1. BASIC FEASIBLE SOLUTIONS
CONTENTS
Polyhedral Sets and Polytopes
Basic Feasible Solutions and Polytopes
Reference: Chapters 2 and 3 in BJS book.

2. Hyperplanes
A hyperplane in En generalizes the notion of a straight line in E2 and the notion of a plane in E3.
A hyperplane H in En is a set of the form {x:px=k} where p is a nonzero vector in En and k is a scalar.
If x0  H, then px0 = k, and for any other point x  H, px = k.
Hence,
p(x – x0) = 0.
Both hyperplanes and halfspaces are convex sets. Why?

3. Rays
A ray is a collection of points of the form {x0 + d:   0}, where d is a nonzero vector. x0 d
A convex cone C is a convex set with the additional property that x  C for each x  C and for each   0.

4. Polyhedron
A hyperplane divides En into two regions, called halfspaces:
S1 = {x: px  k}
S2 = {x: px  k}
A polyhedral set or a polyhedron is the intersection of a finite number of halfspaces.
A bounded polyhedral set is called a polytope.

5. Polyhedral Sets Consider the polyhedral set defined by the following
inequalities:
-2x1 + x2  4
x1 + x2  3
x  2
x  0
x  0
A polyhedral set can be represented by the set {x: Ax  b}
where A is an mxn matrix.

6. Faces, Edges, Adjacent Extreme Points of a Polyhedra

7. An Important Theorem
Theorem: For any polytope S, every solution x can be represented by a convex combinations of the extreme points of the polytope.
Let x1, x2, x3, … , xk denote the extreme points of a polytope S. Then, for any x  S, there exist 1, 2, 3,.., k
x = 1x1 + 2x2 + 3x3 + …. + kxk, where

8. Another Important Theorem
Consider the following linear programming problem:
Minimize cx
subject to
Ax = b
x  0
Let x1, x2, x3, … , xk be the extreme points of the constraint set S. This linear program can be transformed into the following linear program:
Minimize
1, 2, 3, … , k  0
How to solve this linear program?

9. Another Important Theorem (contd.)
Theorem. If a linear programming problem has a bounded feasible solution space, then there must exist an optimal extreme point solution.
Is every optimal solution of a linear programming problem an extreme point solution?
It can be shown that extreme point solutions correspond to basic feasible solutions (BFS) which are defined algebraically.

10. Standard Form of a Linear Program
Condition 1: All right-hand side coefficients in the constraints are nonnegative.
Condition 2 : All constraints are in the equality form.
Condition 3 : All variables must take nonnegative values.
Maximize c1x1 + c2x2 + c3x3 + …. + cnxn
subject to
a11x1 + a12x2 + a13x3 + …. + a1nxn= b1
a21x1 + a22x2 + a23x3 + …. + a2nxn= b2 :
am1x1 + am2x2 + am3x3 + …. + amnxn = bm
x1, x2, x3 , …., xn  0

11. Restoring Condition 1
Condition 1: All right-hand side coefficients in the constraints are nonnegative.
Constraint:
2x1 - 3x2 + x3  - 10
Alternative constraint:
-2x1 + 3x2 - x3  10

12. Restoring Condition 2
Condition 2 : All constraints are in the equality form.
max z = 4x1+ 3x2 max z = 4x1+ 3x2
s.t. x1 + x2  x1 + x2 + s1 = 40
2x1+ x2  x1+ x2 + s2 = 60
x1, x2  0 x1, x2, s1, s2  0
s.t. x1 + x2  x1 + x2 - e1 = 40
2x1+ x2  x1+ x2 - e2 = 60
x1, x2  0 x1, x2, e1, e2  0
si = Slack variables
ei = Surplus variables

13. Restoring Condition 3
Condition 3 : All variables must take nonnegative values.
max z = 30x1 - 4x2 + 8x3
s.t.
5x1 - x2 + x3  30
x  5
x1  0, x2 unrestricted, x3  0
Variable transformation: x2 = , = - x3
max z = 30x1 - 4( ) - 8
5x  30
x  5
x1, , ,  0

14. Basic Feasible Solutions
Consider the linear program with the constraints:
Ax=b and x  0
Where A is an mxn matrix and b is an m vector. Let rank (A) = m.
Let A= [B,N]
where B is an m x m invertible matrix, and
N is an m x (n-m) matrix.
Then the solution of the system BxB = b is a basic solution of the linear program.

15. Basic Feasible Solutions (contd.)
Alternatively, the solution x = [xB, xN]
xB = B-1 b
xN= 0
is the basic solutions of the system.
If xB  0, then x is called a basic feasible solution (BFS) of the system.
A basic feasible solution x is called a degenerate basic feasible solution if some component of xB is zero.
B is called the basis matrix.
Variables in xB are called basic variables.
Variables in xN are called nonbasic variables.

16. Example 1 Consider a polyhedral set: Standard form:
x1 + x2  x1 + x2 + x = 6
x2  x x4 = 3
x1 , x2  x1 , x2 , x3 , x  0
Constraint Matrix:
A= [a1, a2, a3, a4] =

17. Example 1 (contd.)
The solution space:

18. Example 1 (contd.)
B = [a1,a2] =
xB =
xN =

19. Example 1 (contd.)
B = [a1,a4] =
xB =
xN =

20. Example 1 (contd.)
B = [a2,a3] =
xB =
xN =

21. Example 1 (contd.)
B = [a2,a4] =
xB =
xN =

22. Example 1 (contd.)
B = [a3,a4] =
xB =
xN =

23. Example 1 (contd.) Have we considered all the basic solutions?
How many basic solutions there are?
How many basic feasible solutions there are?

24. Example 1 (contd.)
Four Basic Feasible Solutions:

25. Basic Feasible Solutions
The possible number of basic feasible solutions is bounded by the number of ways of extracting m columns out of n columns.
The maximum number of BFS =

26. Example 2 Consider a polyhedral set: Standard form:
x1 + x2  x1 + x2 + x = 6
x2  x x = 3
x1 + 2x2  x1 + 2x x = 9
x1 , x2  x1 , x2 , x3 , x4, x5  0
Constraint Matrix:
A= [a1, a2, a3, a4] =

27. Example 2 (contd.)
Feasible Solution Space:

28. Example 2 (contd.) B = [a1, a2, a3]; xB = xN = B = [a1, a2, a4]; xB =
Observe that several basis correspond to the same basic feasible solution. Also observe that the basic feasible solution must be degenerate for it to happen.

29. Summary: Basic and Basic Feasible Solutions
Identify m linearly independent (basic) variables (BV).
Set the remaining n - m (nonbasic) variables to zero value (NBV).
Solve the equations Ax = b in terms of the basic variables. (Observe that there is a unique solution.)
The resulting solution is a basic solution (BS).
A basic solution is a basic feasible solution (BFS) if xj  0 for each j = 1, 2, … , N.
There are at most n!/(n-m)!m! BFS. Hence there are finite number of BFS.

30. Adjacent Basic Feasible Solutions
For any LP with m constraints, two basic feasible solutions are said to be adjacent if their sets of basic variables have m-1 basic variables in common.
Basic Feasible Solution: (x1, x2, x3, x4)
Adjacent Basic
Feasible Solutions:
(x1, x2, x3, x5)
(x1, x2, x6, x4)
(x1, x7, x3, x4)
(x8, x2, x3, x4)