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74 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
n time directions, it is non-minimal and especially for points near the initial
hypercube more efficient paths can be constructed.
• The equalizing method performs best for points having (almost) equal coordinates
(thus, points near the diagonal, for example the points (10, 10), (50, 50) or
(100, 100)). For n = 2 this method generates stable patterns which constitute
an optimal solution to the shortest path problem for y t1,t 2
with t 1 = t 2 . For
other points it is less efficient. Increasing the dimension n for this method does
not influence this behavior.
• The axis method performs well for points near the coordinate axes, (for example
the points (0, 10), (0, 50) or (0, 100)). For n = 2 this method generates stable
patterns which constitute an optimal solution to the shortest path problem for
y t1,t 2
with t 1 =0ort 2 = 0. Increasing the dimension n for this method does
not influence this behavior.
• The diagonal method does not exhibit a linear numerical complexity with respect
to |t| and is highly inefficient.
To further support and visualize these statements, some simulation experiments
have been performed with n = 2 and m = 3, where the requested state vectors are
further away from the origin: w 0,500 , w 125,375 , w 250,250 , w 375,125 and w 500,0 .
For the linear, diagonal, equalizing and axis methods, the points which are required
for computing the state vectors w 0,500 , w 250,250 , and w 125,375 , respectively, are
displayed in three separate plots in the Figures 5.12, 5.13, 5.14 and 5.15.
Figure 5.12: Points required by the Linear method for computing state vectors at
time instants (0, 500), (250, 250), and (125, 375) with a 3 × 3 initial state
From the Figures 5.12, 5.13, 5.14, and 5.15 one can see that the linear method has
the best performance as it computes the fewest numbers of points in each situation
and that the diagonal method has a highly inefficient performance for all locations
of the requested state vectors. The equalizing and axis methods are only efficient in
some specific situations: Figure 5.14 shows that the equalizing method is only efficient