Linear Programming Lecture Notes - Penn State Personal Web Server

draw parallel lines in the direction of the gradient (7, 6). At some point, these lines will fail
to intersect the feasible region. The last line to intersect the feasible region will do so at a
point that maximizes the profit. In this case, the point that maximizes z(x1, x2) = 7x1 +6x2,
subject to the constraints given, is (x ∗ 1, x ∗ 2) = (16, 72).
Note the point of optimality (x ∗ 1, x ∗ 2) = (16, 72) is at a corner of the feasible region. This
corner is formed by the intersection of the two lines: 3x1 + x2 = 120 and x1 + 2x2 = 160. In
this case, the constraints
3x1 + x2 ≤ 120
x1 + 2x2 ≤ 160
are both binding, while the other constraints are non-binding. In general, we will see that
when an optimal solution to a linear programming problem exists, it will always be at the
intersection of several binding constraints; that is, it will occur at a corner of a higherdimensional
polyhedron.
3. Formalizing The Graphical Method
In order to formalize the method we’ve shown above, we will require a few new definitions.
Definition 2.5. Let r ∈ R, r ≥ 0 be a non-negative scalar and let x0 ∈ R n be a point
in R n . Then the set:
(2.9) Br(x0) = {x ∈ R n | ||x − x0|| ≤ r}
is called the closed ball of radius r centered at point x0 in R n .
In R 2 , a closed ball is just a disk and its circular boundary centered at x0 with radius r.
In R 3 , a closed ball is a solid sphere and its spherical centered at x0 with radius r. Beyond
three dimensions, it becomes difficult to visualize what a closed ball looks like.
We can use a closed ball to define the notion of boundedness of a feasible region:
Definition 2.6. Let S ⊆ R n . Then the set S is bounded if there exists an x0 ∈ R n and
finite r ≥ 0 such that S is totally contained in Br(x0); that is, S ⊂ Br(x0).
Definition 2.6 is illustrated in Figure 2.2. The set S is shown in blue while the ball of
radius r centered at x0 is shown in gray.
We can now define an algorithm for identifying the solution to a linear programing
problem in two variables with a bounded feasible region (see Algorithm 1):
The example linear programming problem presented in the previous section has a single
optimal solution. In general, the following outcomes can occur in solving a linear programming
problem:
(1) The linear programming problem has a unique solution. (We’ve already seen this.)
(2) There are infinitely many alternative optimal solutions.
(3) There is no solution and the problem’s objective function can grow to positive
infinity for maximization problems (or negative infinity for minimization problems).
(4) There is no solution to the problem at all.
Case 3 above can only occur when the feasible region is unbounded; that is, it cannot be
surrounded by a ball with finite radius. We will illustrate each of these possible outcomes in
the next four sections. We will prove that this is true in a later chapter.
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