link to my thesis

68 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
between a point of total time k and nearby points of total time k − 1. (Alternatively,
one may construct minimal stable patterns by deleting points from a stable pattern
of the form N until stability is lost.)
In Figure 5.7 it is shown for the case n = 3 and m = 2 how the points with
total time 4 (blue layer, Figure b) are connected along the directions of the time axes
(red lines) to nearby points of total time 3 (green layer, Figure a). In Figure 5.8b
the points in these two consecutive layers are arranged in convenient triangulated
patterns and shown from two different viewpoints (of which the last one will be used
in the following).
(a)
(b)
Figure 5.7: (a) Points with total time 4. (b) Connections of the points with total
time 4 to nearby points of total time 3
In Figure 5.9 it is visualized how the three different full recursions associated with
f (1) , f (2) and f (3) may assist in the computation of a point with total time 4, and
also which points of total time 3 (red triangle) are required to achieve this.
For n = 3 and m = 2, the stable pattern N in the previous proposition has 10
points. When two of its corner points are deleted, a minimal stable pattern of 8
points remains, which however allows only for a shift in just one of the directions of
the time axes. But when only one corner point is deleted, a stable pattern results
which allows for shifts in all three directions of the time axes.
As a consequence, a solution to the RSPP needs to take into account the coordinate
values of the time instant t =(t 1 ,t 2 ,t 3 ) associated with the terminal node v T . First,
a stable pattern containing 9 points can be used for shifting along the two axes
corresponding to the two smallest values of t 1 , t 2 , and t 3 . Then a cheaper minimal
stable pattern containing 8 points can be used for shifting along the remaining third
axis (this should be a subpattern of the previous pattern of 9 points). Note that such