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October 3, 2012 Jayant Apte. ASPITRG 1
Polyhedral Computation
Jayant Apte ASPITRG
October 3, 2012 Jayant Apte. ASPITRG 2
Part I
October 3, 2012 Jayant Apte. ASPITRG 3
The Three Problems ofPolyhedral Computation
● Representation Conversion● H-representation of polyhedron
● V-Representation of polyhedron
● Projection●
● Redundancy Removal● Compute the minimal representation of a polyhedron
October 3, 2012 Jayant Apte. ASPITRG 4
Outline● Preliminaries
● Representations of convex polyhedra, polyhedral cones● Homogenization - Converting a general polyhedron in to
a cone in
● Polar of a convex cone
● The three problems● Representation Conversion
– Review of LRS
– Double Description Method
● Projection of polyhedral sets
– Fourier-Motzkin Elimination
– Block Elimination
– Convex Hull Method (CHM)
● Redundancy removal
– Redundancy removal using linear programming
October 3, 2012 Jayant Apte. ASPITRG 5
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– Review of LRS– Double Description Method
October 3, 2012 Jayant Apte. ASPITRG 6
Preliminaries
October 3, 2012 Jayant Apte. ASPITRG 7
Convex Polyhedron
October 3, 2012 Jayant Apte. ASPITRG 8
Examples of polyhedra
Bounded- Polytope Unbounded - polyhedron
October 3, 2012 Jayant Apte. ASPITRG 9
H-Representation of a Polyhedron
October 3, 2012 Jayant Apte. ASPITRG 10
V-Representation of a Polyhedron
October 3, 2012 Jayant Apte. ASPITRG 11
Example
(1,0,0)
(0,0,0)
(0,1,0)
(1,1,0)
(0,1,1)
(0.5,0.5,1.5)(1,1,1)
(0,0,1)
October 3, 2012 Jayant Apte. ASPITRG 12
Switching between the two representations : Representation Conversion Problem
● H-representation V-representation: The Vertex Enumeration problem
● Methods:
● Reverse Search, Lexicographic Reverse Search● Double-description method
● V-representation H-representation
The Facet Enumeration Problem
● Facet enumeration can be accomplished by using polarity and doing Vertex Enumeration
October 3, 2012 Jayant Apte. ASPITRG 13
Polyhedral Cone
October 3, 2012 Jayant Apte. ASPITRG 14
A cone in
October 3, 2012 Jayant Apte. ASPITRG 15
Homogenization
October 3, 2012 Jayant Apte. ASPITRG 16
H-polyhedra
October 3, 2012 Jayant Apte. ASPITRG 17
Example(d=2,d+1=3)
October 3, 2012 Jayant Apte. ASPITRG 18
Example
October 3, 2012 Jayant Apte. ASPITRG 19
V-polyhedra
October 3, 2012 Jayant Apte. ASPITRG 20
Polar of a convex cone
October 3, 2012 Jayant Apte. ASPITRG 21
Polar of a convex cone
Looks like an H-representation!!!
Our ticket back to the original cone!!!
October 3, 2012 Jayant Apte. ASPITRG 22
Polar: Intuition
October 3, 2012 Jayant Apte. ASPITRG 23
Polar: Intuition
October 3, 2012 Jayant Apte. ASPITRG 24
Polar: Intuition
October 3, 2012 Jayant Apte. ASPITRG 25
Polar: Intuition
October 3, 2012 Jayant Apte. ASPITRG 26
Use of Polar
● Representation Conversion● No need to have two different algorithms i.e. for
and .
● Redundancy removal● No need to have two different algorithms for
removing redundancies from H-representation
and V-representation
● Safely assume that input is always an H-representation for both these problems
October 3, 2012 Jayant Apte. ASPITRG 27
Polarity: Intuition
October 3, 2012 Jayant Apte. ASPITRG 28
Polar: Intuition
Perform Vertex Enumeration on this cone.
Then take polar again to get facets of this cone
October 3, 2012 Jayant Apte. ASPITRG 29
Problem #1Representation Conversion
October 3, 2012 Jayant Apte. ASPITRG 30
Review of Lexicographic Reverse Search
October 3, 2012 Jayant Apte. ASPITRG 31
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'
October 3, 2012 Jayant Apte. ASPITRG 32
Reverse Search on a Cube
Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm
October 3, 2012 Jayant Apte. ASPITRG 33
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
October 3, 2012 Jayant Apte. ASPITRG 34
Some Terminology
October 3, 2012 Jayant Apte. ASPITRG 35
Double Description Method:The High Level Idea
● 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.
October 3, 2012 Jayant Apte. ASPITRG 36
How it works?
October 3, 2012 Jayant Apte. ASPITRG 37
How it works?
October 3, 2012 Jayant Apte. ASPITRG 38
Initialization
October 3, 2012 Jayant Apte. ASPITRG 39
Initialization
Very trivial and inefficient
October 3, 2012 Jayant Apte. ASPITRG 40
Example: Initialization
October 3, 2012 Jayant Apte. ASPITRG 41
Example: Initialization
October 3, 2012 Jayant Apte. ASPITRG 42
Example : Initialization
4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000
-1 -1 18 1 -1 0 0 1 -4
A B C
October 3, 2012 Jayant Apte. ASPITRG 43
How it works?
October 3, 2012 Jayant Apte. ASPITRG 44
Iteration: Insert a new constraint
October 3, 2012 Jayant Apte. ASPITRG 45
Iteration 1 4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000
A B C
October 3, 2012 Jayant Apte. ASPITRG 46
Iteration 1: Insert a new constraintConstraint 2
October 3, 2012 Jayant Apte. ASPITRG 47
Iteration 1: Insert a new constraint
October 3, 2012 Jayant Apte. ASPITRG 48
Iteration 1: Insert a new constraint
October 3, 2012 Jayant Apte. ASPITRG 49
Main Lemma for DD Method
October 3, 2012 Jayant Apte. ASPITRG 50
Iteration 1: Insert a new constraint
October 3, 2012 Jayant Apte. ASPITRG 51
Iteration 1: Get the new DD pair
10.0000 12.0000 4.0000 9.0000 4.0000 6.0000 4.0000 9.0000 1.0000 1.0000 1.0000 1.0000
-1 -1 18 1 -1 0 0 1 -4 -1 1 6
October 3, 2012 Jayant Apte. ASPITRG 52
Iteration 2
October 3, 2012 Jayant Apte. ASPITRG 53
Iteration 2: Insert new constraint
October 3, 2012 Jayant Apte. ASPITRG 54
Iteration 2: Insert new constraint
October 3, 2012 Jayant Apte. ASPITRG 55
Iteration 2: Insert new constraint
October 3, 2012 Jayant Apte. ASPITRG 56
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
October 3, 2012 Jayant Apte. ASPITRG 57
Iteration 3
October 3, 2012 Jayant Apte. ASPITRG 58
Get the new DD pair
-1 -1 18 1 -1 0 0 1 -4 -1 1 6 0 -1 8 1 1 -12
6.2609 6.4000 6.0000 8.0000 7.2000 9.2000 10.0000 8.0000 10.0000 12.0000 5.7391 5.6000 6.0000 4.0000 4.8000 8.0000 8.0000 8.0000 4.0000 6.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
A B C D E F G H I J
October 3, 2012 Jayant Apte. ASPITRG 59
Termination
October 3, 2012 Jayant Apte. ASPITRG 60
Efficiency Issues
● Is the implementation I just described good enough?
October 3, 2012 Jayant Apte. ASPITRG 61
Efficiency Issues
● Is the implementation I just described good enough?
● Hell no!– The implementation just described suffers from
profusion of redundancy
October 3, 2012 Jayant Apte. ASPITRG 62
Efficiency IssuesWe can totallydo without this one!!
October 3, 2012 Jayant Apte. ASPITRG 63
Efficiency IssuesAnd without all these!!!
October 3, 2012 Jayant Apte. ASPITRG 64
Efficiency Issues
October 3, 2012 Jayant Apte. ASPITRG 65
The Primitive DD method
October 3, 2012 Jayant Apte. ASPITRG 66
What to do?
● Add some more structure● Strengthen the Main Lemma
October 3, 2012 Jayant Apte. ASPITRG 67
Add some more structure
October 3, 2012 Jayant Apte. ASPITRG 68
Some definitions
October 3, 2012 Jayant Apte. ASPITRG 69
October 3, 2012 Jayant Apte. ASPITRG 70
October 3, 2012 Jayant Apte. ASPITRG 71
Algebraic Characterization of adjacency
October 3, 2012 Jayant Apte. ASPITRG 72
Combinatorial Characterization of adjacency (Combinatorial Oracle)
October 3, 2012 Jayant Apte. ASPITRG 73
Example: Combinatorial Adjacency Oracle
October 3, 2012 Jayant Apte. ASPITRG 74
Example: Combinatorial Adjacency Oracle
Only the pairs and will be used to generate new rays
October 3, 2012 Jayant Apte. ASPITRG 75
Strengthened Main Lemma for DD Method
October 3, 2012 Jayant Apte. ASPITRG 76
Procedural Description
October 3, 2012 Jayant Apte. ASPITRG 77
Part 2
October 3, 2012 Jayant Apte. ASPITRG 78
● Projection of polyhedral sets– Fourier-Motzkin Elimination– Block Elimination– Convex Hull Method (CHM)
● Redundancy removal– Redundancy removal using linear programming
October 3, 2012 Jayant Apte. ASPITRG 79
Projection of a Polyhedra
October 3, 2012 Jayant Apte. ASPITRG 80
Fourier-Motzkin Elimination
● Named after Joseph Fourier and Theodore Motzkin
October 3, 2012 Jayant Apte. ASPITRG 81
How it works?
October 3, 2012 Jayant Apte. ASPITRG 82
How it works? Contd...
October 3, 2012 Jayant Apte. ASPITRG 83
Efficiency of FM algorithm
October 3, 2012 Jayant Apte. ASPITRG 84
● 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 Non-Shannon-Type Information Inequalities”
Convex Hull Method(CHM)
October 3, 2012 Jayant Apte. ASPITRG 85
CHM intuition
October 3, 2012 Jayant Apte. ASPITRG 86
CHM intuition
October 3, 2012 Jayant Apte. ASPITRG 87
CHM intuition (12,6,6)
(12,6)
October 3, 2012 Jayant Apte. ASPITRG 88
CHM intuition
October 3, 2012 Jayant Apte. ASPITRG 89
Moral of the story
● We can make decisions about without actually having its H-representation.
● It suffices to have , the original polyhedron● We run linear programs on to make these
decisions
October 3, 2012 Jayant Apte. ASPITRG 90
Input:Faces A of P
Construct an initial Approximation of extreme points of
Perform Facet Enumeration on
to get
Are all inequalities In faces
Of ?
Stop
Take an inequality from that is not a face of
and move it outward To find a new extreme point r
Update by adding r to it
Yes
No
Convex Hull Method:Flowchart
October 3, 2012 Jayant Apte. ASPITRG 91
Example
October 3, 2012 Jayant Apte. ASPITRG 92
Example
October 3, 2012 Jayant Apte. ASPITRG 93
Example
October 3, 2012 Jayant Apte. ASPITRG 94
Convex Hull Method:Flowchart
Input:Faces A of P
Construct an initial Approximation of extreme points of
Perform Facet Enumeration on
to get
Are all inequalities In facets
Of ?
Stop
Take an inequality from that is not a face of
and move it outward To find a new extreme point r
Update by adding r to it
October 3, 2012 Jayant Apte. ASPITRG 95
How to get the extreme points of initial approximation?
October 3, 2012 Jayant Apte. ASPITRG 96
Example: Get the first two points
October 3, 2012 Jayant Apte. ASPITRG 97
Example: Get the first two points
October 3, 2012 Jayant Apte. ASPITRG 98
Example: Get the first two points
October 3, 2012 Jayant Apte. ASPITRG 99
Get rest of the points so you have a total of d+1 points
October 3, 2012 Jayant Apte. ASPITRG 100
Get the rest of the points so you have a total of d+1 points
October 3, 2012 Jayant Apte. ASPITRG 101
Get the rest of the points so you have a total d+1 points
October 3, 2012 Jayant Apte. ASPITRG 102
How to get the initial facets of the projection?
October 3, 2012 Jayant Apte. ASPITRG 103
How to get the initial facets of the projection?
October 3, 2012 Jayant Apte. ASPITRG 104
Input:Faces A of P
Construct an initial Approximation of extreme points of
Incrementally updateThe Convex Hull (Perform Facet
Enumeration on to get )
Are all inequalities In faces
Of ?
Stop
Take an inequality from that is not a face of
and move it outward To find a new extreme point r
Update by adding r to it
Yes
No
Convex Hull Method:Flowchart
October 3, 2012 Jayant Apte. ASPITRG 105
Input:Faces A of P
Construct an initial Approximation of extreme points of
Incrementally updateThe Convex Hull (Perform Facet
Enumeration on to get )
Are all inequalities In faces
Of ?
Stop
Take an inequality from that is not a face of
and move it outward To find a new extreme point r
Update by adding r to it
Yes
No
Convex Hull Method:Flowchart
October 3, 2012 Jayant Apte. ASPITRG 106
Input:Faces A of P
Construct an initial Approximation of extreme points of
Incrementally updateThe Convex Hull (Perform Facet
Enumeration on to get )
Are all inequalities In faces
Of ?
Stop
Take an inequality from that is not a facet of
and move it outward To find a new extreme point r
Update by adding r to it
Yes
No
Convex Hull Method:Flowchart
October 3, 2012 Jayant Apte. ASPITRG 107
Incremental refinement
● Find a facet in current approximation of that is not actually the facet of
● How to do that?
October 3, 2012 Jayant Apte. ASPITRG 108
Incremental refinement
This inequality isNot a facet of
October 3, 2012 Jayant Apte. ASPITRG 109
Incremental refinement
October 3, 2012 Jayant Apte. ASPITRG 110
Input:Faces A of P
Construct an initial Approximation of extreme points of
Incrementally updateThe Convex Hull (Perform Facet
Enumeration on to get )
Are all inequalities In faces
Of ?
Stop
Take an inequality from that is not a facet of
and move it outward To find a new extreme point r
Update by adding r to it
Yes
No
Convex Hull Method:Flowchart
October 3, 2012 Jayant Apte. ASPITRG 111
How to update the H-representationto accommodate new point ?
October 3, 2012 Jayant Apte. ASPITRG 112
Example
October 3, 2012 Jayant Apte. ASPITRG 113
Example
October 3, 2012 Jayant Apte. ASPITRG 114
Example: New candidate facets
October 3, 2012 Jayant Apte. ASPITRG 115
How to update the H-representationto accommodate new point ?
Determine validity of The candidate facet And also its sign
October 3, 2012 Jayant Apte. ASPITRG 116
Example
October 3, 2012 Jayant Apte. ASPITRG 117
Example
October 3, 2012 Jayant Apte. ASPITRG 118
How to update the H-representationto accommodate new point ?
October 3, 2012 Jayant Apte. ASPITRG 119
Example
October 3, 2012 Jayant Apte. ASPITRG 120
Example
October 3, 2012 Jayant Apte. ASPITRG 121
Example
October 3, 2012 Jayant Apte. ASPITRG 122
2nd iteration
Not a facet of
October 3, 2012 Jayant Apte. ASPITRG 123
3rd iteration
Not a facet of
October 3, 2012 Jayant Apte. ASPITRG 124
4th iteration: Done!!!
October 3, 2012 Jayant Apte. ASPITRG 125
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.
October 3, 2012 Jayant Apte. ASPITRG 126
Computation of minimal representation/Redundancy
Removal
October 3, 2012 Jayant Apte. ASPITRG 127
Definitions
October 3, 2012 Jayant Apte. ASPITRG 128
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.
October 3, 2012 Jayant Apte. ASPITRG 129
Minkowski's Theorem for Polyhedral Cones
Weyl's Theorem for Polyhedral Cones
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