A certain subset of the basic solutions to the constraints Ax b of an LP plays an im-portant role in the theory of linear programming.
D E F I N I T I O N ■ Any basic solution to (7) in which all variables are nonnegative is a basic feasible solution (or bfs). ■
Thus, for an LP with the constraints given by (8), the basic solutions x1 2, x2 1, x3 0, and x1 0, x2 3, x3 2 are basic feasible solutions, but the basic solution x1 3, x2 0, x3 1 fails to be a basic solution (because x3 0).
In the rest of this section, we assume that all LPs are in standard form. Recall from Section 3.2 that the feasible region for any LP is a convex set. Let S be the feasible re-gion for an LP in standard form. Recall that a point P is an extreme point of S if all line segments that contain P and are completely contained in S have P as an endpoint. It turns out that the extreme points of an LP’s feasible region and the LP’s basic feasible solutions are actually one and the same. More formally,
A point in the feasible region of an LP is an extreme point if and only if it is a ba-sic feasible solution to the LP.
See Luenburger (1984) for a proof of Theorem 1.
To illustrate the correspondence between extreme points and basic feasible solutions outlined in Theorem 1, let’s look at the Leather Limited example of Section 4.1. Recall that the LP was
max z 4x1 3x2
s.t. x1 x2 40 (LP 1)
s.t. 2x1 x2 60 (1)
x1, x2 0 (2)
By adding slack variables s1and s2, respectively, to (1) and (2), we obtain LP 1 in stan-dard form:
max z 4x1 3x2
s.t. x1 x2 s1 s2 40
(LP 1) s.t. 2x1 x2 s1 s2 60
x1, x2, s1, s2 0
The feasible region for the Leather Limited problem is graphed in Figure 1. Both in-equalities are satisfied: (1) by all points below or on the line AB(x1 x2 40), and (2) by all points on or below the line CD(2x1 x2 60). Thus, the feasible region for LP 1 is the shaded region bounded by the quadrilateral BECF. The extreme points of the feasible region are B (0, 40), C (30, 0), E (20, 20), and F (0, 0).
T H E O R E M 1
Table 1 shows the correspondence between the basic feasible solutions to LP 1 and the extreme points of the feasible region for LP 1. This example should make it clear that the basic feasible solutions to the standard form of an LP correspond in a natural fashion to the LP’s extreme points.
In the context of the Leather Limited example, it is easy to show why any bfs is an ex-treme point. The converse is harder! We now show that for the LL problem, any bfs is an extreme point. Any point in the feasible region for LL may be specified as a four-dimensional column vector with the four elements of the vector denoting x1, x2, s1, and s2, respectively. Consider the bfs B with BV {x2, s2}. If B is not an extreme point, then there exists two distinct feasible points v1and v2and non-negative numbers 1and 2 sat-isfying 0 i 1 and 1 2 1 such that
1v1 2v2
Clearly, both v1and v2must both have x1 s2 0. But because v1and v2are both feasible, the values of x2and s2for both v1and v2can be determined by solving x2 40 and x2 s2 60. These equations have a unique solution (because columns corre-sponding to basic variables x2and s2are linearly independent). This shows that v1 v2, so B is indeed an extreme point.
0 40 0 20
E
F D
B
C A
x2
x1 60
50
10 20 30 40 50 60
40
30
20
F I G U R E 1 10
Feasible Region for Leather Limited
T A B L E 1
Correspondence between Basic Feasible Solutions and Corner Points for Leather Limited
Basic Nonbasic Basic Corresponds to
Variables Variables Feasible Solution Corner Point
x1, x2 s1, s2 s1 s2 0, x1 x2 20 E x1, s1 x2, s2 x2 s2 0, x1 30, s1 10 C
x1, s2 x2, s1 x2 s1 0, x1 40, s2 20 Not a bfs because s2 0 x2, s1 x1, s2 x1 s2 0, s1 20, x2 60 Not a bfs because s1 0 x2, s2 x1, s1 x1 s1 0, x2 40, s2 20 B
s1, s2 x1, x2 x1 x2 0, s1 40, s2 60 F
We note that more than one set of basic variables may correspond to a given extreme point. If this is the case, then we say the LP is degenerate. See Section 4.11 for a dis-cussion of the impact of degeneracy on the simplex algorithm.
We will soon see that if an LP has an optimal solution, then it has a bfs that is optimal. This is important because any LP has only a finite number of bfs’s. Thus we can find the optimal solution to an LP by searching only a finite number of points.
Because the feasible region for any LP contains an infinite number of points, this helps us a lot!
Before explaining why any LP that has an optimal solution has an optimal bfs, we need to define the concept of a direction of unboundedness.
4.3 Direction of Unboundedness
Consider an LP in standard form with feasible region S and constraints Ax b and x 0. Assuming that our LP has n variables, 0 represents an n-dimensional column vector consisting of all 0’s. A nonzero vector d is a direction of unboundedness if for all xS and any c 0, x cdS. In short, if we are in the LP’s feasible region, then we can move as far as we want in the direction d and remain in the feasible region. Figure 2 dis-plays the feasible region for the Dorian Auto example (Example 2 of Chapter 3). In stan-dard form, the Dorian example is
min z 50x1 100x2
7x1 2x2 e1 28 2x1 12x2 e2 24 x1, x2, e1, e2 0
Looking at Figure 2 it is clear that if we start at any feasible point and move up and to the right at a 45-degree angle, we will remain in the feasible region. This means that
z = 600 z = 320
(10)
(4, 4) B
E A C D x2
x1 4
2
(11) 6
8 10 12 14
2 4 6 8 10 12 14
F I G U R E 2 Graphical Solution of Dorian Problem
d
is a direction of unboundedness for this LP. It is easy to show (see Problem 6) that d is a direction of unboundedness if and only if Ad 0 and d 0.
The following Representation Theorem [for a proof, see Nash and Sofer (1996)] is the key insight needed to show why any LP with an optimal solution has an optimal bfs.
Consider an LP in standard form, having bfs b1, b2, . . . , bk. Any point x in the LP’s feasible region may be written in the form
x d
iki1ibi
where d is 0 or a direction of unboundedness and iki1i 1 and i 0.
If the LP’s feasible region is bounded, then d 0, and we may write x iki1ibi, where the iare nonnegative weights adding to 1. In this case, we see that any feasible x may be written as a convex combination of the LP’s bfs. We now give two illustrations of Theorem 2.
Consider the Leather Limited example. The feasible region is bounded. To illustrate Theorem 2, we can write the point G (20, 10) (G is not a bfs!) in Figure 3 as a convex combination of the LP’s bfs. Note from Figure 3 that point G may be written as
1
6 F 56 H [here H (24, 12)]. Then note that point H may be written as .6E .4C. Putting these two relationships together, we may write point G as 1
6 F 56 (.6E .4C) 16 F
1
2 E 13 C. This expresses point G as a convex combination of the LP’s extreme points.
To illustrate Theorem 2 for an unbounded LP, let’s consider Example 2 of Chapter 3 (the Dorian example; see Figure 4) and try to express the point F (14, 4) in the repre-sentation given in Theorem 2. Recall that in standard form the constraints for the Dorian example are given by
7x1 2x2 e1 28 1 1 9 14
T H E O R E M 2
E
H
F D
B
C A
x1 60
50
10 20 30 40 50 60
40
30
20
10 G
F I G U R E 3 Writing (20, 10) as a Convex Combination
of bfs
2x1 12x2 e2 24
From Figure 4, we see that to move from bfs C to point F we need to move up and to the right along a line having slope 1
4 4
0
12 2. This line corresponds to the direction of unboundedness
d
Letting
b1
and x
we may write x d b1, which is the desired representation.
4.4 Why Does an LP Have an Optimal bfs?
Consider an LP with objective function max cx and constraints Ax b. Suppose this LP
has an optimal solution. We now sketch a proof of the fact that the LP has an optimal bfs.
If an LP has an optimal solution, then it has an optimal bfs.
Proof Let x be an optimal solution to our LP. Because x is feasible, Theorem 2 tells us that we may write x d iki1ibi, where d is 0 or a direction of unbound-edness and b1, b2, . . . , bkare the LP’s bfs. Also, iki1i 1 and i 0. If cd
14 4 78 52 12
0 56 0
2 4 22 52
T H E O R E M 3
z = 600 z = 320
(10)
(4, 4) B
E
C F
A D x2
x1 4
2
(11) 6
8 10 12 14
2 4 6 8 10 12 14
F I G U R E 4 Expressing F (14, 4) Using Theorem 2
0, then for any k 0, kd iik1ibiis feasible, and as k grows larger and larger, the objective function value approaches infinity. This contradicts the fact that the LP has an optimal solution. If cd 0, then the feasible point iik1ibihas a larger ob-jective function value than x. This contradicts the optimality of x. In short, we have shown that if x is optimal, then cd 0. Now the objective function value for x is given by
cx cd iik1icbi iik1icbi
Suppose that b1is the bfs with the largest objective function value. Because iik1
i 1 and i 0,
cb1 cx
Because x is optimal, this shows that b1is also optimal, and the LP does indeed have an optimal bfs.