jayant chm
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TRANSCRIPT
October 3, 2012 Jayant Apte. ASPITRG 1
Polyhedral Computation
Jayant Apte ASPITRG
October 3, 2012 Jayant Apte. ASPITRG 2
Part I
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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
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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
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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– 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 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
(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)
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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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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● 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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Polarity: Intuition
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Polar: Intuition
Perform Vertex Enumeration on this cone.
Then take polar again to get facets of this cone
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Problem #1Representation 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
● 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
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
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How it works?
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Iteration: Insert a new constraint
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Iteration 1 4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000
A B C
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Iteration 1: Insert a new constraintConstraint 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
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
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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
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
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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!– The implementation just described suffers from
profusion of redundancy
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Efficiency IssuesWe can totallydo without this one!!
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Efficiency IssuesAnd without all these!!!
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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 and will be used to generate new rays
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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– Fourier-Motzkin Elimination– Block Elimination– Convex Hull Method (CHM)
● Redundancy removal– 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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● 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)
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CHM intuition
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CHM intuition
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CHM intuition (12,6,6)
(12,6)
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CHM intuition
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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
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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
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Example
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Example
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Example
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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
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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
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?
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Incremental refinement
This inequality isNot 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
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 ?
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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-representationto 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-representationto accommodate new point ?
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Example
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Example
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Example
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2nd iteration
Not a facet of
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3rd iteration
Not a facet of
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4th 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.
October 3, 2012 Jayant Apte. ASPITRG 126
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.
October 3, 2012 Jayant Apte. ASPITRG 129
Minkowski's Theorem for Polyhedral Cones
Weyl's Theorem for Polyhedral Cones