Polyhedral Computation
Jayant Apte
ASPITRG
October 3, 2012
Jayant Apte. ASPITRG
1
Part I
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The Three Problems of
Polyhedral Computation
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Representation Conversion
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H-representation of polyhedron
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V-Representation of polyhedron
Projection
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Redundancy Removal
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Compute the minimal representation of a polyhedron
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Outline
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Preliminaries
● Representations of convex polyhedra, polyhedral cones
●
●
●
Homogenization - Converting a general polyhedron in
a cone in
Polar of a convex cone
to
The three problems
●
Representation Conversion
–
–
●
Projection of polyhedral sets
–
–
–
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Fourier-Motzkin Elimination
Block Elimination
Convex Hull Method (CHM)
Redundancy removal
–
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Review of LRS
Double Description Method
Redundancy removal using linear programming
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Outline for Part 1
●
Preliminaries
●
●
●
●
Representations of convex polyhedra, polyhedral cones
Homogenization – Converting a general polyhedron in
to a cone in
Polar of a convex cone
The first problem
●
Representation Conversion
–
–
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Review of LRS
Double Description Method
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Preliminaries
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Convex Polyhedron
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Examples of polyhedra
Bounded- Polytope
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Unbounded - polyhedron
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H-Representation of a Polyhedron
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V-Representation of a Polyhedron
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Example
(0.5,0.5,1.5)
(1,1,1)
(0,1,1)
(0,0,1)
(1,1,0)
(0,1,0)
(1,0,0)
(0,0,0)
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Switching between the two representations :
Representation Conversion Problem
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●
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H-representation
V-representation: The
Vertex Enumeration problem
Methods:
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Reverse Search, Lexicographic Reverse Search
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Double-description method
V-representation
H-representation
The Facet Enumeration Problem
●
Facet enumeration can be accomplished by using
polarity and doing Vertex Enumeration
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Polyhedral Cone
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A cone in
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Homogenization
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H-polyhedra
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Example(d=2,d+1=3)
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Example
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V-polyhedra
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Polar of a convex cone
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Polar of a convex cone
Looks like an H-representation!!!
Our ticket back to the original cone!!!
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Polar: Intuition
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Polar: Intuition
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Polar: Intuition
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Polar: Intuition
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Use of Polar
●
Representation Conversion
●
●
No need to have two different algorithms i.e. for
and
.
Redundancy removal
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No need to have two different algorithms for
removing redundancies from H-representation
and V-representation
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Safely assume that input is always an Hrepresentation for both these problems
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Polarity: Intuition
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Polar: Intuition
Then take polar again to get
facets of this cone
Perform Vertex Enumeration
on this cone.
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Problem #1
Representation Conversion
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Review of Lexicographic Reverse
Search
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Reverse Search:
High Level Idea
●
●
●
Start with dictionary corresponding to the optimal
vertex
Ask yourself 'What pivot would have landed me at this
dictionary if i was running simplex?'
Go to that dictionary by applying reverse pivot to
current dictionary
●
Ask the same question again
●
Generate the so-called 'reverse search tree'
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Reverse Search on a Cube
Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration
Algorithm
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Double Description Method
●
First introduced in:
“ Motzkin, T. S.; Raiffa, H.; Thompson, G. L.; Thrall, R. M.
(1953). "The double description method". Contributions to the
theory of games. Annals of Mathematics Studies. Princeton, N.
J.: Princeton University Press. pp. 51–73”
●
The primitive algorithm in this paper is very inefficient.
(How? We will see later)
●
●
Several authors came up with their own efficient
implementation(viz. Fukuda, Padberg)
Fukuda's implementation is called cdd
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Some Terminology
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Double Description Method:
The High Level Idea
●
●
●
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An Incremental Algorithm
Starts with certain subset of rows of H-representation
of a cone
to form initial H-representation
Adds rest of the inequalities one by one constructing
the corresponding V-representation every iteration
Thus, constructing the V-representation incrementally.
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How it works?
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How it works?
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Initialization
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Initialization
Very trivial and inefficient
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Example: Initialization
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Example: Initialization
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Example : Initialization
-1
1
0
-1
-1
1
18
0
-4
A
B
4.0000 14.0000
4.0000 4.0000
1.0000 1.0000
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C
9.0000
9.0000
1.0000
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How it works?
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Iteration: Insert a new constraint
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Iteration 1 A
B
4.0000 14.0000
4.0000 4.0000
1.0000 1.0000
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C
9.0000
9.0000
1.0000
45
Iteration 1: Insert a new constraint
Constraint 2
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Iteration 1: Insert a new constraint
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Iteration 1: Insert a new constraint
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Main Lemma for DD Method
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Iteration 1: Insert a new constraint
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Iteration 1: Get the new DD pair
-1 -1 18
1 -1 0
0 1 -4
-1 1 6
10.0000 12.0000
4.0000 6.0000
1.0000 1.0000
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4.0000
4.0000
1.0000
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9.0000
9.0000
1.0000
51
Iteration 2
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Iteration 2: Insert new constraint
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Iteration 2: Insert new constraint
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Iteration 2: Insert new constraint
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Get the new DD pair
-1 -1 18
1 -1 0
0 1 -4
-1 1 6
0 -1 8
9.2000 10.0000 8.0000 10.0000 12.0000 4.0000
8.0000 8.0000 8.0000 4.0000 6.0000 4.0000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
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Iteration 3
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Get the new DD pair
-1 -1 18
1 -1 0
0 1 -4
-1 1 6
0 -1 8
1 1 -12
A
6.2609
5.7391
1.0000
B
6.4000
5.6000
1.0000
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C
6.0000
6.0000
1.0000
D
8.0000
4.0000
1.0000
E
F
7.2000
4.8000
1.0000
9.2000
8.0000
1.0000
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G
H
I
J
10.0000 8.0000 10.0000 12.0000
8.0000 8.0000 4.0000 6.0000
1.0000 1.0000 1.0000 1.0000
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Termination
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Efficiency Issues
●
Is the implementation I just described good
enough?
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Efficiency Issues
●
Is the implementation I just described good
enough?
●
Hell no!
–
October 3, 2012
The implementation just described suffers from
profusion of redundancy
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Efficiency Issues
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We can
totally
do
without
this one!!
62
Efficiency Issues
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And without
all these!!!
63
Efficiency Issues
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The Primitive DD method
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What to do?
●
Add some more structure
●
Strengthen the Main Lemma
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Add some more structure
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Some definitions
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Algebraic Characterization of adjacency
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Combinatorial Characterization of adjacency
(Combinatorial Oracle)
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Example: Combinatorial Adjacency
Oracle
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Example: Combinatorial Adjacency
Oracle
Only the pairs
generate new rays
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and
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will be used to
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Strengthened Main Lemma
for DD Method
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Procedural Description
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Part 2
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●
Projection of polyhedral sets
–
–
–
●
Redundancy removal
–
October 3, 2012
Fourier-Motzkin Elimination
Block Elimination
Convex Hull Method (CHM)
Redundancy removal using linear programming
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Projection of a Polyhedra
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Fourier-Motzkin Elimination
●
Named after Joseph Fourier and Theodore
Motzkin
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How it works?
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How it works? Contd...
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Efficiency of FM algorithm
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Convex Hull Method(CHM)
●
●
●
First appears in “C. Lassez and J.-L. Lassez, Quantifier
elimination for conjunctions of linear constraints via
aconvex hull algorithm, IBM Research Report, T.J. Watson
Research Center (1991)”
Found to be better than most other existing algorithms
when dimension of projection is small
Cited by Weidong Xu, Jia Wang, Jun Sun in their ISIT
2008 paper “A Projection Method for Derivation of NonShannon-Type Information Inequalities”
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CHM intuition
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CHM intuition
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CHM intuition
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(12,6,6)
(12,6)
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CHM intuition
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Moral of the story
●
●
●
We can make decisions about
actually having its H-representation.
It suffices to have
,
the original polyhedron
We run linear programs on
decisions
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without
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to make these
89
Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Perform Facet
Enumeration on
to get
Update
by adding r to it
Take an inequality from
that is not a face of
and move it outward
To find a new extreme point r
Are all inequalities
In
faces
Of
?
No
Yes
Stop
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Example
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Example
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Example
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Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Perform Facet
Enumeration on
to get
Update
by adding r to it
Take an inequality from
that is not a face of
and move it outward
To find a new extreme point r
Are all inequalities
In
facets
Of
?
Stop
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How to get the extreme points of
initial approximation?
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Example: Get the first two points
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Example: Get the first two points
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Example: Get the first two points
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Get rest of the points so you have a
total of d+1 points
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Get the rest of the points so you
have a total of d+1 points
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Get the rest of the points so you
have a total d+1 points
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How to get the initial facets of the
projection?
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How to get the initial facets of the
projection?
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Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Incrementally update
The Convex Hull
(Perform Facet
Enumeration on
to get
)
Are all inequalities
In
faces
Of
?
Update
by adding r to it
Take an inequality from
that is not a face of
and move it outward
To find a new extreme point r
No
Yes
Stop
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Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Incrementally update
The Convex Hull
(Perform Facet
Enumeration on
to get
)
Are all inequalities
In
faces
Of
?
Update
by adding r to it
Take an inequality from
that is not a face of
and move it outward
To find a new extreme point r
No
Yes
Stop
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105
Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Incrementally update
The Convex Hull
(Perform Facet
Enumeration on
to get
)
Are all inequalities
In
faces
Of
?
Update
by adding r to it
Take an inequality from
that is not a facet of
and move it outward
To find a new extreme point r
No
Yes
Stop
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Incremental refinement
●
●
Find a facet in current approximation of
that is not actually the facet of
How to do that?
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Incremental refinement
This inequality is
Not a facet of
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Incremental refinement
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Input:
Faces A of P
Construct an initial
Approximation
of
extreme points of
Convex Hull Method:
Flowchart
Incrementally update
The Convex Hull
(Perform Facet
Enumeration on
to get
)
Are all inequalities
In
faces
Of
?
Update
by adding r to it
Take an inequality from
that is not a facet of
and move it outward
To find a new extreme point r
No
Yes
Stop
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110
How to update the H-representation
to accommodate new point ?
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Example
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Example
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Example: New candidate facets
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How to update the H-representation
to accommodate new point ?
Determine validity of
The candidate facet
And also its sign
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Example
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Example
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How to update the H-representation
to accommodate new point ?
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Example
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Example
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Example
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nd
2 iteration
Not a facet of
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rd
3 iteration
Not a facet of
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th
4 iteration: Done!!!
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Comparison of Projection
Algorithms
●
●
●
Fourier Motzkin Elimination and Block
Elimination doesn't work well when used on big
problems
CHM works very well when the dimension of
projection is relatively small as compared to the
dimention of original polyhedron.
Weidong Xu, Jia Wang, Jun Sun have already
used CHM to get non-Shannon inequalities.
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Computation of minimal
representation/Redundancy
Removal
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Definitions
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References
K. Fukuda and A. Prodon. Double description method revisited. Technical report, Department of Mathematics,
Swiss Federal Institute of Technology, Lausanne, Switzerland, 1995
G. Ziegler, Lectures on Polytopes, Springer, 1994, revised 1998. Graduate Texts in Mathematics.
●
Tien Huynh, Catherine Lassez, and Jean-Louis Lassez. Practical issues on the projection of polyhedral sets.
●
Annals of mathematics and artificial intelligence, November 1992
Catherine Lassez and Jean-Louis Lassez. Quantifier elimination for conjunctions of linear constraints via a
●
convex hull algorithm. In Bruce Donald, Deepak Kapur, and Joseph Mundy, editors, Symbolic and Numerical
Computation for Artificial Intelligence. Academic Press, 1992.
R. T. Rockafellar. Convex Analysis (Princeton Mathematical Series). Princeton Univ Pr.
●
W. Xu, J. Wang, J. Sun. A projection method for derivation of non-Shannon-type information inequalities. In
●
IEEE International Symposiumon Information Theory (ISIT), pp. 2116 – 2120, 2008.
●
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Minkowski's Theorem for
Polyhedral Cones
Weyl's Theorem for
Polyhedral Cones
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