Introduction to Unconstrained Optimization - Scilab

x 1
x 2
0
Figure 9: A cone
x 1
x 2
0
Figure 10: A nonconvex cone
Definition 2.3. (Cone) A set C is a cone if for every x ∈ C and θ ≥ 0, we have
θx ∈ C.
A set C is a convex cone if it is convex and a cone, which means that for any
x 1 , x 2 ∈ C and θ 1 , θ 2 ≥ 0, we have
θ 1 x 1 + θ 2 x 2 ∈ C. (28)
A cone is presented in figure 9.
An example of a cone which is nonconvex is presented in figure 10. This cone is
nonconvex because if we pick a point x 1 on the ray, and another point x 2 in the grey
area, there exists a θ with 0 ≤ θ ≤ 1 so that a convex combination θx 1 + (1 − θ)x 2
is not in the set.
A conic combination of points x 1 , . . . , x k ∈ C is a point of the form θ 1 x 1 + . . . +
θ k x k , with θ 1 , . . . , θ k ≥ 0.
The conic hull of a set C is the set of all conic combinations of points in C, i.e.
{θ 1 x 1 + . . . + θ k x k / x i ∈ C, θ i ≥ 0, i = 1, k} . (29)
The conic hull is the smallest convex cone that contains C.
Conic hulls are presented in figure 11.
2.2 Convex functions
Definition 2.4. (Convex function) A function f : C ⊂ R n → R is convex if C is a
convex set and if, for all x, y ∈ C, and for all θ with 0 ≤ θ ≤ 1, we have
f (θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y). (30)
A function f is strictly convex if the inequality 30 is strict.
18