Chapter 4
OPTIMIZATION OVER SETS DEFINED BY INEQUALITY
CONSTRAINTS: LINEAR PROGRAMMING
In the first section we study in detail Example 2 of Chapter I, and then we define the general linear programming problem. In the second section we present the duality theory for linear program-
ming and use it to obtain some sensitivity results. In Section 3 we present the Simplex algorithm which is the main procedure used to solve linear programming problems. In section 4 we apply
the results of Sections 2 and 3 to study the linear programming theory of competitive economy. Additional miscellaneous comments are collected in the last section. For a detailed and readily ac-
cessible treatment of the material presented in this chapter see the companion volume in this Series Sakarovitch [1971].
4.1 The Linear Programming Problem
4.1.1 Example.
Recall Example 2 of Chapter I. Let g and u respectively be the number of graduate and undergradu-
ate students admitted. Then the number of seminars demanded per year is
2g+u 20
, and the number of lecture courses demanded per year is
5g+7u 40
. On the supply side of our accounting, the faculty can offer
2750 + 3250 = 2250 seminars and 6750 + 3250 = 5250 lecture courses. Because of his contractual agreements, the President must satisfy
2g+u 20
≤ 2250 or 2g + u ≤ 45, 000 and
5g+7u 40
≤ 5250 or 5g + 7u ≤ 210, 000 . 27
28
CHAPTER 4. LINEAR PROGRAMMING
Since negative g or u is meaningless, there are also the constraints g ≥ 0, u ≥ 0. Formally then the
President faces the following decision problem: Maximize
αg + βu subject to
2g + u ≤ 45, 000 5g + 7u ≤ 210, 000
g ≥ 0, u ≥ 0 . 4.1
It is convenient to use a more general notation. So let x = g, u
′
, c = α, β
′
, b = 45000, 210000, 0, 0
′
and let A be the 4×2 matrix
A =
2
5 −1
1 7
−1
.
Then 4.1 can be rewritten as 4.2
1
Maximize c
′
x subject to
Ax ≤ b . 4.2
Let A
i
, 1 ≤ i ≤ 4, denote the rows of A. Then the set Ω of all vectors x which satisfy the constraints in 4.2 is given by
Ω = {x|A
i
x ≤ b
i
, 1 ≤ i ≤ 4} and is the polygon OP QR in Figure 4.1. For each choice
x, the President receives the payoff c
′
x. Therefore, the surface of constant payoff k say, is the hyperplane πk = {x|c
′
x = k}. These hyperplanes for different values of k are parallel to one another since they have the same normal
c. Furthermore, as k increases πk moves in the direction
c. Obviously we are assuming in this discussion that c 6= 0. Evidently an optimal decision is any point
x
∗
∈ Ω which lies on a hyperplane πk which is farthest along the direction c. We can rephrase this by saying that x∗ ∈ Ω is an optimal decision if and only if the plane π
∗
through x
∗
does not intersect the interior of Ω, and futhermore at x
∗
the direction c points away
from Ω. From this condition we can immediately draw two very important conclusions: i at least
one of the vertices of Ω is an optimal decision, and ii x
∗
yields a higher payoff than all points in the cone
K
∗
consisting of all rays starting at x
∗
and passing through Ω, since K
∗
lies “below” π
∗
. The first conclusion is the foundation of the powerful Simplex algorithm which we present in Section 3. Here we pursue consequences of the second conclusion. For the situation depicted in
Figure 4.1 we can see that x
∗
= Q is an optimal decision and the cone K
∗
is shown in Figure 4.2. Now
x
∗
satisfies A
x
x
∗
= b
1
, A
2
x
∗
= b
2
, and A
3
x
∗
b
3
, A
4
x
∗
b
4
, so that K
∗
is given by K
∗
= {x
∗
+ h|A
1
h ≤ 0 , A
2
h ≤ 0} . Since
c
′
x
∗
≥ c
′
y for all y ∈ K
∗
we conclude that c
′
h ≤ 0 for all h such that A
1
h ≤ 0, A
2
h ≤ 0 . 4.3
We pause to formulate the generalization of 4.3 as an exercise.
1
Recall the notation introduced in 1.1.2, so that x ≤ y means x
i
≤ y
i
for all i.
4.1.
THE LINEAR PROGRAMMING PROBLEM 29
,
- -
- -
- -
- -
- -
x
2
πk = {x|c
′
x = k} π
∗
Q = x
∗
direction of increasing
payoff k
{x|A
2
x = b
2
} x
1
{x|A
1
x = b
1
} R
A
4
O A
3
A
1
⊥ QR c ⊥ π
∗
A
2
⊥ P Q P
Figure 4.1: Ω = OP QR.
Exercise 1:
Let A
i
, 1 ≤ i ≤ k, be n-dimensional row vectors. Let c ∈ R
n
, and let b
i
, 1 ≤ i ≤ k, be real numbers. Consider the problem
Maximize c
′
x subject to
A
i
x ≤ b
i
, 1 ≤ i ≤ k . For any
x satisfying the constraints, let Ix ⊂ {1, . . . , n} be such that A
i
x = b
i
, i ∈ Ix, A
i
x b
i
, i ∈ Ix. Suppose x
∗
satisfies the constraints. Show that x
∗
is optimal if an only if c
′
h ≤ 0 for all h such that A
i
h ≤ 0 , i ∈ Ix
∗
. Returning to our problem, it is clear that 4.3 is satisfied as long as
c lies between A
1
and A
2
. Mathematically this means that 4.3 is satisfied if and only if there exist
λ
∗ 1
≥ 0, λ
∗ 2
≥ 0 such that
2
c
′
= λ
∗ 1
, A
1
+ λ
∗ 2
A
2
. 4.4
As c varies, the optimal decision will change. We can see from our analysis that the situation is as
follows see Figure 4.1:
2
Although this statement is intuitively obvious, its generalization to n dimensions is a deep theorem known as Farkas’
lemma see Section 2.
30
CHAPTER 4. LINEAR PROGRAMMING
P x
∗
= Q K
∗
π
∗
R
A
4
O A
3
A
2
c A
1
Figure 4.2: K
∗
is the cone generated by Ω at x
∗
. 1.
x
∗
= Q is optimal iff c lies between A
1
and A
2
iff c
′
= λ
∗ 1
A
1
+ λ
∗ 2
A
2
for some λ
∗ 1
≥ 0, λ
∗ 2
≥ 0,
2. x
∗
∈ QP is optimal iff c lies along A
2
iff c
′
= λ
∗ 2
A
2
for some λ
∗ 2
≥ 0, 3.
x
∗
= P is optimal iff c lies between A
3
and A
2
iff c
′
= λ
∗ 2
A
2
+ λ
∗ 3
A
3
for some λ
∗ 2
≥ 0, λ
∗ 3
≥ 0, etc.
These statements can be made in a more elegant way as follows: x
∗
∈ Ω is optimal iff there exists λ
∗ i
≥ 0 , 1 ≤ i ≤ 4, such that a
c
′
=
4
X
i=1
λ
∗ i
a
i
, b if A
i
x
∗
b
i
then λ
∗ i
= 0 . 4.5
For purposes of application it is useful to separate those constraints which are of the form x
i
≥ 0, from the rest, and to reformulate 4.5 accordingly We leave this as an exercise.
Exercise 2: Show that 4.5 is equivalent to 4.6, below. Here
A
i
= a
i1
, a
i2
. x
∗
∈ Ω is optimal iff there exist
λ
∗ 1
≥ 0 , λ
∗ 2
≥ 0 such that a c
i
≤ λ
∗ 1
a
1i
+ λ
∗ 2
a
2i
, i = 1, 2, b if a
j1
x
∗ 1
+ a
j2
x
∗ 2
b
j
then x
∗ j
= 0, j = 1, 2. c if c
i
λ
∗ 1i
+ λ
∗ 2
a
2i
then x
∗ i
= 0, i = 1, 2. 4.6
4.1.
THE LINEAR PROGRAMMING PROBLEM 31
4.1.2 Problem formulation.
