03 geometry 2011
TRANSCRIPT
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15.053 February 8, 2011
The Geometry of Linear Programs
the geometry of LPs illustrated
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Quotes of the day
You don't understand anything until youlearn it more than one way.
Marvin Minsky
One finds limits by pushingthem.
Herbert Simon
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Overview
Views of linear programming Geometry
Algebra
Economic interpretations
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What does the feasible region of an LP
look like?
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Three 2-dimensional examples
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Some 3-dimensional LPs
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mathworld.wolfram.com/ ConvexPolyhedron.html
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Goal of this Lecture: understand the geometrical
nature of 2 and 3 dimensional LPs
What properties does the feasible region have?
What properties does an optimal solution have?
How can one find the optimal solution:
the geometric method
The simplex method
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A Two Variable Linear Program
(a variant of the DTC example)
x, y 0
2x + 3y 10
x + 2y 6x + y 5
y 3x 4
z = 3x + 5yobjective
(1)
(2)
(3)
(4)
(5)
(6)
Co
nstra
ints
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Finding an optimal solution
Introduce yourself to your partner
Try to find an optimal solution to the linear
program, without looking ahead.
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Inequalities
x
yAn inequality with two variables
determines a unique half-plane
x + 2y 6
1 2 3 4 5 6
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2
3
4
5
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101 2 3 4 5 6
1
2
3
4
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Graph the Constraints:2x+ 3y 10 (1)
x 0 , y 0. (6)
x
y
2x + 3y = 10
Graphing the Feasible Region
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111 2 3 4 5 6
1
2
3
4
5
Add the Constraint:
x + 2y 6 (2)
x
y
x + 2y = 6
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131 2 3 4 5 6
1
2
3
4
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Add the Constraints:
x 4; y 3
x
y
We have now
graphed the
feasible
region.
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x
y
1 2 3 4
1
2
3
Find the maximum value p such that there is a
feasible solution with 3x + 5y = p.
Move the line with profit p parallel as much as
possible.
3x + 5y = 8
3x + 5y = 11
3x + 5y = 16The optimal
solution
occurs at a
corner point.
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What types of Linear Programs are there?
There is a feasible
solution and an
optimal solution.
There is a feasiblesolution and the
objective value is
unbounded from
above.
There is no feasiblesolution.
max xs.t. x + 2y -1
x 0, y 0
max xs.t. x + 2y 1
x 0, y 0
max x
s.t. x - y = 1
x 0, y 0
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Any other types
Is it possible to have an LP such that the feasibleregion is bounded, and such that there is no
optimal solution?
No. But it could happen if we
permitted strict inequality
constraints.
Maximize x
subject to 0 < x < 1
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Corner Points
A corner point(also called an extreme point) of the feasible region is apoint that is not the midpoint of two other points of the feasible region.
(They are only defined for convex sets, to be described later.)
Where are the
corner points of
this feasibleregion?
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Some examples of LP feasible regions.
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Region 1.
No corner point. Unbounded feasible region
1 2 3 4 5 6
1
2
3
4
5
Region 2.
Two corner points. Unbounded feasible region
Region 3.
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Facts about corner points.
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If a feasible region is non-empty, and if it does not
contain a line, then it has at least one corner point.
If every variable is non-negative, and if the feasible
region is non-empty, then there is a corner point.
In two dimensions, a corner point is at the intersection of
two equality constraints.
Region 1.
No corner point. Unbounded feasible region
Region 2.
Two corner points. Unbounded feasible region
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Optimality at corner points
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If a feasible region has a corner point, and if it has anoptimal solution, then there is an optimal solution that
is a corner point.
Region 2
Region 3 Example 1: minimize x
Example 2: minimize y
x
y
S LP h f ibl l ti
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Suppose an LP has a feasible solution.
Which of the following is not
possible?
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1. The LP has no corner point.
2. The LP has a corner point that is
optimal.
3. The LP has a corner point, but
there is no optimal solution.
4. The LP has a corner point and anoptimal solution, but no corner
point is optimal.
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Mental Break
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15.053 Trivia
Trivia about US
Presidents
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Towards the simplex algorithm
More geometrical notions edges and rays
Then the simplex algorithm
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Edges of the feasible region
In two dimensions, an edge of the feasible region is one of
the line segments making up the boundary of the feasibleregion. The endpoints of an edge are corner points.
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Anedge
An edge
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Edges of the feasible region
In three dimensions, an edge of the feasible region is one of the line segments
making up the framework of a polyhedron. The edges are where the facesintersect each other. A face is a flat region of the feasible region.
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A face
A face
An edge
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Extreme Rays
An extreme ray is like an edge, but it starts
at a corner point and goes on infinitely.
28x
y
1 2 3 4 5 6
1
2
3
4
5
Two extreme
rays.
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The Simplex Method
291 2 3 4 5 6
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2
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4
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x
y
Start at any feasible corner point.
Move along an edge (or extreme ray) in which theobjective value is continually improving. Stop at the next
corner point. (If moving along an extreme ray, the
objective value is unbounded.)
Continue until no adjacent corner point has a better
objective value. Max z = 3 x + 5 y
3 x + 5 y = 19
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The Simplex Method
Pentagonal prism
Note: in three dimensions, the
edges are the intersections of
two constraints. The corner pointsare the intersection of three
constraints.
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Solving for the Corner Point
x
y
1 2 3 4
1
2
3
In two dimensions, a corner point lies at theintersection of two lines.
2x +3y = 10
x +2y = 6
x + 2y = 6
2x + 3y = 10
x = 2
y = 2
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Each corner points of a 2-variable linear program is thesolution of two equations. Each
corner point of a 3-variablelinear program is the solutionof three equations.
Is each corner points of a 4-
variable linear program thesolution of four equations?
Yes. And eachcorner points ofan n-variable
linear programis the solutionsof n equations.
Cool !!
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341 2 3 4 5 6
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An important difference between the
geometry and the algebra
Usually, corner points can bedescribed in a unique way as an
intersection of two lines
(constraints). But not always.
2x + y = 9
x + 2y = 9
2x + y 9
x + 2y 9
0 x 3
0 y 3The point (3, 3) can be
written as the
intersection of two lines
in 6 ways.
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Degeneracy
When a corner point is the solution oftwo or more different sets of equalityconstraints, then this is calleddegeneracy. This will turn out to beimportant for the simplex algorithm.
I was oncetold thatdegeneracykilled the
dinosaurs.
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Suppose an LP has a feasible solution
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Suppose an LP has a feasible solution.
Which of the following is not possible for the
simplex algorithm (maximization problem)?
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1. It terminates with an optimal
solution.
2. It does not terminate.
3. It terminates with a proof that the
objective value is unbounded
from above.
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38x
y
1 2 3 4
1
2
3
Convex Sets
A set S is convexif for every two points in the set,
the line segment joining the points is also in the set;that is,
p1
p2
Theorem. The feasible regionof a linear program is convex.
if p1, p2 S, then so is (1 - )p 1 + p2 for [0,1].
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391 2 3 4 5 6
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x
y
notnotnotnotconvexconvex
More on Convexity
convexWhich of the following are ?convexconvex or notnot ?
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The feasible region of a linear program
is convex
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x
y
1 2 3 4 5 6
1
2
3
4
5
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2 Di i l LP d
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2-Dimensional LPs and
Sensitivity Analysis
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Mita, an MIT BeaverAmit, an MIT Beaver
Hi, we have a tutorial
for you stored at the
subject web site. We
hope to see you there.
Its on sensitivity analysis intwo dimensions. We know
that youll find it useful for
doing the problem set.
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Summary: Geometry helps guide the
intuition
Note for Thursdays lecture: please read the
tutorial on solving equations prior to lecture.