1. Polyhedral Computation
Jayant Apte
ASPITRG
October 3, 2012 Jayant Apte. ASPITRG 1

2. Part I
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3. The Three Problems of
Polyhedral 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 3

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 4

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
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6. Preliminaries
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7. Convex Polyhedron
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8. Examples of polyhedra
Bounded- Polytope Unbounded - polyhedron
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9. H-Representation of a Polyhedron
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10. V-Representation of a Polyhedron
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11. 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)
October 3, 2012 Jayant Apte. ASPITRG 11

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
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13. Polyhedral Cone
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14. A cone in
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15. Homogenization
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16. H-polyhedra
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17. Example(d=2,d+1=3)
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18. Example
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19. V-polyhedra
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20. Polar of a convex cone
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21. Polar of a convex cone
Looks like an H-representation!!!
Our ticket back to the original cone!!!
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22. Polar: Intuition
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23. Polar: Intuition
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24. Polar: Intuition
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25. Polar: Intuition
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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
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27. Polarity: Intuition
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28. Polar: Intuition
Then take polar again to get
facets of this cone
Perform Vertex Enumeration
on this cone.
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29. Problem #1
Representation Conversion
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30. Review of Lexicographic Reverse
Search
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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 31

32. Reverse Search on a Cube
Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration
Algorithm
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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
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34. Some Terminology
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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.
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36. How it works?
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37. How it works?
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38. Initialization
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39. Initialization
Very trivial and inefficient
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40. Example: Initialization
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41. Example: Initialization
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42. Example : Initialization
-1 -1 18
1 -1 0
0 1 -4
A B C
4.0000 14.0000 9.0000
4.0000 4.0000 9.0000
1.0000 1.0000 1.0000
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43. How it works?
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44. Iteration: Insert a new constraint
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45. Iteration 1 A B C
4.0000 14.0000 9.0000
4.0000 4.0000 9.0000
1.0000 1.0000 1.0000
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46. Iteration 1: Insert a new constraint
Constraint 2
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47. Iteration 1: Insert a new constraint
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48. Iteration 1: Insert a new constraint
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49. Main Lemma for DD Method
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50. Iteration 1: Insert a new constraint
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51. 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 9.0000
4.0000 6.0000 4.0000 9.0000
1.0000 1.0000 1.0000 1.0000
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52. Iteration 2
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53. Iteration 2: Insert new constraint
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54. Iteration 2: Insert new constraint
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55. Iteration 2: Insert new constraint
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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
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57. Iteration 3
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58. Get the new DD pair
-1 -1 18
1 -1 0
0 1 -4
-1 1 6
0 -1 8
1 1 -12
A B C D E F G H I J
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
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59. Termination
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60. Efficiency Issues
● Is the implementation I just described good
enough?
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61. Efficiency Issues
● Is the implementation I just described good
enough?
● Hell no!
– The implementation just described suffers from
profusion of redundancy
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62. Efficiency Issues We can
totally
do
without
this one!!
October 3, 2012 Jayant Apte. ASPITRG 62

63. Efficiency Issues And without
all these!!!
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64. Efficiency Issues
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65. The Primitive DD method
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66. What to do?
● Add some more structure
● Strengthen the Main Lemma
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67. Add some more structure
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68. Some definitions
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69. October 3, 2012 Jayant Apte. ASPITRG 69

70. October 3, 2012 Jayant Apte. ASPITRG 70

71. Algebraic Characterization of adjacency
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72. Combinatorial Characterization of adjacency
(Combinatorial Oracle)
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73. Example: Combinatorial Adjacency
Oracle
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74. Example: Combinatorial Adjacency
Oracle
Only the pairs and will be used to
generate new rays
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75. Strengthened Main Lemma
for DD Method
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76. Procedural Description
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77. Part 2
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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 78

79. Projection of a Polyhedra
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80. Fourier-Motzkin Elimination
● Named after Joseph Fourier and Theodore
Motzkin
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81. How it works?
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82. How it works? Contd...
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83. Efficiency of FM algorithm
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84. 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 Non-
Shannon-Type Information Inequalities”
October 3, 2012 Jayant Apte. ASPITRG 84

85. CHM intuition
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86. CHM intuition
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87. CHM intuition (12,6,6)
(12,6)
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88. CHM intuition
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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 89

90. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Perform Facet Update by adding r to it
Enumeration on
to get
Convex Hull Method:
Flowchart
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
No
Of
?
Yes
Stop
October 3, 2012 Jayant Apte. ASPITRG 90

91. Example
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92. Example
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93. Example
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94. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Perform Facet Update by adding r to it
Enumeration on
Convex Hull Method: to get
Flowchart
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
October 3, 2012 Jayant Apte. ASPITRG 94

95. How to get the extreme points of
initial approximation?
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96. Example: Get the first two points
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97. Example: Get the first two points
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98. Example: Get the first two points
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99. Get rest of the points so you have a
total of d+1 points
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100. Get the rest of the points so you
have a total of d+1 points
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101. Get the rest of the points so you
have a total d+1 points
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102. How to get the initial facets of the
projection?
October 3, 2012 Jayant Apte. ASPITRG 102

103. How to get the initial facets of the
projection?
October 3, 2012 Jayant Apte. ASPITRG 103

104. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Incrementally update Update by adding r to it
The Convex Hull
Convex Hull Method: (Perform Facet
Enumeration on
Flowchart to get )
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 No
Of
?
Yes
Stop
October 3, 2012 Jayant Apte. ASPITRG 104

105. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Incrementally update Update by adding r to it
The Convex Hull
Convex Hull Method: (Perform Facet
Enumeration on
Flowchart to get )
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 No
Of
?
Yes
Stop
October 3, 2012 Jayant Apte. ASPITRG 105

106. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Incrementally update Update by adding r to it
The Convex Hull
Convex Hull Method: (Perform Facet
Enumeration on
Flowchart to get )
Take an inequality from
that is not a facet of
and move it outward
To find a new extreme point r
Are all inequalities
In faces No
Of
?
Yes
Stop
October 3, 2012 Jayant Apte. ASPITRG 106

107. Incremental refinement
● Find a facet in current approximation of
that is not actually the facet of
● How to do that?
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108. Incremental refinement
This inequality is
Not a facet of
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109. Incremental refinement
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110. Input:
Faces A of P
Construct an initial
Approximation of
extreme points of
Incrementally update Update by adding r to it
The Convex Hull
Convex Hull Method: (Perform Facet
Enumeration on
Flowchart to get )
Take an inequality from
that is not a facet of
and move it outward
To find a new extreme point r
Are all inequalities
No
In faces
Of
?
Yes
Stop
October 3, 2012 Jayant Apte. ASPITRG 110

111. How to update the H-representation
to accommodate new point ?
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112. Example
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113. Example
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114. Example: New candidate facets
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115. How to update the H-representation
to accommodate new point ?
Determine validity of
The candidate facet
And also its sign
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116. Example
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117. Example
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118. How to update the H-representation
to accommodate new point ?
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119. Example
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120. Example
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121. Example
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122. nd
2 iteration
Not a facet of
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123. rd
3 iteration
Not a facet of
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124. th
4 iteration: Done!!!
October 3, 2012 Jayant Apte. ASPITRG 124

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 125

126. Computation of minimal
representation/Redundancy
Removal
October 3, 2012 Jayant Apte. ASPITRG 126

127. Definitions
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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 128

129. Minkowski's Theorem for
Polyhedral Cones
Weyl's Theorem for
Polyhedral Cones
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