Linear Programming Lecture Notes - Penn State Personal Web Server

Example 4.5. Figure 4.1 illustrates a convex and non-convex set. Non-convex sets have
x 1
X
x 2
x 1 x 2
Convex Set Non-Convex Set
Figure 4.1. Examples of Convex Sets: The set on the left (an ellipse and its
interior) is a convex set; every pair of points inside the ellipse can be connected by
a line contained entirely in the ellipse. The set on the right is clearly not convex as
we’ve illustrated two points whose connecting line is not contained inside the set.
some resemblance to crescent shapes or have components that look like crescents.
Theorem 4.6. The intersection of a finite number of convex sets in R n is convex.
Proof. Let C1, . . . , Cn ⊆ Rn be a finite collection of convex sets. Let
n
(4.3) C =
i=1
Ci
be the set formed from the intersection of these sets. Choose x1, x2 ∈ C and λ ∈ [0, 1].
Consider x = λx1 + (1 − λ)x2. We know that x1, x2 ∈ C1, . . . , Cn by definition of C. By
convexity, we know that x ∈ C1, . . . , Cn by convexity of each set. Therefore, x ∈ C. Thus
C is a convex set.
2. Convex and Concave Functions
Definition 4.7 (Convex Function). A function f : R n → R is a convex function if it
satisfies:
(4.4) f(λx1 + (1 − λ)x2) ≤ λf(x1) + (1 − λ)f(x2)
for all x1, x2 ∈ R n and for all λ ∈ [0, 1].
This definition is illustrated in Figure 4.2. When f is a univariate function, this definition
can be shown to be equivalent to the definition you learned in Calculus I (Math 140) using
first and second derivatives.
Definition 4.8 (Concave Function). A function f : R n → R is a concave function if it
satisfies:
(4.5) f(λx1 + (1 − λ)x2) ≥ λf(x1) + (1 − λ)f(x2)
for all x1, x2 ∈ R n and for all λ ∈ [0, 1] 1 .
To visualize this definition, simply flip Figure 4.2 upside down. The following theorem
is a powerful tool that can be used to show sets are convex. It’s proof is outside the scope
of the class, but relatively easy.
1 Thanks to Greg Ference and Veselka Kafedzhieva for catching a typo in this definition.
52
X