Model Predictive Control

20 3 Polyhedra, Polytopes and Simplices
Example 3.1 The set
F = {x ∈ R 2 : x1 + x2 = 1}
is an affine set in R 2 of dimension one. The points x 1 = [0, 1] ′ and x 2 = [1, 0] ′ belong
to the set F. The point ¯x = −0.2x 1 + 1.2x 2 = [1.2, −0.2] ′ is an affine combination of
points x 1 and x 2 . The affine hull of x 1 and x 2 , aff({x 1 , x 2 }), is the set F.
Convex sets have been defined in Section 1.2
The convex combination of a finite set of points x1 , . . .,x k belonging to Rn is
defined as the point λ1x1 + . . . + λkxk where k i=1 λi = 1 and λi ≥ 0, i = 1, . . . , k
The convex hull of a set K ⊆ Rn is the set of all convex combinations of points
in K and it is denoted as conv(K):
conv(K) {λ 1 x 1 + . . . + λ k x k : xi ∈ K, λ i ≥ 0, i = 1, . . .,k,
k
λ i = 1}. (3.3)
The convex hull of K is the smallest convex set that contains K, in the following
sense: if S is any convex set with K ⊆ S, then conv(K)⊆ S.
Example 3.2 Consider three points x 1 = [1, 1] ′ , x 2 = [1, 0] ′ , x 3 = [0, 1] ′ in R 2 . The
point ¯x = λ 1 x 1 +λ 2 x 2 +λ 3 x 3 with λ 1 = 0.2, λ 2 = 0.2, λ 3 = 0.6 is ¯x = [0.4, 0.8] ′ and it
is a convex combination of the points {x 1 , x 2 , x 3 }. The convex hull of {x 1 , x 2 , x 3 }
is the triangle plotted in Figure 3.1. Note that any set in R 2 strictly contained in the
triangle and containing {x 1 , x 2 , x 3 } is non-convex.
x2
2
1.5
1
0.5
0
¯x
-0.5
-0.5 0 0.5 1 1.5 2
Figure 3.1 Illustration of a convex hull of three points x 1 = [1, 1] ′ , x 1 = [1, 0] ′ ,
x 3 = [0,1] ′
A cone spanned by a finite set of points K = {x 1 , . . . , x k } is defined as
cone(K) = {
x1
i=1
k
λ i x i , λ i ≥ 0, i = 1, . . .,k}. (3.4)
i=1
We define cone(K) = {0} if K is the empty set.