Introduction to Unconstrained Optimization - Scilab

x 1
x 2
x 1
x 2
Figure 7: Convex sets - The left set is convex. The middle set is not convex, because
the segment joining the two points is not inside the set. The right set is not convex
because parts of the edges of the rectangle are not inside the set.
x 1 x 2
x 2
x 2
x 1
x 1
Figure 8: Convex hull - The left set is the convex hull of a given number of points.
The middle (resp. right) set is the convex set of the middle (resp. right) set in figure
7.
Definition 2.1. (Convex set) A set C in R n is convex if for every x 1 , x 2 ∈ C and
every real number α so that 0 ≤ α ≤ 1, the point x = αx 1 + (1 − α)x 2 is in C.
Convex sets and nonconvex sets are presented in figure 7.
A point x ∈ R n of the form x = θ 1 x 1 + . . . + θ k x k where θ 1 + . . . + θ k = 1
and θ i ≥ 0, for i = 1, k, is called a convex combination of the points {x i } i=1,k in
the convex set C. A convex combination is indeed a weighted average of the points
{x i } i=1,k . It can be proved that a set is convex if and only if it contains all convex
combinations of its points.
For a given set C, we can always define the convex hull of this set, by considering
the following definition.
Definition 2.2. (Convex hull) The convex hull of C, denoted by conv(C), is the
set of all convex combinations of points in C:
conv(C) = {θ 1 x 1 + . . . + θ k x k / x i ∈ C, θ i ≥ 0, i = 1, k, θ 1 + . . . + θ k = 1} (27)
Three examples of convex hulls are given in figure 8. The convex hull of a given
set C is convex. Obviously, the convex hull of a convex set is the convex set itself,
i.e. conv(C) = C if the set C is convex. The convex hull is the smallest convex set
that contains C.
We conclude this section be defining a cone.
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