Math 438 ACTIVITY 8:
WHY:
Direction Vectors and the Finite Basis Theorem
This activity is designed to help you understand the Finite Basis Theorem better. This theorem (which allows
us to represent all of the feasible region for an LP problem in terms of the extreme points and the direction
vectors) is the key to proving the Extreme Point Theorem –which will allow us to search for max/min values by
looking at a finite collection of values [sort of like the critical points & end points theorem for max/min on a
closed interval, in Calc I]. This is the big deal of Linear Programming – from this we will develop an algorithm
for checking the values (and determining if there actually is an optimal solution).
VOCABULARY:
Extreme Point
An Extreme Point of a polyhedron (we care because LP feasible regions will be polyhedrons) is a point of the
polyhedron which is not be in the interior of a line segment joining any two point in the polyhedron [This is the technical
definition that makes it a "corner point" - an extreme point is not between any two points of the set].
Direction Vector
A Direction Vector for a polyhedron is a vector in the direction of one of the bounding rays of the polyhedron.
A polytope (a bounded feasible region) does not have any direction vectors. Two points on a bounding ray are sufficient
to determine the line containing the ray - the direction comes out of the other constraints (increasing values of one or
more variables).
Finite Basis Theorem
Any convex polyhedron can be written as the (algebraic) sum of 1. the convex hull of its extreme points and 2. the
conical hull of its direction vectors.
That is, every point in the polyhedron can be written in the form X + Y for X in the convex hull of the extreme points
and Y in the conical hull of the direction vectors.
LEARNING OBJECTIVES:
1.
2.
3.
Understand the notions of extreme points and bounding rays for convex polyhedra.
Discover the geometric and algebraic meanings of the Finite Basis Theorem.
Understand the role of visualization in learning.
CRITERIA:
1.
2.
Level of ability to explain and apply the Finite Basis Theorem.
Quality of the answers to the Critical Thinking Questions.
RESOURCES:
1.
2.
3.
Section 2.2: Strategic Mathematics.
20 minutes
Maple 12, running on the campus network and the worksheet Public\Courses\Math\Math438\Activitym8.mw
PLAN:
1.
**Plot the solution region, extreme points, and direction vectors for the system of inequalities given in Model 1 by
executing the Maple worksheet Activitym8.mw located at Public\Courses\Math\Math438- instructor will do this
demonstration
2.
Work through the model - have the Recorder write down what happened and what discoveries the team made as each
step was executed.
3.
Represent the general solution (point in the feasible region) of this problem in terms of the extreme points and direction
vectors.
4.
Answer the critical thinking questions .
MODEL:
Plot the graph of the feasible region determined by the following constraints. Find and plot the direction
vectors of the bounding rays and the extreme points for this region. Write out the finite basis theorem using the
particular points and vectors in this problem[“Every point in the feasible region can be written . . .”].
3x - y ≥ 2
x + 2y ≥ 5
x ≥ 3, y ≥ 0
CRITICAL THINKING QUESTIONS:
1.
What does it mean to add the two sets of vectors [ the convex hull of the extreme points and the conical hull of the
direction vectors]?
2.
Is the feasible region in your graph a polyhedron? A polytope? A simplex? A convex hull? Describe or sketch on
paper the conical hull of the feasible region. (Note: this is different from the conical hull of the direction vectors).
What is its affine hull?
3.
Call the extreme points (in the order they appear on the worksheet) X 1, X2, X3 and the direction vectors (in order) Y1,
Y2 .
Give each of these points and determine whether it is in the feasible region:
a. .5X1 + .25X2 + .25X3 + 3Y1
b. .5 X1 + .25X2 + .25X3 + .5Y1+ Y2
c.
X 1 - X3 + Y 2
4.
Can a point in the feasible region be represented in more than one way? That is, if P is in the feasible region, could
there be more than one set of numbers a, b, c, d, e for which P = aX 1 +bX2 + cX3 + dY1+eY2 ?