aarjaneiro/planetmath huggingface dataset:
corner point theorem

corner point theorem

Let

P

be a primal

linear programming problem

with

f

the objective function and

polyhedron

R

as its feasible region. A

corner point

, or extreme point of

P

is a of

R

.

Corner Point Theorem.

If

P

has an optimal solution

a

<

∞

, then there is a corner point

p

of

P

such that

f

⁢

(

p

)

=

a

. If another corner point

q

also

satisfies

f

⁢

(

x

)

=

a

, then

f

⁢

(

p

⁢

q

¯

)

=

{

a

}

. If

r

is a third corner point such that

f

⁢

(

r

)

=

a

, then

f

⁢

(

△

⁢

p

⁢

q

⁢

r

)

=

{

a

}

.

Note that the

line segment

and

triangle

mentioned above are necessarily subsets of

R

.

A cousin of the above

theorem

is the following:

Theorem.

If

P

has an optimal solution

a

<

∞

occurring at an

interior point

of an edge

E

(or a face

F

) on the

boundary

of the feasible region

R

, then

f

⁢

(

E

)

=

{

a

}

(or

f

⁢

(

F

)

=

{

a

}

). In fact, if the solution occurs at an interior point of

R

, then the solution is satisfied by all points of

R

:

f

⁢

(

R

)

=

{

a

}

. In other words, the objective function

f

is

constant

on

R

.

On way to visualize the above theorems is to simplify them into the case when the objective function is a line

f

⁢

(

x

)

=

m

⁢

x

+

b

on the “

x

-

y

plane” and the feasible region is a line segment

I

=

[

x

0

,

x

1

]

on the

x

-axis. It is easy to see now that the maximum (or minimum) of

f

occurs at either

x

0

or

x

1

. If the optimal value occurs either at both

end points

, or at an interior point

x

2

∈

(

x

0

,

x

1

)

, then

f

is a horizontal line segment on

I

.

An application of the above theorems can be demonstrated by the following example: If the feasible region

R

is a

unit square

and if corner points

(

0

,

0

)

,

(

1

,

1

)

satisfy the optimal solution of

P

, then all points on

{

(

x

,

y

)

∣

x

=

y

}

∩

R

satisfy the solution. In particular,

(

1

2

,

1

2

)

, an interior point of

R

, satisfies the solution. As a result, all points of

R

satisfy the solution.

Title

corner point theorem

Canonical name

CornerPointTheorem

Date of creation

2013-03-22 15:39:16

Last modified on

2013-03-22 15:39:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 90C05

Synonym

fundamental theorem of linear programming

Defines

corner point

Generated on Sat Feb 10 13:36:45 2018 by

LaTeXML
linear programming

linear programming

A

linear programming problem

, or

LP

,
is a problem of optimizing a given
linear objective function over some polyhedron. The standard

maximization

LP, sometimes called the
primal problem, is

maximize

c

T

⁢

x

s.t.

A

⁢

x

≤

b

(P)

x

≥

0

Here

c

T

⁢

x

is the objective function and the remaining conditions
define the polyhedron which is the feasible region over which the
objective function is to be optimized. The dual of

(

(P)

)

is the LP

minimize

y

T

⁢

b

s.t.

y

T

⁢

A

≥

c

T

(D)

y

≥

0

The linear constraints for a linear programming problems define a

convex polyhedron

, called the

feasible region

for the problem. The weak duality theorem states that if

x

^

is feasible (i.e. lies in the feasible region) for

(

(P)

)

and

y

^

is feasible for

(

(D)

)

,
then

c

T

⁢

x

^

≤

y

^

T

⁢

b

. This follows readily from the above:

c

T

⁢

x

^

≤

(

y

^

T

⁢

A

)

⁢

x

^

=

y

^

T

⁢

(

A

⁢

x

^

)

≤

y

T

⁢

b

.

The strong duality theorem states that if both LPs are feasible,
then the two objective functions have the same optimal value. As a
consequence, if
either LP has unbounded objective function value, the other must
be infeasible. It is also possible for both LP to be infeasible.

The

simplex method

(

http://planetmath.org/SimplexAlgorithm

) of G. B. Dantzig is
the

algorithm

most commonly used to solve LPs; in practice it runs in

polynomial time

,
but the worst-case running time is exponential. Two polynomial-time
algorithms for solving LPs are the ellipsoid method of L. G. Khachian
and the interior-point method of N. Karmarkar.

References

1

Chvátal, V.,

Linear programming

, W. H. Freeman and Company, 1983.

2

Cormen, T. H., Leiserson, C. E., Rivest, R. L., and C. Stein,

Introduction to algorithms

, MIT Press, 2001.

3

Korte, B. and J. Vygen,

Combinatorial optimization

: theory and
algorithms

, Springer-Verlag, 2002.

Title

linear programming

Canonical name

LinearProgramming

Date of creation

2013-03-22 13:41:41

Last modified on

2013-03-22 13:41:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Topic

Classification

msc 90C05

Synonym

LP

Related topic

DualityInMathematics

Defines

linear programming problem

Defines

objective function

Defines

feasible region

Defines

feasible

Generated on Sat Feb 10 13:36:47 2018 by

LaTeXML
assignment problem

assignment problem

The

assignment problem

is a kind of optimization problem in which the goal is to reduce the total cost of a number of tasks by choosing a particular assignment of those tasks to a number of agents, or task executors. One example might be an algorithm for assigning pieces of work to individual processors of different capacities in a multiprocessor computer. A more real-world example might be Amazon deciding, given a finite number of airplanes, which

distribution

center should be used, with its fleet, to service a particular set of delivery requests in order to minimize the usage of fuel.

Mathematically, the assignment problem starts with a finite set of agents,

A

, and a finite set of tasks,

T

, where

|

A

|

=

|

T

|

. The cost of a particular agent executing a particular task is a cost, or ,

function

C

:

A

×

T

→

ℝ

. The goal is to find the bijective mapping

f

:

A

→

T

assigning agents to tasks such that the cost function

∑

a

∈

A

C

⁢

(

a

,

f

⁢

(

a

)

)

is minimized.

The cost function can also be viewed as a square real-valued matrix

C

, where

C

i

,

j

is the cost of having agent

i

execute task

t

, so that the

objective function

can be written as

∑

a

∈

A

C

a

,

f

⁢

(

a

)

The problem is “linear” because the cost function to be optimized as well as all the constraints contain only linear terms.

This entry was adapted from the Wikipedia article

http://en.wikipedia.org/wiki/Assignment_problem

Assignment problem as of January 13, 2007.

Title

assignment problem

Canonical name

AssignmentProblem

Date of creation

2013-03-22 16:33:40

Last modified on

2013-03-22 16:33:40

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 90C27

Synonym

linear assignment problem

Generated on Sat Feb 10 13:36:52 2018 by

LaTeXML
blackjack

blackjack

Blackjack

or

twenty-one

or

vingt-et-un

is a multi-player game of skill and luck played with standard playing cards and colored chips. The game may be played with two decks of cards instead of just one, especially if there are more than five players. The game begins with the dealer dealing each player a card face down and the players placing their initial bets. After that, players receive additional cards from the dealer, the goal being to collect a total value of exactly 21, no more.

The aces may be worth 1 or 11, as the player wishes, while the royal cards (jack, queen, king) are all worth 10 each.

There are 792 unrestricted

integer partitions

of 21, but we can disregard those containing

integers

greater than 11, as well as those containing more than four instances of an integer from 1 to 5 (assuming only one deck of cards is being used).

References

1

A. Morehead & G. Mott-Smith

Play According to Hoyle: Hoyle’s Rules of Games

New York: Signet (1963): 174 - 177

Title

blackjack

Canonical name

Blackjack

Date of creation

2013-03-22 16:41:02

Last modified on

2013-03-22 16:41:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Feature

Classification

msc 91A05

Synonym

vingt-et-un

Generated on Sat Feb 10 13:37:17 2018 by

LaTeXML
dominant strategy

dominant strategy

For any player

i

, a

strategy

s

*

∈

S

i

weakly dominates

another strategy

s

′

∈

S

i

if:

∀

s

-

i

∈

S

-

i

[

u

i

(

s

*

,

s

-

i

)

≥

u

i

(

s

′

,

s

-

i

)

]

(Remember that

S

-

i

represents the product of all strategy sets other than

i

’s)

s

*

strongly dominates

s

′

if:

∀

s

-

i

∈

S

-

i

[

u

i

(

s

*

,

s

-

i

)

>

u

i

(

s

′

,

s

-

i

)

]

Title

dominant strategy

Canonical name

DominantStrategy

Date of creation

2013-03-22 12:51:39

Last modified on

2013-03-22 12:51:39

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 91A10

Defines

weakly dominant strategy

Defines

dominates

Defines

weakly dominates

Defines

strongly dominates

Defines

dominant

Defines

strongly dominant

Generated on Sat Feb 10 13:37:24 2018 by

LaTeXML
examples of normal form games

examples of normal form games

A few example

normal form games

:

Prisoner’s

dilemma

Probably the most famous

game theory

example, the prisoner’s dilemma is a two player game where

S

1

=

S

2

=

C

,

D

and:

u

1

⁢

(

s

1

,

s

2

)

=

{

5

if

s

1

=

C

and

s

2

=

C

10

if

s

1

=

D

and

s

2

=

C

-

5

if

s

1

=

C

and

s

2

=

D

0

if

s

1

=

D

and

s

2

=

D

u

2

⁢

(

s

1

,

s

2

)

=

{

5

if

s

1

=

C

and

s

2

=

C

10

if

s

1

=

C

and

s

2

=

D

-

5

if

s

1

=

D

and

s

2

=

C

0

if

s

1

=

D

and

s

2

=

D

Traditionally this is interpreted as the case of two criminal partners separately being interrogated and asked to give up the other partner.

C

stands for cooperating (with their partners) by refusing to give up information, and

D

stands for defecting and agreeing to testify against the partner. The different payoffs reflect different jail

sentences

, ranging from nothing (+10) to a long jail sentence (-5), with amounts in between depending on the evidence against them.

The (much more convenient)

normal form

is:

C

D

C

5,5

-5,10

D

10,-5

0,0

Notice that

(

C

,

C

)

Pareto dominates

(

D

,

D

)

, however

(

D

,

D

)

is the only

Nash equilibrium

.

Battle of the Sexes

Another traditional two player game. The normal form is:

O

F

O

2,1

0,0

F

0,0

1,2

A Degenerate Example

One more, rather pointless, example which illustrates a game where one player has no choice:

X

Y

Z

A

2,100

1,7

14,-5

Undercut

A game which illustrates an

infinite

(indeed,

uncountable

)

strategy

space. There are two players and

S

1

=

S

2

=

ℝ

+

.

u

1

⁢

(

s

1

,

s

2

)

=

{

1

if

s

1

<

s

2

0

if

s

1

≥

s

2

u

2

⁢

(

s

1

,

s

2

)

=

{

1

if

s

2

<

s

1

0

if

s

2

≥

s

1

Title

examples of normal form games

Canonical name

ExamplesOfNormalFormGames

Date of creation

2013-03-22 12:51:27

Last modified on

2013-03-22 12:51:27

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

9

Author

Henry (455)

Entry type

Example

Classification

msc 91A10

Classification

msc 91A06

Classification

msc 91A05

Defines

prisoners dilemma

Defines

battle of the sexes

Generated on Sat Feb 10 13:37:27 2018 by

LaTeXML
normal form game

normal form game

A

normal form game

is a game of

complete information

in which there is a list of

n

players, numbered

1

,

…

,

n

. Each player has a

strategy

set,

S

i

and a

utility function

u

i

:

∏

i

≤

n

S

i

→

ℝ

.

In such a game each player simultaneously selects a move

s

i

∈

S

i

and receives

u

i

⁢

(

(

s

1

,

…

,

s

n

)

)

.

Normal form games with two players and finite strategy sets can be represented in

normal form

, a matrix where the rows each stand for an element of

S

1

and the columns for an element of

S

2

. Each cell of the matrix contains an

ordered pair

which states the payoffs for each player. That is, the cell

i

,

j

contains

(

u

1

⁢

(

s

i

,

s

j

)

,

u

2

⁢

(

s

i

,

s

j

)

)

where

s

i

is the

i

-th element of

S

1

and

s

j

is the

j

-th element of

S

2

.

Title

normal form game

Canonical

name

NormalFormGame

Date of creation

2013-03-22 12:51:24

Last modified on

2013-03-22 12:51:24

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 91A10

Classification

msc 91A06

Classification

msc 91A05

Synonym

strategic form game

Related topic

Game

Defines

normal form game

Defines

normal form

Generated on Sat Feb 10 13:37:28 2018 by

LaTeXML
extensive form game

extensive form game

A game in

extensive form

is one that can be represented as a tree, where each node corresponds to a choice by one of the players. Unlike a

normal form game

, in an extensive form game players make choices sequentially. However players do not necessarily always know which node they are at (that is, what moves have already been made).

Formally, an extensive form game is a set of nodes together with a

function

for each

non-terminal

node. The function specifies which player moves at that node, what actions are available, and which node comes next for each action. For each terminal node, there is instead a function defining

utilities

for each player when that node is the one the game results in. Finally the nodes are partitioned into information sets, where any two nodes in the same information set must have the same actions and the same moving player.

A pure

strategy

for each player is a function which, for each information set, selects one of the available actions. That is, if player

i

’s information sets are

h

1

,

h

2

,

…

,

h

m

with corresponding sets actions

a

1

,

a

2

,

…

,

a

m

then

S

i

=

∏

x

h

x

→

∏

x

a

x

.

Title

extensive form game

Canonical name

ExtensiveFormGame

Date of creation

2013-03-22 12:51:41

Last modified on

2013-03-22 12:51:41

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 91A18

Defines

extensive form

Defines

information set

Generated on Sat Feb 10 13:37:30 2018 by

LaTeXML
15 Puzzle

15 Puzzle

The

15 Puzzle

is a square tablet containing 15 smaller square tiles labeled with the integers 1 to 15, set so that only one square may be moved at a time into the only available empty square by a move up or down or left or right (but never diagonally). The goal of the puzzle is to take a puzzle in an unsorted initial state, such as

7

8

1

2

3

4

5

6

9

10

11

12

13

14

15

and set each tile in its proper order.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

The puzzle was invented by Noyes Chapman, who also created a famously unsolvable version with 14 and 15 switched. His original idea was to construct a puzzle with 16 tiles that would be moved to form a

magic square

with 34 as its

magic constant

. The 15 Puzzle was initially made of wood; today they are almost always made of plastic. Darling calls it “the Rubik’s cube of its day.”

References

1

D. Darling, “15 Puzzle” in

The Universal Book of Mathematics: From Abracadabra To Zeno’s paradoxes

. Hoboken, New Jersey: Wiley (2004)

Title

15 Puzzle

Canonical

name

15Puzzle

Date of creation

2013-03-22 16:46:21

Last modified on

2013-03-22 16:46:21

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 91A24

Classification

msc 00A08

Synonym

Fifteen Puzzle

Synonym

Game of Fifteen

Generated on Sat Feb 10 13:37:32 2018 by

LaTeXML
Rubik’s cube

Rubik’s cube

A mechanical puzzle in the shape of a cube, invented by the Hungarian mathematician ErnÃÂ¶ Rubik. Each of the six faces is subdivided into nine squares, which may be painted in any of six different colors, and any of the sides may be rotated independently of the others. The

central

square of each face rotates with the sides, but relative to each other, the central squares are

stationary

.

A Rubik’s cube is said to be solved when the nine squares of each face all have the same color. When any face has two or more colors on it, the cube is said to be scrambled.

Hypothesis

. The maximum number of moves from a scrambled to a solved cube is 22. (A quarter rotation and a half rotation both count as one move).

It is known for a fact that for a given cube there are more than 43 quintillion possible

combinations

, but the maximum number of moves hypothesis remains to be proven.

Currently, the best (though not necessarily speediest) method to solve the cube is considered to be the solve-by-layers method. Instead of trying to solve one face at a time, one tries to first solve the “top” layer. E.g., if the solver chooses white to be the top face, and his cube has blue as the opposite color of white, the solver solves the white face so that the top three squares of the red face are all red, the top three squares of the blue face are all blue, etc. Then he solves the middle layer so that between the central squares of any two faces (except the top and bottom face) there is a piece with the appropriate

matching

colors. Lastly, the bottom layer is solved first by putting the corners in their place and lastly the middle pieces.

White is usually the face that the licensed manufacturer puts their logo on. The license only requires that the six colors be distinct. One popular choice of colors is white, red, green, orange, blue and yellow. In the United States, Rubik’s cube is usually implemented in black plastic with colored stickers on the squares of the faces. This permits someone to solve the cube by removing the stickers and replacing them elsewhere on the cube (this is not considered a legitimate solution).

The standard Rubik’s cube is sometimes referred to as 3 by 3 cube. Variations range from 2 by 2 cubes to 6 by 6 cubes.

Title

Rubik’s cube

Canonical

name

RubiksCube

Date of creation

2013-03-22 15:37:45

Last modified on

2013-03-22 15:37:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 91A24

Synonym

Rubik cube

Generated on Sat Feb 10 13:37:34 2018 by

LaTeXML
sudoku

sudoku

Sudoku

is a logic puzzle, usually consisting of a 9 by 9 grid subdivided into 3 by 3 squares, some of which are marked with numerals from 1 to 9 (the givens). The goal of the puzzle is to mark the rest of the grid so that the result is a

Latin square

, that is, so that each row, column, and subgrid has all of the numerals 1 to 9 with no duplicates.

Given a partially filled

n

×

n

Latin square, the problem of testing if it extends to a

complete

Latin square is NP-complete.

Thus, a sudoku usually has 81 squares with about 30 givens. A sudoku can have a unique solution with as few as 17 givens (or 18 if

symmetry

of givens is required), but it’s also possible for it to have as many as 77 givens (just four squares short of solving) and still lack a unique solution.

Title

sudoku

Canonical name

Sudoku

Date of creation

2013-03-22 16:15:27

Last modified on

2013-03-22 16:15:27

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

8

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 91A24

Classification

msc 00A08

Generated on Sat Feb 10 13:37:35 2018 by

LaTeXML
common knowledge

common knowledge

In a game, a fact (such as the rules of the game) is

common knowledge

for the player if:

All the players know the fact.

All the players know that all the players know the fact.

All the players know that all the players know that all the players know the fact.

⋯

This is a much stronger condition than merely having all the players know the fact. By way of illustration, consider the following example:

There are three participants and an experimenter. The experimenter informs them that a hat, either blue or red, will be placed on their head so that the other participants can see it but the wearer cannot. The experimenter then puts red hats on each person and asks whether any of them know what color hat they have. Of course, none of them do. The experimenter then whispers to each of them that at least two people have red hats, and then asks out loud whether any of them know what color hat they have. Again, none of them do. Finally the experimenter announces out loud that at least two people have red hats, and asks whether any of them know what color hat they have. After a few seconds, all three realize that they must have red hats, since if they had a blue hat then both of the other people could have figured out that their own hat was red.

The significant thing about this example is that the fact that at least two of the participants have red hats was known to every participant from the beginning, but only once they knew that the other people also knew that could they figure out their own hat’s color. (This is only the second requirement in the list above, but more complicated examples can be constructed for any level in the infinite list).

Title

common knowledge

Canonical name

CommonKnowledge

Date of creation

2013-03-22 12:51:21

Last modified on

2013-03-22 12:51:21

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 91A99

Related topic

Game

Generated on Sat Feb 10 13:37:37 2018 by

LaTeXML
complete information

complete information

A game has

complete information

if each player knows the of the game (that is, the

strategy

sets and the payoff functions for each player). A game without complete information has incomplete information.

Title

complete information

Canonical

name

CompleteInformation

Date of creation

2013-03-22 12:51:30

Last modified on

2013-03-22 12:51:30

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 91A99

Related topic

Game

Defines

complete information

Defines

incomplete information

Defines

complete

Defines

incomplete

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