Linear Programming Lecture Notes - Penn State Personal Web Server

f(x1)+(1− λ)f(x2)
f(λx1 +(1− λ)x2)
Figure 4.2. A convex function: A convex function satisfies the expression f(λx1 +
(1 − λ)x2) ≤ λf(x1) + (1 − λ)f(x2) for all x1 and x2 and λ ∈ [0, 1].
Theorem 4.9. Let f : R n → R be a convex function. Then the set C = {x ∈ R n :
f(x) ≤ c}, where c ∈ R, is a convex set.
Exercise 40. Prove the Theorem 4.9. [Hint: Skip ahead and read the proof of Lemma
4.15. Follow the steps in that proof, but apply them to f.]
3. Polyhedral Sets
Important examples of convex sets are polyhedral sets, the multi-dimensional analogs of
polygons in the plane. In order to understand these structures, we must first understand
hyperplanes and half-spaces.
Definition 4.10 (Hyperplane). Let a ∈ R n be a constant vector in n-dimensional space
and let b ∈ R be a constant scalar. The set of points
(4.6) H = x ∈ R n |a T x = b
is a hyperplane in n-dimensional space. Note the use of column vectors for a and x in this
definition.
Example 4.11. Consider the hyper-plane 2x1+3x2+x3 = 5. This is shown in Figure 4.3.
This hyperplane is composed of the set of points (x1, x2, x3) ∈ R 3 satisfying 2x1+3x2+x3 = 5.
This can be plotted implicitly or explicitly by solving for one of the variables, say x3. We
can write x3 as a function of the other two variables as:
(4.7) x3 = 5 − 2x1 − 3x2
Definition 4.12 (Half-Space). Let a ∈ R n be a constant vector in n-dimensional space
and let b ∈ R be a constant scalar. The sets of points
Hl = x ∈ R n |a T x ≤ b
(4.8)
Hu = x ∈ R n |a T x ≥ b
(4.9)
are the half-spaces defined by the hyperplane a T x = b.
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