Linear Programming Lecture Notes - Penn State Personal Web Server

x
Goat Pen
Figure 1.1. Goat pen with unknown side lengths. The objective is to identify the
values of x and y that maximize the area of the pen (and thus the number of goats
that can be kept).
Thus, x = 25 and y = 50 − x = 25. We further recall from basic calculus how to confirm
that this is a maximum; note:
(1.4) = −2 < 0
d 2 A
dx 2
x=25
Which implies that x = 25 is a local maximum for this function. Another way of seeing this
is to note that A(x) = 50x − x 2 is an “upside-down” parabola. As we could have guessed, a
square will maximize the area available for holding goats.
Exercise 1. A canning company is producing canned corn for the holidays. They
have determined that each family prefers to purchase their corn in units of 12 fluid ounces.
Assuming that metal costs 1 cent per square inch and 1 fluid ounce is about 1.8 cubic inches,
compute the ideal height and radius for a can of corn assuming that cost is to be minimized.
[Hint: Suppose that our can has radius r and height h. The formula for the surface area of
a can is 2πrh + 2πr 2 . Since metal is priced by the square inch, the cost is a function of the
surface area. The volume of the can is πr 2 h and is constrained. Use the same trick we did
in the example to find the values of r and h that minimize cost.
1. A General Maximization Formulation
Let’s take a more general look at the goat pen example. The area function is a mapping
from R 2 to R, written A : R 2 → R. The domain of A is the two dimensional space R 2 and
its range is R.
Our objective in Example 1.1 is to maximize the function A by choosing values for x and
y. In optimization theory, the function we are trying to maximize (or minimize) is called the
objective function. In general, an objective function is a mapping z : D ⊆ R n → R. Here D
is the domain of the function z.
Definition 1.2. Let z : D ⊆ R n → R. The point x ∗ is a global maximum for z if for all
x ∈ D, z(x ∗ ) ≥ z(x). A point x ∗ ∈ D is a local maximum for z if there is a neighborhood
S ⊆ D of x ∗ (i.e., x ∗ ∈ S) so that for all x ∈ S, z(x ∗ ) ≥ z(x).
Remark 1.3. Clearly Definition 1.2 is valid only for domains and functions where the
concept of a neighborhood is defined and understood. In general, S must be a topologically
2
y