Linear Programming Lecture Notes - Penn State Personal Web Server

connected set (as it is in a neighborhood in R n ) in order for this definition to be used or at
least we must be able to define the concept of neighborhood on the set 1 .
Exercise 2. Using analogous reasoning write a definition for a global and local minimum.
[Hint: Think about what a minimum means and find the correct direction for the ≥ sign in
the definition above.]
In Example 1.1, we are constrained in our choice of x and y by the fact that 2x+2y = 100.
This is called a constraint of the optimization problem. More specifically, it’s called an
equality constraint. If we did not need to use all the fencing, then we could write the
constraint as 2x+2y ≤ 100, which is called an inequality constraint. In complex optimization
problems, we can have many constraints. The set of all points in R n for which the constraints
are true is called the feasible set (or feasible region). Our problem is to decide the best values
of x and y to maximize the area A(x, y). The variables x and y are called decision variables.
Let z : D ⊆ R n → R; for i = 1, . . . , m, gi : D ⊆ R n → R; and for j = 1, . . . , l
hj : D ⊆ R n → R be functions. Then the general maximization problem with objective
function z(x1, . . . , xn) and inequality constraints gi(x1, . . . , xn) ≤ bi (i = 1, . . . , m) and
equality constraints hj(x1, . . . , xn) = rj is written as:
⎧
max z(x1, . . . , xn)
(1.5)
⎪⎨
⎪⎩
s.t. g1(x1, . . . , xn) ≤ b1
.
gm(x1, . . . , xn) ≤ bm
h1(x1, . . . , xn) = r1
.
hl(x1, . . . , xn) = rl
Expression 1.5 is also called a mathematical programming problem. Naturally when constraints
are involved we define the global and local maxima for the objective function
z(x1, . . . , xn) in terms of the feasible region instead of the entire domain of z, since we
are only concerned with values of x1, . . . , xn that satisfy our constraints.
Example 1.4 (Continuation of Example 1.1). We can re-write the problem in Example
1.1:
⎧
max A(x, y) = xy
⎪⎨ s.t. 2x + 2y = 100
(1.6)
x ≥ 0
⎪⎩
y ≥ 0
Note we’ve added two inequality constraints x ≥ 0 and y ≥ 0 because it doesn’t really make
any sense to have negative lengths. We can re-write these constraints as −x ≤ 0 and −y ≤ 0
where g1(x, y) = −x and g2(x, y) = −y to make Expression 1.6 look like Expression 1.5.
1 Thanks to Bob Pakzad-Hurson who suggested this remark for versions after 1.1.
3