DOR1 Test 2 Terms

Convex Set

If

∀ x1, x2 ∈ X

λx1 + (1 — λ)x2 ∈ X

∀ λ ∈ [0,1]

X ⊆ R^n

Then X is a ____

Rays

A collection of points of the form

{x0 + λd : λ >=0} where d ≠ 0

(Note d has arrow above)

Convex combination

a linear combination of points where all coefficients are non-negative and sum to 1

Direction of a convex set

If for each x0 ∈ X (convex set)

the ray {x0 + λd : λ >=0} ⊆ X

Then d ≠ 0 is called the direction of X

Extreme point of a convex set

A point x in a convex set X is is this if x cannot be represented as a strict convex combination of two distinct points in X

Linear independence

A collection of vectors a1, a2,…, ak in R^n

is this if

[ Summation from j=1 to k ] λj aj = 0

Implies that λj = 0 for all j = 1,…, k