Quantitative Local Analysis of Nonlinear Systems - University of ...

3.2.2 Hit-and-Run Random Point Generation Algorithm
The following random point generation algorithm is used to generate points in Y [61]:
Hit-and-run random point generation (H&R) algorithm: Given a positive integer N V
and α (0) ∈ Y, set k = 0
i. Generate a random vector ζ (k) = y (k) /‖y (k) ‖, where y (k) ∼ N (0, I nb ).
ii. Compute the minimum value of t (k) ≤ 0 and the maximum value of t (k) ≥ 0 such that
α (k) + t (k) ζ (k) ∈ Y and α (k) + t (k) ζ (k) ∈ Y.
iii. Pick w (k) from a uniform distribution on [0, 1];
iv. α (k+1) = w (k) (α (k) + t (k) ζ (k) ) + (1 − w (k) )(α (k) + t (k) ζ (k) ).
v. k = k + 1.
vi. If k = N V , return A. Otherwise, go to (i).
⊳
Figure 3.1 shows the initial point (big dot), an arbitrary point satisfying (3.6), (3.11) and
(3.12), and the points generated by H&R algorithm (small dots) along with the random
directions from step (i) (dashed line segments) for the illustrative example. Y is convex
and points are generated as convex combinations of points in Y (step (iv)); therefore, the
algorithm generates N V
points in Y. Since Y is compact, for sufficiently large N V , points
of A become approximately uniformly distributed in Y [61].
Step (ii) of the H&R algorithm involves solving a linear SOS optimization problem and
a LMI problem
t (k)
SOS :=
max t≥0 t s.t. z(x) T (α (k) + tζ (k) ) ∈ Σ[x],
t (k)
lin := max t≥0 t s.t. L(Q (k) + tΛ (k) ) ≼ −ɛI,
(3.13)
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