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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

3

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What does the feasible region of an LP

look like?

4

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

1

2

3

4

5

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101 2 3 4 5 6

1

2

3

4

5

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

5

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

1

2

3

4

5

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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2

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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

1

23

4

5

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.