Polyhedral Computation                        Jayant Apte                         ASPITRGOctober 3, 2012          Jayant A...
Part IOctober 3, 2012   Jayant Apte. ASPITRG   2
The Three Problems of                  Polyhedral Computation ●   Representation Conversion     ●   H-representation of po...
Outline           ●   Preliminaries               ● Representations of convex polyhedra, polyhedral cones               ● ...
Outline for Part 1●     Preliminaries      ●    Representations of convex polyhedra, polyhedral cones      ●    Homogeniza...
PreliminariesOctober 3, 2012      Jayant Apte. ASPITRG   6
Convex PolyhedronOctober 3, 2012        Jayant Apte. ASPITRG   7
Examples of polyhedra                  Bounded- Polytope                          Unbounded - polyhedronOctober 3, 2012   ...
H-Representation of a PolyhedronOctober 3, 2012   Jayant Apte. ASPITRG   9
V-Representation of a PolyhedronOctober 3, 2012   Jayant Apte. ASPITRG   10
Example                                           (0.5,0.5,1.5)                                                           ...
Switching between the two representations :    Representation Conversion Problem●   H-representation         V-representat...
Polyhedral ConeOctober 3, 2012       Jayant Apte. ASPITRG   13
A cone inOctober 3, 2012    Jayant Apte. ASPITRG   14
HomogenizationOctober 3, 2012       Jayant Apte. ASPITRG   15
H-polyhedraOctober 3, 2012     Jayant Apte. ASPITRG   16
Example(d=2,d+1=3)October 3, 2012         Jayant Apte. ASPITRG   17
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   18
V-polyhedraOctober 3, 2012     Jayant Apte. ASPITRG   19
Polar of a convex coneOctober 3, 2012          Jayant Apte. ASPITRG   20
Polar of a convex cone                                                Looks like an H-representation!!!                   ...
Polar: IntuitionOctober 3, 2012       Jayant Apte. ASPITRG   22
Polar: IntuitionOctober 3, 2012       Jayant Apte. ASPITRG   23
Polar: IntuitionOctober 3, 2012       Jayant Apte. ASPITRG   24
Polar: IntuitionOctober 3, 2012       Jayant Apte. ASPITRG   25
Use of Polar ●   Representation Conversion      ●    No need to have two different algorithms i.e. for                    ...
Polarity: IntuitionOctober 3, 2012        Jayant Apte. ASPITRG   27
Polar: Intuition                                                Then take polar again to get                              ...
Problem #1                  Representation ConversionOctober 3, 2012           Jayant Apte. ASPITRG   29
Review of Lexicographic Reverse                SearchOctober 3, 2012   Jayant Apte. ASPITRG   30
Reverse Search:                   High Level Idea ●   Start with dictionary corresponding to the optimal     vertex ●   As...
Reverse Search on a Cube   Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration   Algori...
Double Description Method ●   First introduced in:      “ Motzkin, T. S.; Raiffa, H.; Thompson, G. L.; Thrall, R. M.     (...
Some TerminologyOctober 3, 2012       Jayant Apte. ASPITRG   34
Double Description Method:                     The High Level Idea●   An Incremental Algorithm●   Starts with certain subs...
How it works?October 3, 2012      Jayant Apte. ASPITRG   36
How it works?October 3, 2012      Jayant Apte. ASPITRG   37
InitializationOctober 3, 2012      Jayant Apte. ASPITRG   38
Initialization                                            Very trivial and inefficientOctober 3, 2012      Jayant Apte. AS...
Example: InitializationOctober 3, 2012          Jayant Apte. ASPITRG   40
Example: InitializationOctober 3, 2012          Jayant Apte. ASPITRG   41
Example : Initialization                                      -1   -1    18                                       1   -1  ...
How it works?October 3, 2012      Jayant Apte. ASPITRG   43
Iteration: Insert a new constraintOctober 3, 2012     Jayant Apte. ASPITRG    44
Iteration 1 A                   B         C                                          4.0000 14.0000   9.0000              ...
Iteration 1: Insert a new constraint                  Constraint 2October 3, 2012   Jayant Apte. ASPITRG     46
Iteration 1: Insert a new constraintOctober 3, 2012   Jayant Apte. ASPITRG     47
Iteration 1: Insert a new constraintOctober 3, 2012   Jayant Apte. ASPITRG     48
Main Lemma for DD MethodOctober 3, 2012       Jayant Apte. ASPITRG   49
Iteration 1: Insert a new constraintOctober 3, 2012   Jayant Apte. ASPITRG     50
Iteration 1: Get the new DD pair                            -1 -1 18                             1 -1 0                   ...
Iteration 2October 3, 2012    Jayant Apte. ASPITRG   52
Iteration 2: Insert new constraintOctober 3, 2012    Jayant Apte. ASPITRG    53
Iteration 2: Insert new constraintOctober 3, 2012    Jayant Apte. ASPITRG    54
Iteration 2: Insert new constraintOctober 3, 2012    Jayant Apte. ASPITRG    55
Get the new DD pair                                 -1 -1 18                                   1 -1 0                     ...
Iteration 3October 3, 2012    Jayant Apte. ASPITRG   57
Get the new DD pair                                          -1 -1 18                                            1 -1 0   ...
TerminationOctober 3, 2012     Jayant Apte. ASPITRG   59
Efficiency Issues ●   Is the implementation I just described good     enough?October 3, 2012        Jayant Apte. ASPITRG  ...
Efficiency Issues ●   Is the implementation I just described good     enough?      ●    Hell no!             –    The impl...
Efficiency Issues           We can                                              totally                                   ...
Efficiency Issues           And without                                              all these!!!October 3, 2012        Ja...
Efficiency IssuesOctober 3, 2012        Jayant Apte. ASPITRG   64
The Primitive DD methodOctober 3, 2012           Jayant Apte. ASPITRG   65
What to do? ●   Add some more structure ●   Strengthen the Main LemmaOctober 3, 2012      Jayant Apte. ASPITRG   66
Add some more structureOctober 3, 2012           Jayant Apte. ASPITRG   67
Some definitionsOctober 3, 2012       Jayant Apte. ASPITRG   68
October 3, 2012   Jayant Apte. ASPITRG   69
October 3, 2012   Jayant Apte. ASPITRG   70
Algebraic Characterization of adjacencyOctober 3, 2012              Jayant Apte. ASPITRG           71
Combinatorial Characterization of adjacency                  (Combinatorial Oracle)October 3, 2012             Jayant Apte...
Example: Combinatorial Adjacency              OracleOctober 3, 2012   Jayant Apte. ASPITRG   73
Example: Combinatorial Adjacency              Oracle            Only the pairs            and              will be used to...
Strengthened Main Lemma                    for DD MethodOctober 3, 2012       Jayant Apte. ASPITRG   75
Procedural DescriptionOctober 3, 2012          Jayant Apte. ASPITRG   76
Part 2October 3, 2012   Jayant Apte. ASPITRG   77
●    Projection of polyhedral sets                  –   Fourier-Motzkin Elimination                  –   Block Elimination...
Projection of a PolyhedraOctober 3, 2012            Jayant Apte. ASPITRG   79
Fourier-Motzkin Elimination ●   Named after Joseph Fourier and Theodore     MotzkinOctober 3, 2012         Jayant Apte. AS...
How it works?October 3, 2012      Jayant Apte. ASPITRG   81
How it works? Contd...October 3, 2012          Jayant Apte. ASPITRG   82
Efficiency of FM algorithmOctober 3, 2012            Jayant Apte. ASPITRG   83
Convex Hull Method(CHM) ●   First appears in “C. Lassez and J.-L. Lassez, Quantifier     elimination for conjunctions of l...
CHM intuitionOctober 3, 2012      Jayant Apte. ASPITRG   85
CHM intuitionOctober 3, 2012      Jayant Apte. ASPITRG   86
CHM intuition             (12,6,6)                                                       (12,6)October 3, 2012      Jayant...
CHM intuitionOctober 3, 2012      Jayant Apte. ASPITRG   88
Moral of the story ●   We can make decisions about                         without     actually having its H-representatio...
Input:                          Faces A of P                       Construct an initial                      Approximation...
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   91
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   92
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   93
Input:                         Faces A of P                      Construct an initial                      Approximation  ...
How to get the extreme points of           initial approximation?October 3, 2012   Jayant Apte. ASPITRG   95
Example: Get the first two pointsOctober 3, 2012   Jayant Apte. ASPITRG    96
Example: Get the first two pointsOctober 3, 2012    Jayant Apte. ASPITRG   97
Example: Get the first two pointsOctober 3, 2012    Jayant Apte. ASPITRG   98
Get rest of the points so you have a          total of d+1 pointsOctober 3, 2012   Jayant Apte. ASPITRG   99
Get the rest of the points so you           have a total of d+1 pointsOctober 3, 2012     Jayant Apte. ASPITRG    100
Get the rest of the points so you            have a total d+1 pointsOctober 3, 2012     Jayant Apte. ASPITRG   101
How to get the initial facets of the                projection?October 3, 2012    Jayant Apte. ASPITRG     102
How to get the initial facets of the                projection?October 3, 2012    Jayant Apte. ASPITRG     103
Input:                          Faces A of P                       Construct an initial                      Approximation...
Input:                          Faces A of P                       Construct an initial                      Approximation...
Input:                          Faces A of P                       Construct an initial                      Approximation...
Incremental refinement ●   Find a facet in current approximation of     that is not actually the facet of ●   How to do th...
Incremental refinement                                                This inequality is                                  ...
Incremental refinementOctober 3, 2012          Jayant Apte. ASPITRG   109
Input:                          Faces A of P                        Construct an initial                       Approximati...
How to update the H-representation      to accommodate new point ?October 3, 2012   Jayant Apte. ASPITRG   111
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   112
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   113
Example: New candidate facetsOctober 3, 2012    Jayant Apte. ASPITRG   114
How to update the H-representation      to accommodate new point ?                                         Determine valid...
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   116
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   117
How to update the H-representation      to accommodate new point ?October 3, 2012   Jayant Apte. ASPITRG   118
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   119
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   120
ExampleOctober 3, 2012   Jayant Apte. ASPITRG   121
nd                  2 iteration                    Not a facet ofOctober 3, 2012     Jayant Apte. ASPITRG   122
rd                  3 iteration                        Not a facet ofOctober 3, 2012     Jayant Apte. ASPITRG   123
th                  4 iteration: Done!!!October 3, 2012         Jayant Apte. ASPITRG   124
Comparison of Projection                       Algorithms ●   Fourier Motzkin Elimination and Block     Elimination doesnt...
Computation of minimal              representation/Redundancy                       RemovalOctober 3, 2012        Jayant A...
DefinitionsOctober 3, 2012    Jayant Apte. ASPITRG   127
References●  K. Fukuda and A. Prodon. Double description method revisited. Technical report, Department of Mathematics,  S...
Minkowskis Theorem for                     Polyhedral Cones                    Weyls Theorem for                     Polyh...
Upcoming SlideShare
Loading in …5
×

Jayant chm

341 views
274 views

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
341
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Jayant chm

  1. 1. Polyhedral Computation Jayant Apte ASPITRGOctober 3, 2012 Jayant Apte. ASPITRG 1
  2. 2. Part IOctober 3, 2012 Jayant Apte. ASPITRG 2
  3. 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 polyhedronOctober 3, 2012 Jayant Apte. ASPITRG 3
  4. 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 programmingOctober 3, 2012 Jayant Apte. ASPITRG 4
  5. 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 MethodOctober 3, 2012 Jayant Apte. ASPITRG 5
  6. 6. PreliminariesOctober 3, 2012 Jayant Apte. ASPITRG 6
  7. 7. Convex PolyhedronOctober 3, 2012 Jayant Apte. ASPITRG 7
  8. 8. Examples of polyhedra Bounded- Polytope Unbounded - polyhedronOctober 3, 2012 Jayant Apte. ASPITRG 8
  9. 9. H-Representation of a PolyhedronOctober 3, 2012 Jayant Apte. ASPITRG 9
  10. 10. V-Representation of a PolyhedronOctober 3, 2012 Jayant Apte. ASPITRG 10
  11. 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. 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 EnumerationOctober 3, 2012 Jayant Apte. ASPITRG 12
  13. 13. Polyhedral ConeOctober 3, 2012 Jayant Apte. ASPITRG 13
  14. 14. A cone inOctober 3, 2012 Jayant Apte. ASPITRG 14
  15. 15. HomogenizationOctober 3, 2012 Jayant Apte. ASPITRG 15
  16. 16. H-polyhedraOctober 3, 2012 Jayant Apte. ASPITRG 16
  17. 17. Example(d=2,d+1=3)October 3, 2012 Jayant Apte. ASPITRG 17
  18. 18. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 18
  19. 19. V-polyhedraOctober 3, 2012 Jayant Apte. ASPITRG 19
  20. 20. Polar of a convex coneOctober 3, 2012 Jayant Apte. ASPITRG 20
  21. 21. Polar of a convex cone Looks like an H-representation!!! Our ticket back to the original cone!!!October 3, 2012 Jayant Apte. ASPITRG 21
  22. 22. Polar: IntuitionOctober 3, 2012 Jayant Apte. ASPITRG 22
  23. 23. Polar: IntuitionOctober 3, 2012 Jayant Apte. ASPITRG 23
  24. 24. Polar: IntuitionOctober 3, 2012 Jayant Apte. ASPITRG 24
  25. 25. Polar: IntuitionOctober 3, 2012 Jayant Apte. ASPITRG 25
  26. 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 problemsOctober 3, 2012 Jayant Apte. ASPITRG 26
  27. 27. Polarity: IntuitionOctober 3, 2012 Jayant Apte. ASPITRG 27
  28. 28. Polar: Intuition Then take polar again to get facets of this cone Perform Vertex Enumeration on this cone.October 3, 2012 Jayant Apte. ASPITRG 28
  29. 29. Problem #1 Representation ConversionOctober 3, 2012 Jayant Apte. ASPITRG 29
  30. 30. Review of Lexicographic Reverse SearchOctober 3, 2012 Jayant Apte. ASPITRG 30
  31. 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 treeOctober 3, 2012 Jayant Apte. ASPITRG 31
  32. 32. Reverse Search on a Cube Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration AlgorithmOctober 3, 2012 Jayant Apte. ASPITRG 32
  33. 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) ● Fukudas implementation is called cddOctober 3, 2012 Jayant Apte. ASPITRG 33
  34. 34. Some TerminologyOctober 3, 2012 Jayant Apte. ASPITRG 34
  35. 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 35
  36. 36. How it works?October 3, 2012 Jayant Apte. ASPITRG 36
  37. 37. How it works?October 3, 2012 Jayant Apte. ASPITRG 37
  38. 38. InitializationOctober 3, 2012 Jayant Apte. ASPITRG 38
  39. 39. Initialization Very trivial and inefficientOctober 3, 2012 Jayant Apte. ASPITRG 39
  40. 40. Example: InitializationOctober 3, 2012 Jayant Apte. ASPITRG 40
  41. 41. Example: InitializationOctober 3, 2012 Jayant Apte. ASPITRG 41
  42. 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 1.0000October 3, 2012 Jayant Apte. ASPITRG 42
  43. 43. How it works?October 3, 2012 Jayant Apte. ASPITRG 43
  44. 44. Iteration: Insert a new constraintOctober 3, 2012 Jayant Apte. ASPITRG 44
  45. 45. Iteration 1 A B C 4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000October 3, 2012 Jayant Apte. ASPITRG 45
  46. 46. Iteration 1: Insert a new constraint Constraint 2October 3, 2012 Jayant Apte. ASPITRG 46
  47. 47. Iteration 1: Insert a new constraintOctober 3, 2012 Jayant Apte. ASPITRG 47
  48. 48. Iteration 1: Insert a new constraintOctober 3, 2012 Jayant Apte. ASPITRG 48
  49. 49. Main Lemma for DD MethodOctober 3, 2012 Jayant Apte. ASPITRG 49
  50. 50. Iteration 1: Insert a new constraintOctober 3, 2012 Jayant Apte. ASPITRG 50
  51. 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 1.0000October 3, 2012 Jayant Apte. ASPITRG 51
  52. 52. Iteration 2October 3, 2012 Jayant Apte. ASPITRG 52
  53. 53. Iteration 2: Insert new constraintOctober 3, 2012 Jayant Apte. ASPITRG 53
  54. 54. Iteration 2: Insert new constraintOctober 3, 2012 Jayant Apte. ASPITRG 54
  55. 55. Iteration 2: Insert new constraintOctober 3, 2012 Jayant Apte. ASPITRG 55
  56. 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 1.0000October 3, 2012 Jayant Apte. ASPITRG 56
  57. 57. Iteration 3October 3, 2012 Jayant Apte. ASPITRG 57
  58. 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 J6.2609 6.4000 6.0000 8.0000 7.2000 9.2000 10.0000 8.0000 10.0000 12.00005.7391 5.6000 6.0000 4.0000 4.8000 8.0000 8.0000 8.0000 4.0000 6.00001.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 58
  59. 59. TerminationOctober 3, 2012 Jayant Apte. ASPITRG 59
  60. 60. Efficiency Issues ● Is the implementation I just described good enough?October 3, 2012 Jayant Apte. ASPITRG 60
  61. 61. Efficiency Issues ● Is the implementation I just described good enough? ● Hell no! – The implementation just described suffers from profusion of redundancyOctober 3, 2012 Jayant Apte. ASPITRG 61
  62. 62. Efficiency Issues We can totally do without this one!!October 3, 2012 Jayant Apte. ASPITRG 62
  63. 63. Efficiency Issues And without all these!!!October 3, 2012 Jayant Apte. ASPITRG 63
  64. 64. Efficiency IssuesOctober 3, 2012 Jayant Apte. ASPITRG 64
  65. 65. The Primitive DD methodOctober 3, 2012 Jayant Apte. ASPITRG 65
  66. 66. What to do? ● Add some more structure ● Strengthen the Main LemmaOctober 3, 2012 Jayant Apte. ASPITRG 66
  67. 67. Add some more structureOctober 3, 2012 Jayant Apte. ASPITRG 67
  68. 68. Some definitionsOctober 3, 2012 Jayant Apte. ASPITRG 68
  69. 69. October 3, 2012 Jayant Apte. ASPITRG 69
  70. 70. October 3, 2012 Jayant Apte. ASPITRG 70
  71. 71. Algebraic Characterization of adjacencyOctober 3, 2012 Jayant Apte. ASPITRG 71
  72. 72. Combinatorial Characterization of adjacency (Combinatorial Oracle)October 3, 2012 Jayant Apte. ASPITRG 72
  73. 73. Example: Combinatorial Adjacency OracleOctober 3, 2012 Jayant Apte. ASPITRG 73
  74. 74. Example: Combinatorial Adjacency Oracle Only the pairs and will be used to generate new raysOctober 3, 2012 Jayant Apte. ASPITRG 74
  75. 75. Strengthened Main Lemma for DD MethodOctober 3, 2012 Jayant Apte. ASPITRG 75
  76. 76. Procedural DescriptionOctober 3, 2012 Jayant Apte. ASPITRG 76
  77. 77. Part 2October 3, 2012 Jayant Apte. ASPITRG 77
  78. 78. ● Projection of polyhedral sets – Fourier-Motzkin Elimination – Block Elimination – Convex Hull Method (CHM) ● Redundancy removal – Redundancy removal using linear programmingOctober 3, 2012 Jayant Apte. ASPITRG 78
  79. 79. Projection of a PolyhedraOctober 3, 2012 Jayant Apte. ASPITRG 79
  80. 80. Fourier-Motzkin Elimination ● Named after Joseph Fourier and Theodore MotzkinOctober 3, 2012 Jayant Apte. ASPITRG 80
  81. 81. How it works?October 3, 2012 Jayant Apte. ASPITRG 81
  82. 82. How it works? Contd...October 3, 2012 Jayant Apte. ASPITRG 82
  83. 83. Efficiency of FM algorithmOctober 3, 2012 Jayant Apte. ASPITRG 83
  84. 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. 85. CHM intuitionOctober 3, 2012 Jayant Apte. ASPITRG 85
  86. 86. CHM intuitionOctober 3, 2012 Jayant Apte. ASPITRG 86
  87. 87. CHM intuition (12,6,6) (12,6)October 3, 2012 Jayant Apte. ASPITRG 87
  88. 88. CHM intuitionOctober 3, 2012 Jayant Apte. ASPITRG 88
  89. 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 decisionsOctober 3, 2012 Jayant Apte. ASPITRG 89
  90. 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 getConvex 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 StopOctober 3, 2012 Jayant Apte. ASPITRG 90
  91. 91. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 91
  92. 92. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 92
  93. 93. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 93
  94. 94. Input: Faces A of P Construct an initial Approximation of extreme points of Perform Facet Update by adding r to it Enumeration onConvex 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 ? StopOctober 3, 2012 Jayant Apte. ASPITRG 94
  95. 95. How to get the extreme points of initial approximation?October 3, 2012 Jayant Apte. ASPITRG 95
  96. 96. Example: Get the first two pointsOctober 3, 2012 Jayant Apte. ASPITRG 96
  97. 97. Example: Get the first two pointsOctober 3, 2012 Jayant Apte. ASPITRG 97
  98. 98. Example: Get the first two pointsOctober 3, 2012 Jayant Apte. ASPITRG 98
  99. 99. Get rest of the points so you have a total of d+1 pointsOctober 3, 2012 Jayant Apte. ASPITRG 99
  100. 100. Get the rest of the points so you have a total of d+1 pointsOctober 3, 2012 Jayant Apte. ASPITRG 100
  101. 101. Get the rest of the points so you have a total d+1 pointsOctober 3, 2012 Jayant Apte. ASPITRG 101
  102. 102. How to get the initial facets of the projection?October 3, 2012 Jayant Apte. ASPITRG 102
  103. 103. How to get the initial facets of the projection?October 3, 2012 Jayant Apte. ASPITRG 103
  104. 104. Input: Faces A of P Construct an initial Approximation of extreme points of Incrementally update Update by adding r to it The Convex HullConvex 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 StopOctober 3, 2012 Jayant Apte. ASPITRG 104
  105. 105. Input: Faces A of P Construct an initial Approximation of extreme points of Incrementally update Update by adding r to it The Convex HullConvex 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 StopOctober 3, 2012 Jayant Apte. ASPITRG 105
  106. 106. Input: Faces A of P Construct an initial Approximation of extreme points of Incrementally update Update by adding r to it The Convex HullConvex 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 StopOctober 3, 2012 Jayant Apte. ASPITRG 106
  107. 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 107
  108. 108. Incremental refinement This inequality is Not a facet ofOctober 3, 2012 Jayant Apte. ASPITRG 108
  109. 109. Incremental refinementOctober 3, 2012 Jayant Apte. ASPITRG 109
  110. 110. Input: Faces A of P Construct an initial Approximation of extreme points of Incrementally update Update by adding r to it The Convex HullConvex 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 StopOctober 3, 2012 Jayant Apte. ASPITRG 110
  111. 111. How to update the H-representation to accommodate new point ?October 3, 2012 Jayant Apte. ASPITRG 111
  112. 112. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 112
  113. 113. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 113
  114. 114. Example: New candidate facetsOctober 3, 2012 Jayant Apte. ASPITRG 114
  115. 115. How to update the H-representation to accommodate new point ? Determine validity of The candidate facet And also its signOctober 3, 2012 Jayant Apte. ASPITRG 115
  116. 116. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 116
  117. 117. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 117
  118. 118. How to update the H-representation to accommodate new point ?October 3, 2012 Jayant Apte. ASPITRG 118
  119. 119. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 119
  120. 120. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 120
  121. 121. ExampleOctober 3, 2012 Jayant Apte. ASPITRG 121
  122. 122. nd 2 iteration Not a facet ofOctober 3, 2012 Jayant Apte. ASPITRG 122
  123. 123. rd 3 iteration Not a facet ofOctober 3, 2012 Jayant Apte. ASPITRG 123
  124. 124. th 4 iteration: Done!!!October 3, 2012 Jayant Apte. ASPITRG 124
  125. 125. Comparison of Projection Algorithms ● Fourier Motzkin Elimination and Block Elimination doesnt 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. 126. Computation of minimal representation/Redundancy RemovalOctober 3, 2012 Jayant Apte. ASPITRG 126
  127. 127. DefinitionsOctober 3, 2012 Jayant Apte. ASPITRG 127
  128. 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. 129. Minkowskis Theorem for Polyhedral Cones Weyls Theorem for Polyhedral ConesOctober 3, 2012 Jayant Apte. ASPITRG 129

×