Linear Programming Lecture Notes - Penn State Personal Web Server

CHAPTER 4
Convex Sets, Functions and Cones and Polyhedral Theory
In this chapter, we will cover all of the geometric prerequisites for understanding the
theory of linear programming. We will use the results in this section to prove theorems
about the Simplex Method in other sections.
1. Convex Sets
Definition 4.1 (Convex Set). Let X ⊆ R n . Then the set X is convex if and only if for
all pairs x1, x2 ∈ X we have λx1 + (1 − λ)x2 ∈ X for all λ ∈ [0, 1].
The definition of convexity seems complex, but it is easy to understand. First recall that
if λ ∈ [0, 1], then the point λx1 +(1−λ)x2 is on the line segment connecting x1 and x2 in R n .
For example, when λ = 1/2, then the point λx1 + (1 − λ)x2 is the midpoint between x1 and
x2. In fact, for every point x on the line connecting x1 and x2 we can find a value λ ∈ [0, 1]
so that x = λx1 + (1 − λ)x2. Then we can see that, convexity asserts that if x1, x2 ∈ X,
then every point on the line connecting x1 and x2 is also in the set X.
then
Definition 4.2 (Positive Combination). Let x1, . . . , xm ∈ R n . If λ1, . . . , λm > 0 and
(4.1) x =
m
i=1
λixi
is called a positive combination of x1, . . . , xm.
then
Definition 4.3 (Convex Combination). Let x1, . . . , xm ∈ R n . If λ1, . . . , λm ∈ [0, 1] and
m
λi = 1
i=1
(4.2) x =
m
i=1
λixi
is called a convex combination of x1, . . . , xm. If λi < 1 for all i = 1, . . . , m, then Equation
4.2 is called a strict convex combination.
Remark 4.4. If you recall the definition of linear combination, we can see that we move
from the very general to the very specific as we go from linear combinations to positive
combinations to convex combinations. A linear combination of points or vectors allowed us
to choose any real values for the coefficients. A positive combination restricts us to positive
values, while a convex combination asserts that those values must be non-negative and sum
to 1.
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