Model Predictive Control

3.4 Basic Operations on Polytopes 27
x2
15
10
5
0
-5
-10
R1
x1
R2
-15
-6 -4 -2 0 2 4 6
x2
(a) P-collection R =
i∈{1,2} Ri.
15
10
5
0
-5
-10
R1
x2
15
10
5
0
-5
-10
conv(R)
-15
-6 -4 -2 0 2 4 6
x1
(b) Convex hull of R.
Figure 3.5 Illustration of the convex hull operation.
x1
R2
-15
-6 -4 -2 0 2 4 6
(a) P-collection R =
i∈{1,2} Ri.
3.4.3 Vertex Enumeration
x2
15
10
5
0
-5
-10
env(R)
-15
-6 -4 -2 0 2 4 6
x1
(b) Envelope env(R).
Figure 3.6 Illustration of the envelope operation.
The operation of extracting the vertices of a polytope P given in H-representation
is referred to as vertex enumeration. This operation is the dual of the convex hull
operation and the algorithmic implementation is identical to a convex hull com-
putation, i.e., given a set of extreme points V = {Vi} NV
i=1
= vert(P) of a polytope
P given in H-representation it holds that P = conv(V ), where the operator vert
denotes the vertex enumeration. The necessary computational effort is exponential
in the number of input facets. Two different approaches for vertex enumeration
are commonly used: the double description method [99] and reverse search [11].
An efficient implementation of the double description method is available in [96].