xi on R[x 1 ,...,x n ]/I are introduced,
which represent multiplication by x i modulo the ideal I, generated by the
first-order conditions d (i) . More generally, a similar set-up of the Stetter-Möller matrix
method for a polynomial r(x 1 ,...,x n ) can be used to introduce a linear operator
A r(x1,...,x n). It turns out that this operator is of importance for the global optimization
of a Minkowski dominated polynomial. The operator A r(x1,...,x n) is defined
as:
A r(x1,...,x n) : R[x 1 ,...,x n ]/I ↦→ R[x 1 ,...,x n ]/I : g ↦→ r(x 1 ,...,x n ) · g (4.7)
A crucial observation now is that also in this case polynomial multiplication within
R[x 1 ,...,x n ]/I is a linear operation. Therefore, given any basis for R[x 1 ,...,x n ]/I,
for instance the basis B introduced in (4.5), it is possible to represent this operator
A r by a matrix: the matrix A T r(x of dimension N × N = 1,...,x n) mn × m n . This
matrix is then associated with the linear operation of multiplication by a polynomial
r(x 1 ,...,x n ) within R[x 1 ,...,x n ]/I. It turns out that A T r(x = 1,...,x n) r(AT x 1
,...,A T x n
)
and that the matrix A T r also commutes with the matrices A T x i
(see also [20]). Depending
on the choice of r, such a matrix A r is usually highly sparse and structured.
If v is a common eigenvector of the matrices A T x 1
,...,A T x n
for (λ 1 ,...,λ n ), then
v is also an eigenvector of A T r(x with corresponding eigenvalue r(λ 1,...,x n) 1,...,λ n ):
A T r v = r(A T x 1
,...,A T x n
)v = r(λ 1 ,...,λ n )v.