8
CHAPTER 2. OPTIMIZATION OVER AN OPEN SET 2.1.4
If f : R
n
→ R
m
is a function, its ith component is written f
i
, i = 1, . . . , m. Note that f
i
: R
n
→ R. Sometimes we describe a function by specifying a rule to calculate
f x for every x. In this case we write
f : x 7→ f x. For example, if A is an m × n matrix, we can write F : x 7→ Ax to denote the function
f : R
n
→ R
m
whose value at a point x ∈ R
n
is Ax.
2.1.5
If f : R
n
7→ R is a differentiable function, the derivative of f at ˆx is the row vector ∂f ∂x
1
ˆ x, . . . , ∂f ∂x
n
ˆ x.
This derivative is denoted by ∂f ∂xˆ
x or f
x
ˆ x or ∂f ∂x|
x=ˆ x
or f
x
|
x=ˆ x
, and if the argument ˆ
x is clear from the context it may be dropped. The column vector
f
x
ˆ x
′
is also denoted ∇
x
f ˆ x,
and is called the gradient of f at ˆ
x. If f : x, y 7→ f x, y is a differentiable function from R
n
× R
m
into R, the partial derivative of f with respect to x at the point ˆ
x, ˆ y is the n-dimensional
row vector f
x
ˆ x, ˆ
y = ∂f ∂xˆ x, ˆ
y = ∂f ∂x
1
ˆ x, ˆ
y, . . . , ∂f ∂x
n
ˆ x, ˆ
y, and similarly f
y
ˆ x, ˆ
y = ∂f ∂yˆ x, ˆ
y = ∂f ∂y
1
ˆ x, ˆ
y, . . . , ∂f ∂y
m
ˆ x, ˆ
y. Finally, if f : R
n
→ R
m
is a differentiable function with components
f
1
, . . . , f
m
, then its derivative at ˆ
x is the m × n matrix
∂f ∂x
ˆ x = f
x
ˆ x =
f
1x
ˆ x
.. .
f
mx
ˆ x
=
∂f
1
∂x
1
ˆ x
.. .
∂f
m
∂x
1
ˆ x
. . . . . .
∂f
1
∂x
n
ˆ x
.. .
∂f
m
∂x
n
ˆ x
2.1.6
If f : R
n
→ R is twice differentiable, its second derivative at ˆx is the n×n matrix ∂
2
f ∂x∂xˆ x =
f
xx
ˆ x where f
xx
ˆ x
j i
= ∂
2
f ∂x
j
∂x
i
ˆ x. Thus, in terms of the notation in Section 2.1.5 above,
f
xx
ˆ x = ∂∂xf
x ′
ˆ x.
2.2 Example
We consider in detail the first example of Chapter 1. Define the following variables and functions: α = fraction of mary-john in proposed mixture,
p = sale price per pound of mixture, v = total amount of mixture produced,
f α, p = expected sales volume as determined by market research of mixture as a function ofα, p.
2.2.
EXAMPLE 9
Since it is not profitable to produce more than can be sold we must have: v = f α, p,
m = amount in pounds of mary-john purchased, and t = amount in pounds of tobacco purchased.
Evidently, m = αv, and
t = l − αv. Let
P
1
m = purchase price of m pounds of mary-john, and P
2
= purchase price per pound of tobacco. Then the total cost as a function of
α, p is Cα, p = P
1
αf α, p + P
2
1 − αf α, p. The revenue is
Rα, p = pf α, p, so that the net profit is
N α, p = Rα, p − Cα, p. The set of admissible decisions is
Ω, where Ω = {α, p|0 α
1 2
, 0 p ∞}. Formally, we have the the following decision problem:
Maximize subject to
N α, p, α, p ∈ Ω.
Suppose that α
∗
, p∗ is an optimal decision, i.e., α
∗
, p
∗
∈ Ω N α
∗
, p
∗
≥ Nα, p and
for all α, p ∈ Ω.
2.1 We are going to establish some properties of
a
∗
, p
∗
. First of all we note that Ω is an open subset of
R
2
. Hence there exits ε 0 such that
α, p ∈ Ω whenever |α, p − α
∗
, p
∗
| ε 2.2
In turn 2.2 implies that for every vector h = h
1
, h
2 ′
in R
2
there exists η 0 η of course
depends on h such that
α
∗
, p
∗
+ δh
1
, h
2
∈ Ω for 0 ≤ δ ≤ η 2.3
10
CHAPTER 2. OPTIMIZATION OVER AN OPEN SET
|
.
α
∗
, p
∗
+ δh
1
, h
2
ǫ α
1 2
Ω δh
h p
a
∗
, p
∗
Figure 2.1: Admissable set of example. Combining 2.3 with 2.1 we obtain 2.4:
N α
∗
, p
∗
≥ Nα
∗
+ δh
1
, p
∗
+ δh
2
for 0 ≤ δ ≤ η 2.4
Now we assume that the function N is differentiable so that by Taylor’s theorem
N α
∗
+ δh
1
, p
∗
+ δh
2
= N α
∗
, p
∗
+δ[
∂N ∂α
δ
∗
, p
∗
h
1
+
∂N ∂p
α
∗
, p
∗
h
2
] +oδ,
2.5 where
oδ δ
→ 0 as δ → 0. 2.6
Substitution of 2.5 into 2.4 yields 0 ≥ δ[
∂N ∂α
α
∗
, p
∗
h
1
+
∂N ∂p
α
∗
, p
∗
h
2
] + oδ. Dividing by
δ 0 gives 0 ≥ [
∂N ∂α
α
∗
, p
∗
h
1
+
∂N ∂p
α
∗
, p
∗
h
2
] +
oδ δ
. 2.7
Letting δ approach zero in 2.7, and using 2.6 we get
0 ≥ [
∂N ∂α
α
∗
, p
∗
h
1
+
∂N ∂p
α
∗
, p
∗
h
2
]. 2.8
Thus, using the facts that N is differentiable, α
∗
, p
∗
is optimal, and δ is open, we have concluded that the inequality 2.9 holds for every vector
h ∈ R
2
. Clearly this is possible only if
∂N ∂α
α
∗
, p
∗
= 0,
∂N ∂p
α
∗
, p
∗
= 0. 2.9
Before evaluating the usefulness of property 2.8, let us prove a direct generalization.
2.3.
THE MAIN RESULT AND ITS CONSEQUENCES 11
2.3 The Main Result and its Consequences
