Name:
Extreme points, basic feasible solutions, linear programming
SA305, Spring 2013
1. Consider the set S = {x ∈ R : 1 ≤ x ≤ 5}. Show that S is convex.
P
>
2. Consider the set S = x ∈ R5 :
=
i∈{1,...,5} xi = 1, xi ≥ 0, i ∈ {1, . . . , 5} . Show that the point y
(1/5, 1/2, 0, 0, 3/10) can be expressed as a convex combination of two different vectors in S.
3. Note how many points in a convex set can be expressed as a convex combination of points in the convex set.
However, there are points that cannot be expressed as convex combinations. We refer to these points as extreme
points of a set. Note that, geometrically, extreme points of a set are not on any line segment drawn between
any two different points in the set. However, not all sets have extreme points. For the following figures, circle
the extreme points in each set, if there are any. Also, indicate how many extreme points each set has.
y
y
x
x
(a)
# of ext. pts:
(b)
# of ext. pts:
y
y
x
(c)
# of ext. pts:
x
(d)
# of ext. pts:
4. It turns out that extreme points and basic feasible solutions are identical for linear programming feasible regions.
To develop some intuition for this, calculate the basic feasible solutions to the following sets. Then, plot each of
the sets and label the basic feasible solutions on the axes given.
S = {(x, y) : x = y, 1 ≤ x ≤ 2}
S = {(x, y) : 1 ≤ x ≤ 3, 1 ≤ y ≤ 3, x + y ≤ 4}
y
y
5
5
4
4
3
3
2
2
1
1
x
1
2
3
4
5
x
1
2
3
4
5
5. The fundamental theorem of linear programming can be stated as follows: If a linear program is over nonnegative
variables, then exactly one of the following three statements is true: (1) The linear program has an optimal
solution that is an extreme point (i.e., basic feasible solution). (2) The linear program is unbounded. (3) The
linear program is infeasible. To develop intuition for why this theorem is true, consider the following linear
program:
max x + y
(P) s.t.
1≤x≤3
1≤y≤3
x + y ≤ 4.
Note that the optimal objective value is 4. Now, consider the following set:
S = {(x, y) : 1 ≤ x ≤ 3, 1 ≤ y ≤ 3, x + y = 4}
Graph the feasible region of the linear program (P) and S using the axes given:
y
5
4
3
2
1
x
1
2
3
4
5
Circle the extreme points of S.
In a few sentences explain why the basic feasible solutions of S are also basic feasible solutions to the feasible
region of (P).
Bonus: Explain why nonnegativity implies that the feasible region of a linear program has to have extreme points.