candidacy exam talk
TRANSCRIPT
![Page 1: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/1.jpg)
April 10, 2013
Polyhedral Computationfor Characterization of Region of Entropic Vectors
and Computation of Rate Regions of Coded Networks
Jayant ApteASPITRG
![Page 2: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/2.jpg)
April 10, 2013
Introduction
![Page 3: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/3.jpg)
April 10, 2013
Why do we care about this object?
Kolmogorov Complexity
GroupTheory
Network Coding
Combinatorics
Probability Theory
Quantum Mechanics
Matrix Theory
![Page 4: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/4.jpg)
April 10, 2013
Region of entropic vectors and Network Coding
● Achievable Information Rate Region of multi-source network coding problem is the set of all possible rates at which multiple information sources can be multicast simultaneously on a network
● Most general of all network coding problems● Implicit characterization in terms of region of
entropic vectors is available
![Page 5: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/5.jpg)
April 10, 2013
Where does polyhedral computation come into picture?
● Finding better polyhedral inner and outer bounds on the region of entropic vectors
● Finding the the Achievable Information Rate Region of multi-source network coding problem by substituting in these better inner and outer bounds in place of exact region of entropic vectors in the implicit characterization.
● Both the problems above become problems of polyhedral computation
![Page 6: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/6.jpg)
April 10, 2013
Outline
● Background on Polyhedra● Representation Conversion
– Lexicographic Reverse Search
– Double Description Method
● Polyhedral Projection– Convex Hull Method(As implemented in chm0.1)
![Page 7: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/7.jpg)
7Jayant Apte. ASPITRGApril 10, 2013
Convex Polyhedron
![Page 8: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/8.jpg)
8Jayant Apte. ASPITRGApril 10, 2013
Examples of polyhedra
Bounded- Polytope Unbounded - polyhedron
![Page 9: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/9.jpg)
9Jayant Apte. ASPITRGApril 10, 2013
H-Representation of a Polyhedron
![Page 10: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/10.jpg)
10Jayant Apte. ASPITRGApril 10, 2013
V-Representation of a Polyhedron
![Page 11: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/11.jpg)
11Jayant Apte. ASPITRGApril 10, 2013
Representation conversion
● Given the H-representation of a polyhedron, compute V-representation: vertex enumeration
● Given the V-representation of a polyhedron, compute the H-representation: facet enumeration
![Page 12: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/12.jpg)
12Jayant Apte. ASPITRGApril 10, 2013
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)
H-rep V-rep
![Page 13: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/13.jpg)
13Jayant Apte. ASPITRGApril 10, 2013
Polyhedral Cone
![Page 14: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/14.jpg)
14Jayant Apte. ASPITRGApril 10, 2013
A cone in
![Page 15: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/15.jpg)
15Jayant Apte. ASPITRGApril 10, 2013
Homogenization
![Page 16: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/16.jpg)
16Jayant Apte. ASPITRGApril 10, 2013
H-polyhedra
![Page 17: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/17.jpg)
17Jayant Apte. ASPITRGApril 10, 2013
Example(d=2,d+1=3)
![Page 18: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/18.jpg)
18Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 19: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/19.jpg)
19Jayant Apte. ASPITRGApril 10, 2013
V-polyhedra
![Page 20: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/20.jpg)
20Jayant Apte. ASPITRGApril 10, 2013
Polar of a convex cone
![Page 21: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/21.jpg)
21Jayant Apte. ASPITRGApril 10, 2013
Polar of a convex cone
![Page 22: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/22.jpg)
22Jayant Apte. ASPITRGApril 10, 2013
Polar of a convex cone
H-representation V-representation
H-representationV-representation
Original space Polar/dual space
![Page 23: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/23.jpg)
23Jayant Apte. ASPITRGApril 10, 2013
Equivalence of vertex-enumeration and facet-enumeration
![Page 24: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/24.jpg)
24Jayant Apte. ASPITRGApril 10, 2013
Equivalence of vertex-enumeration and facet-enumeration
Perform Vertex Enumeration on this cone.
![Page 25: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/25.jpg)
25Jayant Apte. ASPITRGApril 10, 2013
Equivalence of vertex-enumeration and facet-enumeration
Then take polar again to get facets of this cone
Perform Vertex Enumeration on this cone.
![Page 26: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/26.jpg)
26Jayant Apte. ASPITRGApril 10, 2013
Minimality of H-representation
● If an inequality can be removed from an H-representation of a polyhedron without changing the polyhedron, then that inequality is said to be redundant.
● An H-representation is minimal if there are no redundant inequalities
![Page 27: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/27.jpg)
27Jayant Apte. ASPITRGApril 10, 2013
Minimality of H-representation• Magenta inequality can be removed
without changing the polyhedron• Magenta inequality is redundant
![Page 28: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/28.jpg)
28Jayant Apte. ASPITRGApril 10, 2013
Minimality of V-representation
● If an extreme point/extreme ray can be removed from a V-representation of a polyhedron without changing the polyhedron, then that extreme point/extreme ray is said to be redundant.
● A V-representation is minimal if there are no redundant extreme points/extreme rays
![Page 29: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/29.jpg)
29Jayant Apte. ASPITRGApril 10, 2013
Minimality of V-representation
![Page 30: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/30.jpg)
30Jayant Apte. ASPITRGApril 10, 2013
Minimality of V-representation
The red points are redundant
![Page 31: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/31.jpg)
31Jayant Apte. ASPITRGApril 10, 2013
Algorithm ILexicographic Reverse Search
![Page 32: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/32.jpg)
32Jayant Apte. ASPITRGApril 10, 2013
Lexicographic Reverse Search
● A pivoting algorithm● Based on variant of Simplex Method called
Lexicographic Simplex Method
![Page 33: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/33.jpg)
33Jayant Apte. ASPITRGApril 10, 2013
A linear program
(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)
(1,0,1)
![Page 34: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/34.jpg)
34Jayant Apte. ASPITRGApril 10, 2013
Add slack variables
No. of variables=n=12No. of dimensions=d=3
![Page 35: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/35.jpg)
35Jayant Apte. ASPITRGApril 10, 2013
Co-basis(N) and Basis(B)d-subset of slack variables that are 0={ 9,10,11}: Co-basisRemaining n-d variables can be grouped together: Basis
![Page 36: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/36.jpg)
36Jayant Apte. ASPITRGApril 10, 2013
Co-basis(N) and Basis(B)
(0,0,1)
d-subset of slack variables that are 0={ 7,9,11}
![Page 37: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/37.jpg)
37Jayant Apte. ASPITRGApril 10, 2013
Degeneracy
(0,0,1)
Vertex (0,0,1) has more than one co-bases It is called a degenerate extreme point
![Page 38: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/38.jpg)
38Jayant Apte. ASPITRGApril 10, 2013
Lexicographic Simplex MethodOverview
● Simplex Method maximizes/minimizes a linear objective function over a polytope/polyhedron
● Uses dictionary as a primary data structure: Every basis-cobasis pair has a dictionary corresponding to it
● Choose entering basis using least subscript rule. If none is found, we've reached optimum
● Choose leaving the basis and going into co-basis using lexicographic pivot selection rule. If none is found, problem is unbounded
● Obtain the next dictionary corresponding to new basis-cobasis pair by doing the pivot operation denoted as pivot(r,s)
![Page 39: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/39.jpg)
39Jayant Apte. ASPITRGApril 10, 2013
Lexicographic simplex on our 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)
(1,0,1)
![Page 40: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/40.jpg)
40Jayant Apte. ASPITRGApril 10, 2013
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(r,s): pivot(r,s)
![Page 41: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/41.jpg)
41Jayant Apte. ASPITRGApril 10, 2013
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=(0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(r,s): pivot(r,s)
![Page 42: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/42.jpg)
42Jayant Apte. ASPITRGApril 10, 2013
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(r,s): pivot(r,s)
![Page 43: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/43.jpg)
43Jayant Apte. ASPITRGApril 10, 2013
P(9,5)
V=(1 1 1)N=(6 8 5)
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(7,6)
V=(0.5 0.5 1.5)N=(6 8 9)
P(11,8)
V=(0.5 0.5 1.5)N=(7 8 9)
P(10,7)V=(0 0 1)N=(7 11 9)
P(12,9)
V=(0 0 1)N=(10 11 9)
P(r,s): pivot(r,s)
![Page 44: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/44.jpg)
44Jayant Apte. ASPITRGApril 10, 2013
P(9,5)
V=(1 1 1)N=(6 8 5)
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(7,6)
V=(0.5 0.5 1.5)N=(6 8 9)
P(11,8)
V=(0 0 1)N=(7 8 9)
P(10,7)V=(0 0 1)N=(7 11 9)
P(12,9)
V=(0 0 1)N=(10 11 9)
P(9,8)
V=(1 0 1)N=(8 12 9)
P(r,s): pivot(r,s)
![Page 45: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/45.jpg)
45Jayant Apte. ASPITRGApril 10, 2013
P(9,5)
V=(1 1 1)N=(6 8 5)
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(7,6)
V=(0.5 0.5 1.5)N=(6 8 9)
P(11,8)
V=(0 0 1)N=(7 8 9)
P(10,7)V=(0 0 1)N=(7 11 9)
P(12,9)
V=(0 0 1)N=(10 11 9)
P(9,8)
V=(1 0 1)N=(8 12 9)
P(11,6)
V=(0 1 1)N=(10 6 9)
P(r,s): pivot(r,s)
![Page 46: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/46.jpg)
46Jayant Apte. ASPITRGApril 10, 2013
P(9,5)
V=(1 1 1)N=(6 8 5)
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(7,6)
V=(0.5 0.5 1.5)N=(6 8 9)
P(11,8)
V=(0 0 1)N=(7 8 9)
P(10,7)V=(0 0 1)N=(7 11 9)
P(12,9)
V=(0 0 1)N=(10 11 9)
P(9,8)
V=(1 0 1)N=(8 12 9)
P(11,6)
V=(0 1 1)N=(10 6 9)
P(r,s): pivot(r,s)
![Page 47: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/47.jpg)
47Jayant Apte. ASPITRGApril 10, 2013
P(9,5)
V=(1 1 1)N=(6 8 5)
P(12,6)
V=(0 1 1)N=(10 5 6)
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
P(10,4)
P(12,8)
P(11,5)
V=(1 0 1)N=(4 11 8)
V=(1 1 1)N=(4 5 8)
P(11,5)
P(10,4)
V=0 1 0)N=(10 5 12)
V=(1 1 0)N=(4 5 12)
P(7,6)
V=(0.5 0.5 1.5)N=(6 8 9)
P(11,8)
V=(0 0 1)N=(7 8 9)
P(10,7)V=(0 0 1)N=(7 11 9)
P(12,9)
V=(0 0 1)N=(10 11 9)
P(9,8)
V=(1 0 1)N=(8 12 9)
P(11,6)
V=(0 1 1)N=(10 6 9)
P(r,s): pivot(r,s)
●Tree formed by tracing all possible pathsof simplex method
●Reverse the direction of edges to get the reverse search tree
![Page 48: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/48.jpg)
48Jayant Apte. ASPITRGApril 10, 2013
ЯEVERSE Search
1. Start with dictionary corresponding to optimum vertex
2. Let current basis be B
3. For a certain and any is there a valid simplex pivot from dictionary corresponding to to the current dictionary?
4. Denoted as reverse(s), for and returns if answer is yes else returns 0
5. If do pivot(r,s), go down the reverse search tree by recursively performing 2-5
6. If reverse(s) returns 0 for all go back 1 level up the tree using ordinary simplex pivot
![Page 49: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/49.jpg)
49Jayant Apte. ASPITRGApril 10, 2013
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
R(10)=4P(4,10)
R(11)=5p(5,11)
R(12)=9P(9,12)
R(12)=8P(8,12)
R(11)=5P(5,11)
R(10)=4P(4,10)
R(12)=6P(6,12)
R(11)=6P(6,11)
R(10)=7P(7,10)
R(9)=8P(8,9)
R(11)=8P(8,11)
R(7)=6P(6,7)
R(9)=5P(5,9)
V=(1 0 1)N=(4 11 8)
V=0 1 0)N=(10 5 12)
V=(0 0 1)N=(10 11 9)
V=(1 1 0)N=(4 5 12)
R(5)=0
R(4)=0
R(11)=0
R(8)=0
R(9)=0
V=(0 1 1)N=(10 5 6)
V=(0.5 0.5 1.5)N=(7 8 9)
V=(1 0 1)N=(8 12 9)
V=(1 1 1)N=(6 8 5)
V=(1 1 1)N=(5 6 9)
V=(0 1 1)N=(10 6 9)
V=(0 0 1)N=(7 11 9)
R(9)=0
R(5)=0 R(6)=0 R(9)=0
R(7)=0
R(8)=0 R(12)=0
R(9)=0
V=(0.5 0.5 1.5)N=(6 8 9)
R(6)=0 R(8)=0 R(5)=0
R(6)=0R(8)=0 R(8)=0 R(12)=0 R(9)=0
R(4)=0 R(11)=0
R(4)=R(5)=R(12)
R(10)=R(5)=R(6)
R(s): reverse(s)P(r,s): pivot(r,s)
![Page 50: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/50.jpg)
50Jayant Apte. ASPITRGApril 10, 2013
V=(0 0 0)N=( 10 11 12)
V=(1 0 0)N=(4 11 12)
R(10)=4P(4,10)
R(11)=5p(5,11)
R(12)=9P(9,12)
R(12)=8P(8,12)
R(11)=5P(5,11)
R(10)=4P(4,10)
R(12)=6P(6,12)
R(11)=6P(6,11)
R(10)=7P(7,10)
R(9)=8P(8,9)
R(11)=8P(8,11)
R(7)=6P(6,7)
R(9)=5P(5,9)
V=(1 0 1)N=(4 11 8)
V=0 1 0)N=(10 5 12)
V=(0 0 1)N=(10 11 9)
V=(1 1 0)N=(4 5 12)
R(5)=0
R(4)=0
R(11)=0
R(8)=0
R(9)=0
V=(0 1 1)N=(10 5 6)
V=(0.5 0.5 1.5)N=(7 8 9)
V=(1 0 1)N=(8 12 9)
V=(1 1 1)N=(6 8 5)
V=(1 1 1)N=(5 6 9)
V=(0 1 1)N=(10 6 9)
V=(0 0 1)N=(7 11 9)
R(9)=0
R(5)=0 R(6)=0 R(9)=0
R(7)=0
R(8)=0 R(12)=0
R(9)=0
V=(0.5 0.5 1.5)N=(6 8 9)
R(6)=0 R(8)=0 R(5)=0
R(6)=0R(8)=0 R(8)=0 R(12)=0 R(9)=0
R(4)=0 R(11)=0
R(4)=R(5)=R(12)
R(10)=R(5)=R(6)
R(s): reverse(s)P(r,s): pivot(r,s)
![Page 51: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/51.jpg)
51Jayant Apte. ASPITRGApril 10, 2013
Problems with pivoting methods
● Degeneracy● Duplicate output of extreme points
![Page 52: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/52.jpg)
52Jayant Apte. ASPITRGApril 10, 2013
How Lexicographic Simplex deals with them
● Degeneracy– Lexicographic Simplex Method visits only a subset of
bases called Lex-positive Bases
● Duplicate output extreme points– Out of the lex-positive basis we can identify a unique basis
called Lex-min Basis corresponding to each extreme point
– Output extreme point only if current basis is lex-min
● These features make Lexicographic simplex best choice for reverse search
![Page 53: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/53.jpg)
53Jayant Apte. ASPITRGApril 10, 2013
Algorithm IIDouble Description Method
![Page 54: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/54.jpg)
54Jayant Apte. ASPITRGApril 10, 2013
Definitions
![Page 55: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/55.jpg)
55Jayant Apte. ASPITRGApril 10, 2013
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.
![Page 56: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/56.jpg)
56Jayant Apte. ASPITRGApril 10, 2013
How it works?
![Page 57: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/57.jpg)
57Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 58: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/58.jpg)
58Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 59: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/59.jpg)
59Jayant Apte. ASPITRGApril 10, 2013
Example
Consider a DD pair:
Insert new constraint:
![Page 60: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/60.jpg)
60Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 61: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/61.jpg)
61Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 62: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/62.jpg)
62Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 63: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/63.jpg)
63Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 64: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/64.jpg)
64Jayant Apte. ASPITRGApril 10, 2013
Compute new rays(DD Lemma)
![Page 65: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/65.jpg)
65Jayant Apte. ASPITRGApril 10, 2013
New DD pair
![Page 66: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/66.jpg)
66Jayant Apte. ASPITRGApril 10, 2013
New cone
![Page 67: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/67.jpg)
67Jayant Apte. ASPITRGApril 10, 2013
Minimality of representation
● New ray AD generated above is redundant● What to do?
– Generate new rays for only those positive-negative ray pairs that are adjacent
– Can check adjacency using either
combinatorial adjacency oracle or algebraic adjacency oracle
● Prevents combinatorial explosion of number of extreme rays
![Page 68: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/68.jpg)
68Jayant Apte. ASPITRGApril 10, 2013
Algorithm IIIConvex Hull Method
![Page 69: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/69.jpg)
69Jayant Apte. ASPITRGApril 10, 2013
Polyhedral Projection
![Page 70: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/70.jpg)
70Jayant Apte. ASPITRGApril 10, 2013
Example
![Page 71: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/71.jpg)
71Jayant Apte. ASPITRGApril 10, 2013
CHM intuition (12,6,6)
(12,6)
![Page 72: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/72.jpg)
72Jayant Apte. ASPITRGApril 10, 2013
How it works...
● If projection dimension=d, first find d+1 extreme points of projection and their convex hull using procedure called initialhull()
● Initialhull() gives us first approximation of projection ● Every iteration find one new extreme point of projection
and compute convex hull corresponding to pre-existing extreme points and the new extreme point
● We stop when all the facets of current approximation are facets of
![Page 73: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/73.jpg)
73Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
initialhull( )
![Page 74: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/74.jpg)
74Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 75: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/75.jpg)
75Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 76: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/76.jpg)
76Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 77: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/77.jpg)
77Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 78: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/78.jpg)
78Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 79: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/79.jpg)
79Jayant Apte. ASPITRGApril 10, 2013
Finding the first d+1 points of projection
![Page 80: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/80.jpg)
80Jayant Apte. ASPITRGApril 10, 2013
Fact
● The cost functions for finding the extreme points of projection can be obtained from facets of that are not the facets of
● Checking whether a facet of is a facet of can be accomplished by simply running a linear program over
![Page 81: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/81.jpg)
81Jayant Apte. ASPITRGApril 10, 2013
CHM
?
?
?
![Page 82: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/82.jpg)
82Jayant Apte. ASPITRGApril 10, 2013
CHMNot a facet of
![Page 83: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/83.jpg)
83Jayant Apte. ASPITRGApril 10, 2013
CHM
![Page 84: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/84.jpg)
84Jayant Apte. ASPITRGApril 10, 2013
CHM
![Page 85: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/85.jpg)
85Jayant Apte. ASPITRGApril 10, 2013
Updating the current hull to include new extreme
point of projectionupdatehull( )
![Page 86: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/86.jpg)
86Jayant Apte. ASPITRGApril 10, 2013
CHM
Existing hull
New Vertex
![Page 87: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/87.jpg)
87Jayant Apte. ASPITRGApril 10, 2013
CHM
Existing hull
New Vertex
![Page 88: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/88.jpg)
88Jayant Apte. ASPITRGApril 10, 2013
Updating hull via iteration of DD Method
Homogenization Polar
DD Iteration
Polar Again
ReverseHomogenization
Old Hull
New Hull
![Page 89: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/89.jpg)
89Jayant Apte. ASPITRGApril 10, 2013
CHM
![Page 90: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/90.jpg)
90Jayant Apte. ASPITRGApril 10, 2013
CHM
![Page 91: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/91.jpg)
91Jayant Apte. ASPITRGApril 10, 2013
CHM
![Page 92: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/92.jpg)
92Jayant Apte. ASPITRGApril 10, 2013
Runtime Comparison
![Page 93: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/93.jpg)
93Jayant Apte. ASPITRGApril 10, 2013
Demonstration
![Page 94: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/94.jpg)
94Jayant Apte. ASPITRGApril 10, 2013
Questions
![Page 95: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/95.jpg)
95Jayant Apte. ASPITRGApril 10, 2013
Vertices of
![Page 96: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/96.jpg)
96Jayant Apte. ASPITRGApril 10, 2013
Vertices of
![Page 97: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/97.jpg)
97Jayant Apte. ASPITRGApril 10, 2013
Vertices of
![Page 98: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/98.jpg)
98Jayant Apte. ASPITRGApril 10, 2013
Vertices of
![Page 99: Candidacy Exam Talk](https://reader034.vdocuments.net/reader034/viewer/2022042715/559c56711a28aba31c8b47e0/html5/thumbnails/99.jpg)
99Jayant Apte. ASPITRGApril 10, 2013
Vertices of