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52 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
Another important advantage of iterative eigenproblem solvers is that they only
require the action of the matrix A T r on given vectors v, which avoids the need to build
the large matrix A T r explicitly. This matrix-free approach to compute the action of
the matrix A T r as the input for iterative eigenproblem solvers, is the main subject of
this chapter.
To avoid building the large matrix A T r one can associate the system of firstorder
derivatives of p λ with an nD-system of difference equations, by interpreting
the variables in the polynomial equations as shift operators σ 1 ,...,σ n working on
a multidimensional time series y t1,t 2,...,t n
. Then calculation of the action of A T r on
a given vector v requires solving for y t1,t 2,...,t n
using the difference equations. The
vector v corresponds to an initial state of the associated nD-system. See [5] and [40]
for similar ideas in the 2D-case. This set-up is presented in the first section of this
chapter.
The usage of the nD-system in global polynomial optimization using the Stetter-
Möller matrix method in combination with an iterative eigenproblem solver is described
in Section 5.2.
The huge number of required iterations is the main reason why the action of a
matrix A T r has to be computed efficiently. This is studied in Section 5.3 and its subsections.
One way to compute efficiently the action of A T r on v with an nD-system is
by first setting up a corresponding shortest path problem and to apply an algorithm,
like Dijkstra’s algorithm [32] or Floyd’s algorithm [38], to solve it. A drawback is that
the computation of an optimal shortest path along these lines can be quite expensive.
On the other hand, the numerical complexity of the computation of the action of A T r
based on a shortest path solution can be shown to depend only linearly on the total
degree of the polynomial r. Interestingly, suboptimal paths can easily be designed
which also achieve a numerical complexity which depends linearly on the total degree
of r. In the case of two-dimensional systems when there is no additional structure in
the first-order derivatives of p λ , the shortest path problem can be solved analytically.
For three-dimensional systems the situation is more complicated but a number of
partial results are available and presented in this chapter.
A numerical example is given in the first section of Chapter 7 where the approach
described in this chapter is demonstrated by means of a worked example
and compared to other approaches available in the literature for global optimization:
SOSTOOLS, GloptiPoly and SYNAPS.
5.1 The nD-system
In this section we pursue a state-space approach with respect to the computation
of the action of the linear operation of multiplication by a polynomial r within
R[x 1 ,...,x n ]/I, i.e., the action of the matrix A r . More precisely, we will be con-