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80 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
(a)
(b)
Figure 5.19: Region of points that occur in (a) 2-dimensional paths and (b) 3-
dimensional paths
Because the paths in the nD-systems approach for polynomial optimization are
short and have much overlap, the use of parallel computations is not advisable. The
overhead of communication is too high in this case, especially for high-dimensional
problems. In this particular case of polynomial optimization the difference between
the performance of the linear method and the least-increments method will not be
that high because of the shortness of the occurring paths.
As said before in the beginning of this section, in a more general case, parallelization
can be an improvement of the efficiency of the nD-system. One can imagine
that in a certain application one wants to compute the action of a matrix A r , where
r is a polynomial of high degree containing a lot of terms, without construction this
matrix explicitly. The nD-system in such a case contains long separate paths with
little overlap and therefore every path can be considered by a single processor. Parallelization
even becomes more suitable when the number of variables increases in such
a situation, but this still requires further research.