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POLYHEDRAL POLARITIES
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JULIAN ARAOZ
Universidad Simón Bolı́var
To Amanda
ii
Preface
I have enjoyed writing this book. It is mainly concerned with my research
work of the past 25 years, beginning in 1972 at the University of Waterloo,
when Jack Edmonds and I learned of the work of Ralph Gomory (which later
became known as Gomory’s Group Problems). D. R. Fulkerson has also had
a profound influence. I admire their contributions to our field and want to
thank them for the inspiration they have given me.
Special thanks to Jack for his guidance and friendship, and to all my
students and co-authors.
Part of this book was written at The Geometry Center, University of
Minnesota at Minneapolis, whom I thank for their support.
Without the support of people at the “Unidad de Gestión de Informática,
FUNINDES-USB” in the last month this work would not have been ready
on time. Accordingly I extend my thanks to Patricia Barra, Claudia Cermeli
and Xiomara Uzcátegui.
This is where authors traditionally insert something like “The chores of
deciphering my untidy handwritten drafts and of retyping endless revisions
were done with the utmost care by . . . ”. Unhappily the advent of PCs and
LATEX means that I have to do all this myself. Perhaps the old way was
better.
During the past six years, due to a combination of circumstances, I have
not developed this material at a constant pace. However the “ESCUELA
VENEZOLANA DE MATEMATICAS” and my preparation of these notes
has revived my enthusiasm. I will certainly remain indebted to the organizers
for giving me a strong push.
Finally, I thank Amanda Salceda de Aráoz for her infinite patience during
the evenings, nights and weekends spent by her husband in writing. I thank
her for her love, and hope to make amends.
JULIAN ARAOZ
iii
iv
Contents
1 Introduction
1.1 Motivations . . . . . . . . . . . . . . .
1.1.1 Linear Programming . . . . . .
1.1.2 Integer Programming . . . . . .
1.1.3 Combinatorial Optimization . .
1.2 Polarities . . . . . . . . . . . . . . . .
1.2.1 Cone Polarity . . . . . . . . . .
1.2.2 Polyhedral Polarity . . . . . . .
1.3 Semigroup Problems and Neopolarities
1.4 Notes . . . . . . . . . . . . . . . . . . .
2 Polyhedra Theory
2.1 Elementary Linear Algebra . . . .
2.1.1 Vectors . . . . . . . . . . .
2.1.2 Matrices . . . . . . . . . .
2.1.3 Linear Systems . . . . . .
2.2 Polyhedra . . . . . . . . . . . . .
2.2.1 Valid Inequalities . . . . .
2.2.2 Convex Sets and Cones . .
2.3 Faces and Facets . . . . . . . . .
2.4 Internal Polyhedra Representation
2.5 Polyhedral Cones . . . . . . . . .
2.5.1 Convex Cone Basis . . . .
2.6 Notes . . . . . . . . . . . . . . . .
2.7 Exercises . . . . . . . . . . . . . .
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1
1
2
3
6
9
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11
12
14
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17
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18
19
20
21
22
24
27
29
30
32
32
vi
3 Convex General Polarity
3.1 General Polarity . . . . . . . . . . . . . . .
3.1.1 Cone Polarity . . . . . . . . . . . .
3.2 General Bilinear Inequality Polarity . . . .
3.2.1 Ω-Polar of Finitely Generated Sets
3.2.2 Ω-Polar Types . . . . . . . . . . . .
3.3 Bilinear System Relation Polarities . . . .
3.4 Notes . . . . . . . . . . . . . . . . . . . . .
3.5 Exercises . . . . . . . . . . . . . . . . . . .
Bibliography
CONTENTS
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37
37
40
42
44
47
51
52
53
56
List of Figures
1.1
Feasible Solutions . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Integer Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Integer Problem Cut . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Minimum Spanning Tree Problem . . . . . . . . . . . . . . . .
7
1.5
Minimum Spanning Tree Solution . . . . . . . . . . . . . . . .
7
1.6
Minimum Matching Problem . . . . . . . . . . . . . . . . . . .
8
1.7
Minimum Matching Solution . . . . . . . . . . . . . . . . . . .
8
1.8
Cone Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9
Polarity xy ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.10 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1
Polyhedron Elements . . . . . . . . . . . . . . . . . . . . . . . 26
vii
viii
LIST OF FIGURES
List of Tables
1.1
Cyclic Semigroup of Order 5 . . . . . . . . . . . . . . . . . . . 13
1.2
Facets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1
Ω-polar Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
ix
x
LIST OF TABLES
Chapter 1
Introduction
Polyhedral Polarity is based in the work of Farkas about Polar Cones.
Polar Cones is in the foundations of the solution of Linear Inequalities
Systems and Farkas’s result about polar cones equivalent to the important
theorems of Alternatives and to the Strong Duality Theorem in Linear Programming and was used to prove the Strong Duality Theorem.
Although some extension have been studied before, the development began with the development of Combinatorial Optimization after the milestone
works of Edmonds “Paths, Trees and Flowers” and “Maximum Matching and
a Polyhedron with 0-1 Vertices”.
The special cases of Fulkerson’s Blocking and Antiblocking and Gomory’s
Group Problems inspired Aráoz and Edmonds work in Polarity Types and
Neopolarities. A general Convex Polarity for polyhedra was then developed
by the works of Aráoz, Edmonds and Griffin.
1.1
Motivations
AlthoughPolyhedra Theory is the study of solution sets to systems of linear
inequalities and these are related to feasible solutions to linear programs,
the main motivation for as to study polyhedra and polyhedral polarities is
1
2
CHAPTER 1. INTRODUCTION
Integer Programming and Combinatorial Optimization.
We introduce these subjects using examples, the formal definitions that
we need are given in later chapters.
1.1.1
Linear Programming
A Linear Programming is to maximize a linear functional over a set of linear
equations and inequalities.
Typically they look like the Example 1.1, however, usually they have
much more variables and inequalities, they could be in the thousands, even
in several hundred thousands.
Example 1.1 A particular case of Linear Programming is:
max z = 2x1 + x2

A) −x1




B)


 C)

+ 2x2
5x1 + x2
2x1 + 2x2
x1 ,
x2
≤ 4
≤ 20
≥ 7
≥ 0
(1.1)
(1.2)
The solutions to the inequalities 1.2 are called feasible solutions. The set
of feasible solutions is a polyhedron.
We show them graphically in Figure 1.1, this is the reason to use only two
variables since then is easy to draw. The points {a, b, c, d} are the vertices and
the segments {[a, b][b, c][c, d][d, a]} are the facets of the polyhedron. Vertices
and facets are among the faces.
A bounded polyhedron is also the convex hull of it vertices. For unbounded polyhedra it is necessary to take in account the extreme rays also.
The inequality x1 ≥ 0 is redundant, since we could delete it without
changing the polyhedron, the others are facet inducing.
1.1. MOTIVATIONS
3
x2
46
A
A D
AD a
t
A
DA
D
DA
A 4.5
D A10.18
3
D
@ A
D
@A
D
@A t
A
D
@
d A@
D
A
D
@
A
D
@
2
D
@
D
@
@
D
D
@
C @
D
D
@
1
D
@
DB
@
D
@
@ 7
0
8 D
A
A D
@ AA
A
A
@A
A
AD b
@At
ADt- x1
At
c
A
A
1
2
3
4A
A
A
A
A
A
Figure 1.1: Feasible Solutions A
In the figure are also shown the value of the functional in the vertices,
the optimum is in vertex a with a value of 10.18. The optimum is always in
a vertex.
The Simplex Algorithm provide an efficient method to solve this problems.
It began in a vertex and move to adjacent vertices with a better value of the
functional until it reach the optimum.
1.1.2
Integer Programming
An Integer Programming is a Linear Programming where some or all the
variables are required to have integer values.
4
CHAPTER 1. INTRODUCTION
Because the flexibility of Integer Programming to model many situations
a rich variety of problems can be represented.
If in the Example 1.1 the variable are required to be integers we obtain
the integer program of Example 1.2
Example 1.2 A particular case of Integer Programming is:
max z = 2x1 + x2


A) −x1





 B) 5x1
C)







2x1
x1
x1
+ 2x2
≤
4
+ x2
≤
20
+ 2x2
≥
7
,
x2
≥
0
,
x2 integer
(1.3)
(1.4)
The system above is equivalent to the system in Example 1.3, which is the
convex hull of the integer feasible solutions.
Example 1.3 The same problem of Example 1.2 could be stated as:
max z = 2x1 + x2

D)



 E)
x1
x2
3x1 + x2
G) x1 + x2

F)



≥ 2
≤ 3
≤ 12
≥ 4
(1.5)
(1.6)
Both polyhedra are shown in Figure 1.2.
As we see, if we have the system 1.6, we could use the Simplex algorithm
to obtain the optimum value 9. However if we have the system in 1.4 we get
1.1. MOTIVATIONS
5
x2
rA
r
A D
AD
t
A
DA
D
A
A
DA
A
A D A10.18 r
r
v
Av
3
D
BA
@
E
D
@
BA
D
@ B
D
A9 D
@
B
@
D
B
D
@
B
r
v
v
F
r
@
2
B D
@
D
@
B
@
@
B D
@
@
B D
@
@
B D
C @
@
B D
@
@
G
B D r
r
r
v
1
@
@
B D
@
@
BDB
@
@
BD
@
@
BD
@ BD
@
@ BD
@
@BDv- x1
@
46
r
r
1
2
3
4
Figure 1.2: Integer Problem
a wrong answer 10.18 when applying the Simplex Algorithm to the problem
in Example 1.2 without the integer condition.
The system 1.4 without the integer condition is call a relaxation of the
integer problem, in the example we obtain the system 1.2.
Several of the methods to solve Integer Programming problems use the
Simplex Algorithm in a relaxation of the problem and when the solution
is not integer look for a valid inequality that cuts the non-integer solution
found. These inequalities are called Cuts, that is valid inequalities for the
integer program which approximate the convex hull at the optimal point.
For example, if we add inequalities A, B and E we obtain the valid in-
6
CHAPTER 1. INTRODUCTION
x2
46
q
q
q
q
DD
D
@@ A D
A
A
q
s q
3
@s A D
@
@ AD
@ @DAr
@
D@
@
DA @H
q
s
s
q
2
@
DA
@
D A9.67
@
D
C @
D
q
q @
s
1
D q
@
DB
@
D
@
D
@
Ds x1
1
2
3
4
Figure 1.3: Integer Problem Cut
equality
4x1 + 4x2 ≤ 27
which is equivalent to
27
= 6.75.
4
Since the variables are integer, then the following inequality has to be valid.
x1 + x2 ≤
H) x1 + x2 ≤ 6 = b
27
c.
4
We could see this cut in Figure 1.3.
The best cuts we could found are the facets of the convex hull of integer
solutions to the problem. Gomory’s Group Problems was developed to study
properties of these facets.
1.1.3
Combinatorial Optimization
Combinatorial Optimization is, mainly, the optimization of a linear functional
over a finite set that admits a combinatorial description, usually, but not
always, over a graph.
1.1. MOTIVATIONS
7
Well known examples are the Traveling Salesman Problem, The Minimum
Spanning Tree, Matchings, Covering and many others.
In Figure 1.4 we give an example of a Minimum Spanning Tree problem
and in Figure 1.5 its solution.
@
@
@
B
9
HH
H
HH
7
2
D
A
7
F
@
@
@
@ @
4
HH
H
H
HH
H
3
@
@
HH
H
6
9
HH
HH
6
C
E
Figure 1.4: Minimum Spanning Tree Problem
HH
H
B
D
@
@
@
2
A
3
@
@
HH
H
4
7
HH
H
HH
HH
C
6H
HH
HH
H
F
@
@
@
@ @
E
Figure 1.5: Minimum Spanning Tree Solution
The development of Combinatorial Optimization began after the milestone works “Paths, Trees and Flowers”, where Edmonds define what is a
“Good Combinatorial Algorithm”, and “Maximum Matching and a Polyhedron with 0-1 Vertices”, where Edmonds relate the solution of combinatorial
problems to the convex hull of characteristic vectors corresponding to feasible
solutions.
8
CHAPTER 1. INTRODUCTION
In Figure 1.6 we give an example of a Minimum Perfect Matching problem
and in Figure 1.7 its solution.
@
@
@
B
9
HH
H
HH
H
HH
H
7
D
2
A
3
@
@
H
HH
7
F
@
@
@
@ @
4
H
HH
6
9
H
HH
H 6
C
E
Figure 1.6: Minimum Matching Problem
HH
H
HH
B
D
@
@
@
@
@
2
A
H
HH
6
H
HH
H
F
@
7
H
HH
H
HH
H C
@
@
@ @
E
Figure 1.7: Minimum Matching Solution
Example 1.4 A combinatorial problem, which is not over a graph, is the
Set Covering Problem.
Given a finite set E, a family F of subsets of E covering E and a function
c : F → IR+ find a subfamily J ⊆ F such that
X
j∈J
is minimum and J still covers E.
c(j)
1.2. POLARITIES
9
To put this problem as an integer program, let A be a matrix with a row
for each element of E and a column for each element of F.
Define aij = 1 if i ∈ j and 0 otherwise, that is, a column of A is the
characteristic vector of the corresponding set in F.
It is not difficult to see that the solution to the covering correspond to
the solution of the integer problem
min
X
c(j)xj
j∈F
Ax ≥ ~1
Where x is a non-negative integer vector and ~1 is a vector of 1’s.
In this way the solution of combinatorial problems is associated to special
Integer Programming problems and the search for the facets of these problems
became relevant.
1.2
Polarities
Polarities is the main subject of this book. We give here a couple of examples.
1.2.1
Cone Polarity
Given a set P ⊆ IRn the polar of P , as study by Farkas, is
P γ = {y ∈ IRn : xy ≥ 0 ∀x ∈ P }.
Farkas proved that:
i) P γ is a convex cone,
ii) P = P γγ if and only if P is a convex cone,
iii) P γ is a polyhedral convex cone when P is a polyhedron.
10
CHAPTER 1. INTRODUCTION
among other properties.
That is, the set of homogeneous inequalities of a set is a convex cone. We
show an example of this.
x2
KA
A
6
γ
C
A
A
A
C
*
A
A
A A
- x1
HH
H
HH
H
HH
j
H
Figure 1.8: Cone Polarity
Example 1.5 Let C be the cone defined by the system:
(
2x1 − x2 ≥ 0
−x1 + 2x2 ≥ 0
Then C γ is the solution set of the system:
(
and C = C γγ (see Figure 1.8).
2x1 + x2 ≥ 0
x1 + 2x2 ≥ 0
1.2. POLARITIES
1.2.2
11
Polyhedral Polarity
Given a set P ⊆ IRn another type of polar of P is
P ω = {y ∈ IRn : xy ≤ 1 ∀x ∈ P }.
That is, P ω is the set of valid inequalities of P of the form xy ≤ 1. As we
see polarities could be used to study inequalities of a given form. We show
an example of this.
x2
1 6
P
@
@
@
@
@
@
@
Pω
-1@@
@
@- x1
1
@
@
@
@
@
@
-1@
Figure 1.9: Polarity xy ≤ 1
Example 1.6 Let P be the polyhedron defined by the system:

x1 ≤ 1




x2 ≤ 1
 x1 ≥ −1



x2 ≥ −1
12
CHAPTER 1. INTRODUCTION
Then P ω is the solution set of the system:

x1 + x2 ≤ 1




−x1 + x2 ≤ 1

x1 − x2 ≤ 1



−x1 − x2 ≤ 1
and P = P ωω (see Figure 1.9).
1.3
Semigroup Problems and Neopolarities
Gomory’s Group Problems was developed to study mathematical properties
of facets for the convex hull of integer solutions of integer problems.
The theory was extended to semigroups, first to Covering Semigroups
and latter to semigroups which model any integer problem with non-negative
coefficients.
The result related to polarity theory is that the facets other than nonnegativity are the minimal extreme rays of the cone of subadditive functions
over the semigroup and in many case we could normalize to the vertices of
a related polyhedron. These sets of subadditive functions are what we call a
Neopolar.
Notice that the Covering Problem given in Example 1.4 is a problem over
a semigroup, because the set union operator define a semigroup over the
subsets. Next we use an example of the Knapsack Problem to show how to
transform it to a Semigroup Problem.
Example 1.7 A Covering Knapsack Problem is an Integer Problem with
only one inequality restriction. Consider the inequality:
x1 + 2x2 + 3x3 + 4x4 + 5x5 ≥ 5
where we look for integer non-negative solutions.
The main question is, what are the facets of the convex hull others that
the non-negative restrictions?.
1.3. SEMIGROUP PROBLEMS AND NEOPOLARITIES
+̂
1
2
3
4
5
1
2
3
4
5
5
2
3
4
5
5
5
3
4
5
5
5
5
4
5
5
5
5
5
13
5
5
5
5
5
5
Table 1.1: Cyclic Semigroup of Order 5
We define the equivalent semigroup given in Table 1.1.
The Subadditivity Theory say that the facets, normalized to right-hand
side 1, correspond to the vertices of the polyhedron:

π(g) + π(h) ≥ π(g +̂h) ∀g, h ∈ {1, 2, 3, 4, 5}



 π(g) + π(5 − g) = 1
∀g ∈ {1, 2, 3, 4, 5}

π(5)
=
1



π(g) ≥ 0
∀g ∈ {1, 2, 3, 4, 5}
The facets of the convex hull others that the non-negative restrictions are
of the form
π1 x1 + π2 x2 + π3 x3 + π4 x4 + π5 x5 ≥ π5
and the coefficients are given in Table 1.2.
a
b
c
d
π1
1
1
1
1
π2
2
2
1
1
π3
3
2
1
2
π4
4
3
1
2
π5
5
4
2
3
Table 1.2: Facets
This coefficients could be obtained using the equations
π(4) = 1 − π(1), π(3) = 1 − π(2), π(5) = 1
to reduce to a polyhedron in π(1), π(2) whose vertices are shown in Figure 1.10.
14
CHAPTER 1. INTRODUCTION
π2
6
br
1
2
2
5
1
3
d
r
ar
HH
H
H
Hr
c
1
5
1
4
1
3
1
2
- π1
Figure 1.10: Vertices
1.4
Notes
For Section 1.1.1 any book in Linear Programming is a good reference, like
[Bazaraa & Jarvis 77], [Murty] and [Schriver 86].
For Section 1.1.2 Integer Programming a classical reference is the book
of [Garfinkel & Nemhauser] or the more recent books of [Schriver 86] and
[Nemhauser & Wolsey 88].
This last two books are also a good reference for Section 1.1.3 Combinatorial Optimization.
Although some extension to polarities have been studied before, the development began with the development of Combinatorial Optimization after
the milestone works of Edmonds “Paths, Trees and Flowers” and “Maximum
Matching and a Polyhedron with 0-1 Vertices”.
This book is, mainly about Section 1.2 Polarities, early results are in
[Rockafellar 69] and [Aráoz 74].
For the last Section 1.3, of this chapter Semigroup Optimization and
1.4. NOTES
15
Neopolarities, in this book we cover mainly the topic related to Neopolarities,
however, Semigroup Optimization was the main motivation to develop de
Polarity Types. In this topic there is a lot more work done. It began with
Gomory’s early works [Gomory 65] and [Gomory 67], but it is in [Gomory 69]
the theory of Gomory’s Group Problems is developed. Aráoz and Edmonds
extend it to Semigroups Problems and introduce the idea of Neopolarity (see
[Aráoz 74]), this idea was defined in [Aráoz & Johnson 92]. A good book
about it is [Johnson 80] and an introduction to the topic is [Aráoz 96].
16
CHAPTER 1. INTRODUCTION
Chapter 2
Polyhedra Theory
In this chapter we introduce the elements of Linear Algebra that we need in
the rest of the book.
2.1
Elementary Linear Algebra
We will work in real space IRN , where N will denote a finite set of indices
and the dimension of IRN is the cardinality |N | of N . However the theory
exposed in this book is still valid for any space over a ring. The elements of
IRN will be called, indistinctly, vectors or points.
2.1.1
Vectors
Notation 2.1 Let x ∈ IRJ and y ∈ IRK , < x, y >∈ IRJ∪K will denote the
concatenation of x with y, in this operation we always assume J ∩ K = ∅.
For any scalar α we denote by α
~ a vector which components are all equal to
α, hence ~0 is the zero vector and ~1 is a vector of 1’s.
δ j is a vector with all the components equal to 0 but the j th equal to 1,
they are the versors. That is
(
δij
=
1 if j = i
0 if j 6= i
17
18
CHAPTER 2. POLYHEDRA THEORY
Definition 2.2 A set of vectors x1 , . . . , xk ∈ IRN is linearly independent if
and only if
k
X
αj xj = 0 ⇒ αj = 0, ∀ j = 1, . . . , k
(2.1)
j=1
Proposition 2.3 The maximum number of linearly independent points in
IRN is |N |.
Definition 2.4 A set of vectors x1 , . . . , xk ∈ IRN is affinely independent if
and only if
k
X
αj xj = 0,
j=1
k
X
αj = 0 ⇒ αj = 0, ∀ j = 1, . . . , k
(2.2)
j=1
Proposition 2.5 The following statements are equivalent:
a. x0 , . . . , xk ∈ IRN are affinely independent.
b. x1 − x0 , . . . , xk − x0 are linearly independent.
c. < x0 , −1 >, . . . , < xk , −1 >are linearly independent.
Note that the maximum number of affinely independent points in IRN is
|N | + 1 (e.g., n linearly independent points and the zero vector).
2.1.2
Matrices
Some times we are interested in seeing a matrix as a set of vectors and some
others times we will form matrices from sets of vectors or we will concatenate
matrices.
Notation 2.6 A matrix AI×J could be considered as two sets:
The set of rows each denoted by either ai ∈ IRI or Ai ∈ IRI .
2.1. ELEMENTARY LINEAR ALGEBRA
19
The set of columns each denoted by Aj ∈ IRJ .
If we have two matrices AI×J and BI×K , we denote by < A, B > the
matrix whose first |J| columns is A and whose last K columns is B, that is,
the concatenation of A and B.
Vectors are considered either row 1 × J matrices or column I × 1 matrices and in general will be clear from the context. Scalars are some times
considered 1 × 1 matrices.
For example consider the system Ax = b, here x and b will be column
vectors of appropriate dimension, < A, b > is the matrix whose rows are
< ai , bi >, that is, each row of A followed by the corresponding element in b.
However in xA = b, x and b will be row vectors of appropriate dimension.
Proposition 2.7 If A is a I × J matrix, the maximum number of linearly
independent vectors in the set of columns equals the maximum number of
linearly independent vectors in the set of rows.
Definition 2.8 The maximum number of linearly independent rows of a
matrix A is the rank of A and is denoted by rank(A).
Proposition 2.9 The following statements are equivalent:
a. {x ∈ IRJ : Ax = b} =
6 ∅.
b. rank(A) = rank(< A, b >).
Proposition 2.10 If {x ∈ IRJ : Ax = b} 6= ∅, the maximum number of
affinely independent solutions x to Ax = b is |J| + 1 − rank(A).
2.1.3
Linear Systems
In this book we are mainly interested in properties of systems of linear equations and linear inequalities
20
CHAPTER 2. POLYHEDRA THEORY
Definition 2.11 A half space is the solution set of a linear inequality, that
is, given the inequality πx ≤ π◦ the solution set is the half space
{x ∈ IRJ : πx ≤ π◦ },
denoted by P(π, π◦ ).
2.2
Polyhedra
Definition 2.12 A polyhedron P is the set of solutions to a finite set of
linear inequalities, that is, {x ∈ IRJ : Ax ≤ b}, where A is a I × J matrix
and b is a vector in IRI , and is denoted by P(A, b) we say that (A, b) is a
defining system for P .
Clearly each inequality could be of type ≤, = or ≥, but we could consider
them normalized to ≤.
Some times we consider the extended I × (J + 1) matrix < A, b >.
We denote P(A, ~0) by C(A) and it is called a polyhedral cone, that is, a
polyhedral cone is the set of solutions to a finite set of homogeneous linear
inequalities.
Note that for a polyhedron there exists many defining systems.
Definition 2.13 A polyhedron is bounded if it is contained in a ball. A
bounded polyhedron is called a polytope.
Definition 2.14 A polyhedron P ⊆ IRJ is of dimension k, denoted by
dim(P ) = k if the maximum number of affinely independent points in P
is k + 1.
We say that a polyhedron P is full dimensional if dim(P ) = |J|. By
definition dim(∅) = −1.
2.2. POLYHEDRA
2.2.1
21
Valid Inequalities
Definition 2.15 Let P be a set in IRJ . The inequality πx ≤ π◦ is called a
valid inequality for P if it is satisfied by all points in P , this inequality is
represented by the vector < π, π◦ >∈ IRJ+1 .
Note that < π, π◦ > is a valid inequality if and only if P lies in the half
space defined by < π, π◦ >, that is,
P ⊆ {x ∈ IRJ : πx ≤ π◦ },
or equivalently if and only if
max{πx} ≤ π◦ .
x∈P
Lemma 2.16 (Farkas) Let P 6= ∅ be a polyhedron P = P(A, b).
Then πx ≤ πo is a valid inequality for P if and only if there exists λ ≥ ~0
such that
λA = π, λb ≤ πo .
There are many equivalent forms of this lemma. Another which is very
well known is:
Theorem 2.17 (of Alternatives) Let A ∈ IRJ×K and b ∈ IRJ .
Exactly one of the following alternatives must be true.
Either there exists y ∈ IRK such that
Ay ≤ b,
or
there exists x ∈ IRJ+ such that
xA = ~0, xb < 0.
22
CHAPTER 2. POLYHEDRA THEORY
Equivalently
Either there exists y ∈ IRK
+ such that
Ay = b,
or
there exists x ∈ IRJ such that
xA ≤ ~0, xb < 0.
2.2.2
Convex Sets and Cones
Definition 2.18 Let X ⊆ IRJ .
The set X is a convex set if x1 , x2 ∈ X implies
λx1 + (1 − λ)x2 ∈ X
for all 0 ≤ λ ≤ 1, that is the segment [x1 , x2 ] is contained in X.
The set X is a cone if x ∈ X implies
λx ∈ X
for all λ ≥ 0. A convex set which is a cone is called a convex cone.
Note that X is a convex cone if and only if x1 , x2 ∈ X implies x1 +x2 ∈ X
and x ∈ X implies λx ∈ X for all λ ≥ 0.
Definition 2.19 Let X ⊆ IRJ 6= ∅, a point y ∈ X is a recessional direction
of X if and only if
∃x ∈ X such that {z : z = x + λy, λ ∈ IR+ } ⊆ X
The set of recessional directions of X is a cone and is denoted by rec(X).
2.2. POLYHEDRA
23
Proposition 2.20 Let X ⊆ IRJ 6= ∅ be a convex set and y be a recessional
direction of X. Then
∀x ∈ X we have {z : z = x + λy, λ ∈ IR+ } ⊆ X
and rec(X) is a convex cone.
Definition 2.21 Given a finite set X ⊆ IRJ , we say that y ∈ IRJ is a convex
P
combination of X if there exists λ ∈ IRX
x∈X λx = 1 such that
+ with
y=
X
λx x,
x∈X
we said that it is a conical combination of X if there exists λ ∈ IRX
+ such that
y=
X
λx x,
x∈X
and we said that it is a linear combination of X if there exists λ ∈ IRX such
that
X
y=
λx x.
x∈X
Let Y ⊆ IRJ , the convex hull of Y is the set Z satisfying z ∈ Z if and only
if there exists a finite subset Y 0 ⊆ Y such that z is a convex combination of
Y 0.
Similarly we define the conical hull of Y ⊆ IRJ as the set Z satisfying
z ∈ Z if and only if there exists a finite subset Y 0 ⊆ Y such that z is a
conical combination of Y 0 , and we define the linear hull of Y ⊆ IRJ as the
set Z satisfying z ∈ Z if and only if there exists a finite subset Y 0 ⊆ Y such
that z is a linear combination of Y 0 .
Note that if |Y | is finite we could use the set Y instead of Y 0 in the
definition of convex, conical and linear hulls.
Proposition 2.22 The polyhedron P(A, b) is a closed convex set and the
polyhedral cone C(A) is a closed convex cone.
The cone C(A) is the recessional cone of P(A, b).
24
CHAPTER 2. POLYHEDRA THEORY
Definition 2.23 Let X ⊆ IRJ be a convex set.
A point x ∈ X is a vertex if and only if x is not a convex combination of
any two others points in X, that is, for any two points x1 , x2 ∈ X, x1 , x2 6= x,
we have
1
1
x 6= x1 + x2
2
2
A point x ∈ X is a exposed point if and only if there exists π ∈ IRJ and
α such that
α = max{πy}
y∈X
and x is the only point in X satisfying α = πy.
Note that any exposed point is a vertex but no necessarily the converse
hold.
Proposition 2.24 Let P ⊆ IRJ be a polyhedron. The vertices of P are
exposed points.
2.3
Faces and Facets
Informally, a facet is a maximal boundary of a polyhedron which is still a
polyhedron. This facet is also a face, we define in general, recursively, a face
as a facet of any face and include the polyhedron and the empty set as faces
also. Vertices, edges, etc. are faces. Next we give a formal definition.
Definition 2.25 A face of a polyhedron P is the intersection of P with the
set of solutions to a valid inequality, that is, if < π, π◦ > is a valid inequality
then F = {x ∈ P : πx = π◦ } is a face of P , and we said that < π, π◦ >
represents F .
A face F is said to be proper if F 6= P and F 6= ∅.
The faces of P form a lattice close under set intersection with
P = P ∩ {x ∈ IRJ : ~0x ≤ 0}
2.3. FACES AND FACETS
25
as maximum and
∅ = P ∩ {x ∈ IRJ : ~0x ≤ −1}
as minimum.
The face F represented by < π, π◦ > is non-empty if and only if
max{πx} = π◦
x∈P
. When F is non-empty, we say that < π, π◦ > supports P .
Notation 2.26 Let P = P(A, b), with matrix AI×J and vector bI . We divide
I in two sets,
I = = {i ∈ I : ∀x ∈ P, ai x = bi }
and
I ≤ = I \ I =.
Note that P is full dimension if and only if I = = ∅.
We denote by < A= , b= > and < A≤ , b≤ > the corresponding rows of
< A, b >.
We refer to them as the equality set and the inequality set of the representation < A, b > of P , that is
P = {x ∈ IRJ : A= x = b= , A≤ x ≤ b≤ }.
.
Note that if i ∈ I ≤ , then < ai , bi > cannot be written as a linear combination of the rows of < A= , b= >.
Proposition 2.27 If P = P(A, b) and F be a non-empty face of P , then F
is the polyhedron:
F = {x ∈ IRJ : ai x = bi , ∀i ∈ IF= , ai x ≤ bi , ∀i ∈ IF≤ }
where IF= ⊇ I = and IF≤ = I = \ IF= .
The number of distinct faces is finite.
(2.3)
26
CHAPTER 2. POLYHEDRA THEORY
Proposition 2.28 If P = P(A, b) ⊆ IRJ , then
dim(P ) + rank(A= , b= ) = |J|.
Note that by this Proposition, if F is a proper face of P , then, we have
that dim(F ) < dim(P ). In particular, the dimension of F is k if and only if
the maximum number of affinely independent points that lie in F is k + 1.
Definition 2.29 A face F of a polyhedron P is called a facet of P if
dim(F ) = dim(P ) − 1.
it is a vertex of P if dim(F ) = 0 and it is an edge of P whenever dim(F ) = 1.
The polyhedron P is pointed when it have at least a vertex, in this case
an unbounded edge F of P defines an extreme ray which is a recessional
direction and any point x lying in that direction is called an extreme point.
x2
6
E
bs
as
H
HH
Hs
FH
c
P
C
b-ar
r
d-c
G
s
d
- x1
Figure 2.1: Polyhedron Elements
An example of the elements of a polyhedron are shown in Figure 2.1, a
and c are vertices, E, F and G are edges, notice than in two dimensions they
are also facets, the extreme rays are generated by the extreme points b-a and
d-c and C is the recessional cone.
2.4. INTERNAL POLYHEDRA REPRESENTATION
27
Proposition 2.30 If F is a proper facet of P , there exists some inequality
ak x ≤ bk for k ∈ I ≤ representing F .
2.4
Internal Polyhedra Representation
Here we consider a representation of polyhedra in terms of vertices and extreme rays or more generally as convex combination of a finite set of points
plus a conical combination of a second finite set of points.
Definition 2.31 Let S ∈ IRM ×J and T ∈ IRN ×J where M , N and J are
finite index sets. We define
~
conv(S) ≡ {x ∈ IRJ : ∃λ ∈ IRM
+ , x = λS, λ1 = 1}
That is, conv(S) is the convex hull of the rows of S.
cone(T ) ≡ {x ∈ IRJ : ∃µ ∈ IRN
+ ; x = µT }
That is, cone(T ) is the conical hull of the rows of T .
lin(T ) ≡ {x ∈ IRJ : ∃µ ∈ IRN ; x = µT }
That is, lin(T ) is the linear hull of the rows of T .
Definition 2.32 We extend the definition of conv(S) to
conv(S, T ) ≡ conv(S) + cone(T )
that is
N
~
conv(S, T ) ≡ {x ∈ IRJ : ∃λ ∈ IRM
+ , µ ∈ IR+ ; x = λS + µT, λ1 = 1}
28
CHAPTER 2. POLYHEDRA THEORY
Thus conv(S, T ) is the set of all vectors expressible as the sum of a convex
combination of rows of matrix S and a non-negative linear combination of
rows of matrix T .
Clearly conv(S, T ) = ∅ if and only if M = ∅ and also we have that
conv(S, T ) = conv(S) when N = ∅.
Notice that we have defined cone, and the others “hull” functions, applying to matrices and some times we will apply them to finite vector sets. But
any finite vector set can be consider a matrix whose rows are the vectors in
the set.
Definition 2.33 Given a set X ⊆ IRJ if there exists a pair of matrices
S ∈ IRM ×J and T ∈ IRN ×J such that X = conv(S, T ) we say that X is a
finitely generated set and the pair of matrices (S, T ) is said to finitely generate
X or to be a generating system for X.
Proposition 2.34 Let X = conv(S, T ) ⊆ IRJ be a finitely generated set for
some matrices S and T .
Then X is a convex set with recession cone cone(T ). The vertices of X
are among the rows of S and if X has a vertex, the extreme rays of X are
non-negative multiples of some rows of T .
The valid inequalities for non-empty conv(S, T ) are characterized by the
next lemma.
Lemma 2.35 Let S ∈ IRM ×J and T ∈ IRN ×J where M , N are finite sets and
M non-empty.
xc ≤ α is a valid inequality for conv(S, T ) if and only if Sm c ≤ α for all
m ∈ M and Tn c ≤ 0 for all n ∈ N .
Proof: If xc ≤ α is a valid inequality for conv(S, T ), then Sm c ≤ α for
all m ∈ M , since such Sm belong to conv(S, T ).
Also (Sm + λTn )c = Sm c + λT c ≤ α for all n ∈ N , m ∈ M , λ ∈ IR+ , hence
Tn c ≤ 0 for all n ∈ N .
2.5. POLYHEDRAL CONES
29
Let Sm c ≤ α for all m ∈ M and Tn c ≤ 0 for all n ∈ N . Then
X
λm Sm c +
m∈M
X
µn Tn c ≤
n∈N
N
for all λ ∈ IRM
+ , µ ∈ IR+ and
P
m∈M
X
λm α = α
m∈M
λm = 1, hence (λS + µT ) ≤ α.
A main result, bearing important consequences in polyhedral theory is
the next theorem.
Theorem 2.36 The set P ⊆ IRJ is a polyhedron if and only if P is a finitely
generated set.
That is, if and only if there exists a matrix S ∈ IRM ×J and a matrix
T ∈ IRN ×J , where M and N are finite index sets, such that
P = conv(S, T ).
Given a polyhedron P = P(A, b) = conv(S, T ) we say that P(A, b) is an
external representation and that conv(S, T ) is an internal representation.
< A, b > and S, T are minimal representations if deleting any row we have
in either the equality do not hold. Such a minimal pair of matrices S, T is
called a basis.
Proposition 2.37 Let P = conv(S, T ) be pointed and S, T be minimal matrices defining conv(S, T ). Then the rows of S correspond to the vertices of P
and the rows of T are extreme points corresponding one to one to the extreme
rays of the recessional cone of P .
2.5
Polyhedral Cones
We usually prefer to work with pointed polyhedra (polyhedra that have vertices), that is one of the reasons why the Simplex Algorithm consider nonnegativity. These polyhedra have a unique basis.
30
CHAPTER 2. POLYHEDRA THEORY
When a polyhedron is not pointed, the recessional cone have a lineality
of dimension greater than zero. Here we extend the concept of extreme rays
to cones without a vertex. Hence in this section we will deal with cones, and
unless otherwise stated, a cone will denote a polyhedral cone.
Even if we could consider a polyhedral cone as a polyhedron, the homogeneous conditions make a big difference an it is convenient to modify some
definitions when we know that a polyhedron is a cone, like the cone of valid
inequalities.
First, we do not need to consider valid inequalities which are not homogeneous.
Second, a cone is never empty, since the origin is always in it.
As a consequence, we do not need to consider the empty set in the face
lattice of the cone. Also we could specialize Farkas’ Lemma.
Lemma 2.38 (Farkas) Let C = C(A) be a polyhedral cone.
Then πx ≤ 0 is a valid inequality for C if and only if there exists λ ≥ ~0
such that
λA = π.
Proof: Since C is not empty, by Farkas’ Lemma 2.16 < π, 0 > is a valid
inequality for C if and only if there exists λ ≥ ~0 such that
λA = π, λ0 ≤ 0.
But the second condition became 0 ≤ 0 and is always satisfied.
2.5.1
Convex Cone Basis
Let C be a closed convex cone in IRn , we denote by LC the linearity of C,
LC = {x ∈ C : −x ∈ C}
2.5. POLYHEDRAL CONES
31
We do not assume that C is pointed, that is LC may be different from {~0}.
Definition 2.39 We extend the definition of extreme point to mean that
x ∈ C is an extreme point if x = x1 + x2 , both x1 and x2 belong to C imply
xi = αi x + `i , αi ≤ 0, `i ∈ LC for either, (and hence both) i = 1 or i = 2.
When LC = {~0} an extreme point is any vector on an extreme ray of C.
When a non-zero linearity is present, any vector in the linearity is extreme,
and adding a vector in LC to an extreme point gives another equivalent
extreme point.
In general, intersecting the cone with the orthogonal complement of the
linearity gives a pointed cone generated by non-negative combinations of its
extreme rays. The original cone is generated by linear combinations of a
basis of the linearity plus non-negative combinations of the extreme rays.
In terms of the original cone, we do not have extreme rays, but instead, we
might say, extreme half-subspaces of dimension two or higher. These extreme
half-subspaces can be formed as an extreme ray plus the linearity. Any vector
in such an extreme half-subspace is an extreme vector, and C is equal to the
non-negative combinations of its extreme vectors.
When C has a linearity LC , this linearity form a vector subspace of IRn ,
hence has a finite basis. The extreme vectors can be taken module de linearity
LC , i.e., two extreme x, y are equivalents if one is a positive multiple of the
other plus a vector in the linearity, in this case we write x ' y when x is
equivalent to y.
Being polyhedral for C means that, in this sense, there are a finite number
of non-equivalent extreme vectors only.
A basis (E,B) of C are two disjoint sets contained in C such that B is
a basis of LC an E is a set of pair wise non-equivalent extreme points such
that for any extreme point not in LC there is a point equivalent to it in E.
In this case we have C = cone(E) + lin(B), where cone(E) is the cone
generated by E and lin(B) is the subspace generated by B (recall that
cone(∅) = lin(∅) = {~0}) and (E,B) is a minimal representation of C.
When LC = {~0}, E correspond to a unique set of rays, in general the
32
CHAPTER 2. POLYHEDRA THEORY
elements of E are one to one equivalents to the unique basis of the intersection
of C with the orthogonal complement of the linearity of C.
2.6
Notes
A good reference for this chapter is the book [Nemhauser & Wolsey 88] from
which we take most of the material. For faces of polyhedra a good study is
in [Pulleyblank 73].
However, the last section is from [Aráoz & Johnson 92].
We also recommend the classic book Convex Analysis [Rockafellar 69].
2.7
Exercises
Exercise 2.1 Prove that the maximum number of linearly independent vectors in IRN is |N |.
Exercise 2.2 Prove that the following statements are equivalent:
a. x0 , . . . , xk ∈ IRN are affinely independent.
b. x1 − x0 , . . . , xk − x0 are linearly independent.
c. < x0 , −1 >, . . . , < xk , −1 >are linearly independent.
Exercise 2.3 Prove that the maximum number of affinely independent vectors in IRN is |N | + 1.
Exercise 2.4 Prove that if A is a I × J matrix, the maximum number
of linearly independent vectors in the set of columns equals the maximum
number of linearly independent vectors in the set of rows.
2.7. EXERCISES
33
Exercise 2.5 Prove that the following statements are equivalent:
a. {x ∈ IRJ : Ax = b} =
6 ∅.
b. rank(A) = rank(< A, b >).
Exercise 2.6 Prove that if {x ∈ IRJ : Ax = b} 6= ∅, the maximum number
of affinely independent solutions x to Ax = b is |J| + 1 − rank(A).
Exercise 2.7 Prove that we could change a system with mixed inequalities
of type ≤, = and ≥ to an equivalent system with all the inequalities of type
≤.
Exercise 2.8 (Theorem of Alternatives) Let A ∈ IRJ×K be a given matrix
and the vector b ∈ IRJ be given.
Prove that exactly one of the following alternatives must be true.
Either there exists y ∈ IRK such that
Ay ≤ b,
or
there exists x ∈ IRJ+ such that
xA = ~0, xb < 0.
Also prove that:
Either there exists y ∈ IRK
+ such that
Ay = b,
or
there exists x ∈ IRJ such that
xA ≤ ~0, xb < 0
is equivalent.
34
CHAPTER 2. POLYHEDRA THEORY
Exercise 2.9 (Farkas’ Lemma) Let the polyhedron P = P(A, b) be nonempty.
Using the Theorem of Alternatives prove that πx ≤ πo is a valid inequality
for P if and only if there exists λ ≥ ~0 such that
λA = π, λb ≤ α.
Exercise 2.10 Let X ⊆ IRJ . Prove that X is a convex cone if and only if
x1 , x2 ∈ X implies x1 + x2 ∈ X and x ∈ X implies λx ∈ X for all λ ≥ 0.
Exercise 2.11 Prove that if X ⊆ IRJ 6= ∅ is a convex set and y is a recessional direction of X, then
∀x ∈ X we have {z : z = x + λy, λ ∈ IR+ } ⊆ X.
Prove that rec(X) is a convex cone.
Exercise 2.12 Prove that the polyhedron P(A, b) is a closed convex set and
the polyhedral cone C(A) is a closed convex cone.
Also prove that the cone C(A) is the recessional cone of P(A, b).
Exercise 2.13 Prove that any exposed point is a vertex but no necessarily
the converse hold.
Give an example of a convex set with a vertex which is not an exposed
point.
Exercise 2.14 Prove that if P ⊆ IRJ is a polyhedron, the vertices of P are
exposed points.
Exercise 2.15 Let P = P(A, b). Prove that P is full dimension if and only
if I = = ∅.
Exercise 2.16 Prove that if i ∈ I ≤ , then < ai , bi > cannot be written as a
linear combination of the rows of < A= , b= >.
2.7. EXERCISES
35
Exercise 2.17 Prove that if P = P(A, b) and F is a non-empty face of P ,
then F is the polyhedron:
F = {x ∈ IRJ : ai x = bi , ∀i ∈ IF= , ai x ≤ bi , ∀i ∈ IF≤ }
where IF= ⊇ I = and IF≤ = I = \ IF= .
Prove also that the number of distinct faces is finite.
Exercise 2.18 Prove that if P = P(A, b) ⊆ IRJ , then
dim(P ) + rank(A= , b= ) = |J|.
Exercise 2.19 Prove that if F is a proper face of a polyhedron P , then
dim(F ) < dim(P ).
In particular, prove that the dimension of F is k if and only if the maximum number of affinely independent points that lie in F is k + 1.
Exercise 2.20 Prove that a face F of P is vertex of P if dim(F ) = 0 and is
an extreme ray of P if dim(F ) = 1.
Exercise 2.21 Prove that if F is a proper facet of P , there exists some
inequality ak x ≤ bk for k ∈ I ≤ representing F .
Exercise 2.22 Let X = conv(S, T ) ⊆ IRJ be a finitely generated set for
some matrices S and T .
Prove that X is a convex set with recession cone cone(T ). The vertices
of X are among the rows of S and if X has a vertex, the extreme rays of X
are non-negative multiples of some rows of T .
Exercise 2.23 Prove that the set P ⊆ IRJ is a polyhedron if and only if P
is a finitely generated set.
36
CHAPTER 2. POLYHEDRA THEORY
That is, if and only if there exists a matrix S ∈ IRM ×J and a matrix
T ∈ IRN ×J , where M and N are finite index sets, such that
P = conv(S, T ).
Also prove that P is a polytope if and only if T is empty and that a
polyhedron is a polytope plus a cone.
Exercise 2.24 Prove that when the polyhedron P is pointed there exists a
unique minimal matrix S whose rows correspond to the vertices of P and a
unique minimal matrix T , unique up to multiplying by a positive coefficient,
whose rows correspond to extreme points one to one in the extreme rays of
P , such that P = conv(S, T ).
Exercise 2.25 Prove that the cone C is pointed if and only if
LC = {~0}.
Exercise 2.26 Prove that given the cone C, when LC = {~0}, a basis E
correspond to a unique set of rays, and that in general the elements of a
basis E are one to one equivalents to the unique basis of the intersection of
C with the orthogonal complement of the linearity of C.
Chapter 3
Convex General Polarity
In this chapter we extend the theory of polar cones to different families of
polyhedra.
We began defining polarity in the most general form, since any relation
between two sets define a polarity relation.
3.1
General Polarity
Let X and Y be given sets and ∗ be a binary relation ∗ ⊆ X × Y . We denote
(x, y) ∈ ∗ by x ∗ y.
Define the ∗-polar of any set P ⊆ X as
P ρ(∗) = {y ∈ Y | x ∗ y ∀x ∈ P }
Likewise define the ∗-polar of any set Q ⊆ Y as
Qσ(∗) = {x ∈ X| x ∗ y ∀y ∈ Q}
The functions ρ(∗) : 2X → 2Y and σ(∗) : 2Y → 2X are called the polarity
defined by ∗ between subsets of X and subsets of Y or ∗-polarity.
37
38
CHAPTER 3. CONVEX GENERAL POLARITY
Symmetric Rule: Notice that the definition of ρ(∗) is identical to that of
σ(∗) with the sets X and Y interchanged. From any proposition concerning
the polarity between X and Y , another proposition may be obtaining by
interchanging X and Y , and interchanging ρ(∗) and σ(∗). Except when it
is necessary to refer explicitly to the function ρ(∗) or the function σ(∗) we
denote the ∗-polar of P by P ∗ where
(
∗
P =
P ρ(∗) if P ⊆ X
P σ(∗) if P ⊆ Y
Lemma 3.1 Let P and P 0 be both subsets of X. Then we have
P ⊆ P 0 ⇒ P ∗ ⊇ P 0∗
Proof: Clearly since if y ∈ P 0∗ then x ∗ y for all x ∈ P 0 , in particular
x ∗ y for all x ∈ P ⊆ P 0 , hence y ∈ P ∗ .
Lemma 3.2 Let P be a subset of X. Then we have P ∗∗ ⊇ P .
Proof: Because, for any x ∈ P and all y ∈ P ∗ we have x ∗ y hence, by
the Symmetric Rule, x ∈ (P ∗ )∗ .
Lemma 3.3 Let P be a subset of X. Then we always have
P ∗∗∗ = P ∗∗ .
Proof: We have (P ∗ )∗∗ ⊇ P ∗ by Symmetric Rule and Lemma 3.2. Let
P 0 = P ∗∗ , hence P ⊆ P 0 by Lemma 3.2, using Lemma 3.1 we obtain
P ∗ ⊇ P 0∗ = P ∗∗∗ .
3.1. GENERAL POLARITY
39
Definition 3.4 For any set P ⊆ X, or any set P ⊆ Y , the set P ∗∗ is called
the ∗-closure of P . If P = P ∗∗ , then P is called ∗-closed .
Using the Symmetric Rule, we could state:
Lemma 3.2 as any subset of either X or Y is contained in its ∗-closure.
Lemma 3.3 as for any subset of either X or Y its ∗-polar is ∗-closed.
This facts illustrate the appropriateness of the term ∗-closed. P ∗ is closed
in the sense that adding any other element to P ∗ would destroy the property
that every element of P ∗ is related to every element of P . P ∗ is the unique
maximal set with this property. The next Lemma characterize the ∗-closed
sets.
Lemma 3.5 The set C ⊆ X is ∗-closed if and only if there exists Q ⊆ Y
such that C = Q∗ .
Proof: If C = Q∗ then C ∗∗ = Q∗∗∗ = Q∗ = C by the Symmetric Rule
and Lemma 3.3. If C = C ∗∗ let Q = C ∗ .
Hence C = C ∗∗ = Q∗ .
With every subset P ⊆ X, there corresponds a unique ∗-closed subset of
Y , namely the ∗-polar of P . More precisely,
Lemma 3.6 The function ρ(∗) is an inclusion reversing bijection between
the family of ∗-closed subsets of X and the family of ∗-closed subsets of Y .
The function σ(∗) is the inverse of ρ(∗).
Proof: The function ρ(∗) is onto since any ∗-closed subset P ⊆ Y is the
∗-polar of P σ(∗) which is a ∗-closed subsets of X by Lemma 3.3.
The inverse of ρ(∗) exists, being σ(∗) , since P = (P ρ(∗) )σ(∗) for all ∗-closed
subsets of X.
40
CHAPTER 3. CONVEX GENERAL POLARITY
Thus ρ(∗) is a bijection. The function ρ(∗) is inclusion reversing by
Lemma 3.1.
We refer to a ∗-closed subset of X and its ∗-polar as a ∗-polar pair since
each is the ∗-polar of the other.
Lemma 3.7 The intersection of any two ∗-closed subsets of either X or Y
is also ∗-closed.
Proof: Let P1 and P2 be two closed subsets of X, that is, P1 = P1∗∗ and
P2 = P2∗∗ .
By Lemma 3.2 P1 ∩ P2 ⊆ Pi , implies (P1 ∩ P2 )∗∗ ⊆ Pi∗∗ , for i = 1, 2. Then
(P1 ∩ P2 )∗∗ ⊆ P1∗∗ ∩ P2∗∗ = P1 ∩ P2
But again by Lemma 3.2, P1 ∩ P2 ⊆ (P1 ∩ P2 )∗∗ , hence
P1 ∩ P2 = (P1 ∩ P2 )∗∗ ,
and P1 ∩ P2 is ∗-closed.
Finally, we note that both X and Y are the ∗-polars of the empty set, so
by Lemma 3.5 we have the following lemma:
Lemma 3.8 Both sets X and Y are ∗-closed. The empty set not allays is
∗-closed.
Given a polarity, the main questions are what are the polars, what are
the closures and what are the polar pairs.
3.1.1
Cone Polarity
Cone Polarity is the most well known of the polarities and the one with more
known properties. We describe it before we pass to more general polarities
3.1. GENERAL POLARITY
41
to compare the general results. Most of the results are corollaries of results
for polyhedral polarities.
Consider de polarity given by the relation
xγy ≡ xy ≥ 0.
For any set Q ∈ IRJ , the polar Qγ is a convex cone, we call Qγ the polar cone
of Q.
Remark 3.9 In this and other cases it is better to work with inequalities
of the form πx ≥ π◦ rather than πx ≤ π◦. Hence we denote as P (C) the
corresponding polyhedra. That is
P(A, b) = {x ∈ IRJ | Ax ≥ b} = {x ∈ IRJ | − Ax ≤ −b} = P(−A, −b).
The vector < π, π◦ > will represent either the inequality πx ≤ π◦ or the
inequality in agreement with the context.
Lemma 3.10 Let C be a polyhedral cone, say C = C(A)C(−A).
Its polar cone is cone(ai | ai is a row of A).
Proof: Let A = (ai ∈ IRJ | i ∈ I). We have y ∈ C γ if and only if
< y, 0 > is a valid inequality for C(A) if and only if exists λ ∈ IRI+ such that
P
y = i∈I λi ai (see Farkas’ Lemma 2.38 for cones) if and only if y ∈ cone(A).
Really this lemma is equivalent to Farkas’ Lemma 2.38 for cones.
Lemma 3.11 Let A = (ai ∈ IRJ : i ∈ I) where I is a finite set and let
C = cone(A). The polar cone of C is C(A).
Proof: Let y ∈ C(A). Then
X
y(
i∈I
λ i ai ) =
X
i∈I
λ i ai y ≥
X
i∈I
λi 0 = 0
42
CHAPTER 3. CONVEX GENERAL POLARITY
for all λ ∈ IRI+ , hence y ∈ C γ .
Let y ∈
/ C(A). Then there exists i ∈ I such that ai y < 0, but ai ∈ C,
hence y ∈
/ Cγ.
A direct consequence of Lemma 3.10 and Lemma 3.11 is the following
theorem.
Theorem 3.12 The polyhedral cones are γ-closed.
As we see, Qγ is the set of valid homogeneous inequalities for the set Q,
therefore, the γ-closure of Q is the cone generated by Q, by the Symmetric
Rule, and the γ-closed sets are the convex cones.
Some times Cone Polarity is defined by the relation xγ 0 y ≡ xy ≤ 0,
0
however, because P γ = −P γ for any set P , the general properties of both
polarities are the same.
3.2
General Bilinear Inequality Polarity
Given a set Q we generalize γ-polarity considering the valid inequalities for
Q of a given pattern.
As we have shown, γ-polarity correspond to the pattern xy ≥ 0, the
first generalization correspond to the pattern xy ≤ 1 and a valid inequality
correspond to the pattern xy ≤ yo .
All these patterns are bilinear inequalities and we will study the polarity
corresponding to a most general bilinear inequality which correspond to the
pattern:
xW y + xu + vy ≤ α
For J and K finite non-empty sets, W a J × K matrix , u ∈ IRJ , v ∈ IRK
and α ∈ IR, let the relation Ω ⊆ IRJ × IRK be:
3.2. GENERAL BILINEAR INEQUALITY POLARITY
xΩy ⇐⇒ xW y + xu + vy ≤ α
43
(3.1)
Our main purposes are to describe the Ω-polar and Ω-closure of a polyhedron and to characterize the Ω-closed polyhedra.
Proposition 3.13 For any set P ⊆ IRJ , the Ω-polar of P ,
P Ω = {y ∈ IRK | xW y + xu + vy ≤ α, ∀x ∈ P }
(3.2)
may be expressed in terms of valid inequalities for P by
P Ω = {y ∈ IRK | x(W y + u) ≤ α − vy is a valid inequality for P }
(3.3)
Proposition 3.14 P Ω may also be expressed as the solution set of linear
inequalities since for each x ∈ P ,
(xW + v)y ≤ α − ux
is a valid inequality for P Ω and P Ω is the solution set of all such inequalities.
Unless P is a finite set, this set of linear inequalities defining P Ω is not
finite. However, we will describe a finite linear system defining the Ω-polar
of any finitely generated set.
Lemma 3.15 Let P be a polyhedron. Points of the recessional cone of P
also determine valid inequalities for P Ω . For each t ∈ rec(P ),
tW y ≤ −tu
is a valid inequality for P Ω .
44
CHAPTER 3. CONVEX GENERAL POLARITY
Proof: If P = ∅, then rec(P ) = ∅ and P Ω = IRK so the lemma is trivially
true.
If P 6= ∅, then there exists x◦ ∈ P such that x◦ + λt ∈ P for all λ ≥ 0,
t ∈ rec(P ) so by Proposition 3.14,
((x◦ + λt)W + v)y ≤ α − u(x◦ + λt)
which is equivalent to
x◦ W y + x◦ u + vy + λ(tW y + tu) ≤ α
must be valid for P Ω for all λ ≥ 0, that is
tW y + tu ≤ 0 ∀ t ∈ P Ω
which implies the lemma.
As we said, we are interested in characterizing the Ω-polars and Ω-closures
of polyhedra as well as the Ω-closed polyhedra.
In particular we will show that these sets are also polyhedra.
3.2.1
Ω-Polar of Finitely Generated Sets
Recall that a finitely generated set is a polyhedron and a polyhedron is a
finitely generated set (see Theorem 2.36).
Let us consider the Ω-polar of the finitely generated set
conv(SM , TN ) ⊆ IRJ
where M , N are finite sets, SM ∈ IRM ×J , and TN ∈ IRM ×K .
Since
Sm ∈ conv(SM , TN ) for all m ∈ M
and
Tn ∈ rec(conv(SM , TN )) for all n ∈ N
3.2. GENERAL BILINEAR INEQUALITY POLARITY
45
we know by Proposition 3.14 and Lemma 3.15 that the inequalities
Sm W y + SM u + vy ≤ α
Tn W y + Tn u ≤ 0
∀m ∈ M
∀n ∈ N
are valid inequalities for conv(SM , TN )Ω .
We denote the system of linear inequalities (3.4) by
(Ω; SM , TN )
Written in standard matrix format (Ω; SM , TN ) is
SM W + ~1v
TN W
!
y≤
α
~ − SM u
−TN u
!
where



~1v = 


v
v
..
.



 ∈ IRM ×J


v
the matrix all of whose rows equal v.
We denote the set of solutions to (Ω; SM , TN ) by
P (Ω; SM , TN )
Theorem 3.16 For any finite sets, M and N , and matrices
SM ∈ IRM ×J , TN ∈ IRM ×K
the Ω-polar of conv(SM , TN ) is the polyhedron
P (Ω; SM , TN )
if conv(SM , TN ) 6= ∅ and IRK otherwise.
(3.4)
46
CHAPTER 3. CONVEX GENERAL POLARITY
Proof: The Ω-polar of the empty set is IRK since every y ∈ IRK satisfies xΩy
for all x ∈ ∅.
Assume conv(SM , TN ) 6= ∅. y ∈ IRK is in conv(SM , TN )Ω if and only if, by
Proposition 3.14,
x(W y + u) ≤ α − vy
is a valid inequality for conv(SM , TN ) if and only if, by Lemma 2.35,
(
Sm (W y + u) ≤ α − vy,
Tn (W y + u) ≤ 0,
∀m∈M
∀n∈N
if and only if
y ∈ P (Ω; SM , TN )
If P ⊆ IRJ is a non-empty polyhedron defined by a linear system, say
P = {x ∈ IRJ | xA ≤ b}
for some A ∈ IRJ×I and b ∈ IRI , then by Farkas’ Lemma 2.16 and Proposition 3.14,
P Ω = {y ∈ IRK | ∃λ ∈ IRI+ such that W y + u = Aλ, α − vy ≥ bλ}
Such a description for P Ω is neither a linear system with variables only
in IRK nor is a generating system, thus it is not easily related to neither the
facets nor the vertices and extreme rays of the polyhedron, however when
we have an internal representation of the polyhedron we can obtain a linear
system defining the Ω-polar.
The Ω-polar function ρ(Ω) is, according to Proposition 3.6, an order reversing bijection between the Ω-closed subsets of IRJ and the Ω-closed subsets
of IRK .
Since by Theorem 3.16 and Theorem 2.36, this bijection sends every polyhedron onto a polyhedron we conclude the following:
3.2. GENERAL BILINEAR INEQUALITY POLARITY
47
Corollary 3.17 The Ω-polar function ρ(Ω) is an order reversing bijection
between the Ω-closed polyhedra of IRJ and the Ω-closed polyhedra of IRK .
This means we can strengthen Lemma 3.5 to
Lemma 3.18 A polyhedron is Ω-closed if and only if it is the Ω-polar of a
polyhedron.
and thus another corollary to Theorem 3.16 is:
Theorem 3.19 (Ω-closed polyhedra in IRK ) A polyhedron
Q ⊆ IRK
is Ω-closed if and only if Q = IRK or there exists finite sets M 6= ∅, N and
matrices SM ∈ IRM ×J , TN ∈ IRN ×J such that
Q = P (Ω; SM , TN ).
3.2.2
Ω-Polar Types
Three well-known results will help to illustrate this theorem.
Example 3.20 Let ω ⊆ IRJ × IRJ where xωy if and only if xy ≤ 1. That is,
let ω = Ω where W = IJ , u = v = ~0 and α = 1.
ω-polarity is, some times, called “polarity of convex sets”.
From Theorem 3.19 we have:
A polyhedron Q ⊆ IRJ is ω-closed if and only if Q is the solution set of a
system of linear inequalities of the form
SM y ≤ ~1
TN y ≤ ~0
!
that is, Q is ω-closed if and only if all inequalities of any linear system
defining Q are satisfied by the origin ~0 ∈ IRJ .
48
CHAPTER 3. CONVEX GENERAL POLARITY
Example 3.21 Let β ⊆ IRJ × IRJ where xβy if and only if xy ≥ 1. That is,
let β = Ω where W = −IJ , u = v = ~0 and α = −1.
β-polarity is, some times, called “Blocking polarity”.
Theorem 3.19 gives:
A polyhedron Q ⊆ IRJ is β-closed if and only if either Q = IRJ or Q is
the solution set of a system of linear inequalities of the form
−SM y ≤ −~1
−TN y ≤ ~0
!
or equivalent, Q is the solution set of a system of linear inequalities of
the form
SM y ≥ ~1
TN y ≥ ~0
!
for some M 6= ∅.
Example 3.22 Let γ ⊆ IRJ × IRJ where xγy if and only if xy ≥ 0, i.e. γ is
the Cone Polarity.
This correspond to γ = Ω with W = IJ , u = v = ~0 and α = 0.
From Theorem 3.19 a polyhedron Q ⊆ IRJ is γ-closed if and only if Q is
the solution set of a homogeneous system of linear inequalities.
Notice that this proves Theorem 3.12.
The above examples show different polar types. We now turn our attention to determining the different polar type of a given Ω-polarity.
Let
XΩ = {x ∈ IRJ | xW + v = ~0}
(3.5)
3.2. GENERAL BILINEAR INEQUALITY POLARITY
YΩ = {y ∈ IRK | W y + u = ~0}
49
(3.6)
Note that if xo ∈ XΩ , then
xo W y + xo u + vy ≤ α
reduces to the true or false statement, xo u ≤ α.
The two sets XΩ and YΩ and the value α − xo u for xo ∈ XΩ are critical
in determining the Ω-polar type.
Lemma 3.23 If YΩ 6= ∅, then we have that xu = 0 for all x ∈ IRJ satisfying
xW = ~0.
Proof: If y ∈ YΩ , that is W y + u = ~0, then xu = −xW y = 0 for any x
satisfying xW = ~0 which proves the Lemma.
Lemma 3.24 If XΩ 6= ∅ and YΩ 6= ∅, then xu = vy for all x ∈ XΩ and
y ∈ YΩ
Proof: This Lemma follows from the fact that
xu = −xW y = vy
for any x ∈ XΩ and any y ∈ YΩ .
Lemma 3.25 If XΩ 6= ∅ and YΩ = ∅, then for any θ ∈ IR there exists
xo ∈ XΩ such that xo u = θ.
50
CHAPTER 3. CONVEX GENERAL POLARITY
Proof: To prove this Lemma, suppose YΩ = ∅, that is, there does not
exist y ∈ IRK such that W y = −u. By Theorem 2.17 of Alternatives, there
thus exists x1 ∈ IRJ such that x1 W = ~0, −x0 u < 0.
For any x2 ∈ XΩ , define xo as
xo = x2 + ((θ − x2 u)x1 /x1 u)
satisfies
xo W + v = x2 W + ((θ − x2 u)x1 W/x1 u) + v = x2 W + ~0 + v = ~0
and
xo u = x2 u + ((θ − x2 u)x1 u/x1 u) = x2 u + (θ − x2 u) = θ
Definition 3.26 When XΩ 6= ∅, we choose any point q ∈ XΩ subject to the
condition that qu = α when YΩ = ∅.
We call q an XΩ -pole. We denote the number α − qu by i(XΩ ).
Lemma 3.24 and Lemma 3.25 ensure that i(XΩ ) is a well defined constant
when XΩ 6= ∅.
Notice that i(XΩ ) = 0 if XΩ 6= ∅ and YΩ = ∅, hence
i(XΩ ) 6= 0 only if XΩ 6= ∅ and YΩ 6= ∅
The sets
XΩ1 = {x ∈ IRJ | xW = ~0, xu ≤ 0}
and
XΩ2 = {x ∈ IRJ | xW = ~0, xu > 0}
(3.7)
3.3. BILINEAR SYSTEM RELATION POLARITIES
51
also play a prominent role in subsequent theorems in the next chapter.
Really, we could identify six different Polar Types according to XΩ , YΩ
and i(XΩ ). These are shown in Table 3.1.
Type
Type
Type
Type
Type
Type
1:
2:
3:
4:
5:
6:
XΩ
XΩ
XΩ
XΩ
XΩ
XΩ
6= ∅
6= ∅
6= ∅
6= ∅
=∅
=∅
YΩ
YΩ
YΩ
YΩ
YΩ
YΩ
6= ∅
6= ∅
6= ∅
=∅
6= ∅
=∅
i(XΩ ) > 0
i(XΩ ) < 0
i(XΩ ) = 0
i(XΩ ) = 0
i(XΩ ) undefined
i(XΩ ) undefined
Table 3.1: Ω-polar Types
3.3
Bilinear System Relation Polarities
Let Θ ∈ IRJ × IRK be the relation defined as xΘy if and only if
xW e y + xue + v e y ≤ α∀e ∈ E
where E, J and K are finite index sets, W e ∈ IRJ×K , ue ∈ IRJ , v e ∈ IRK
and αe ∈ IR, ∀e ∈ E.
Since each inequality comprising the relation Θ is linear in both x and
y, Θ is called a bilinear system relation. Any relation between IRJ and IRK
consisting of inequalities which are linear in both x and y can be written in
this form.
Let as call Ωe ⊆ IRJ × IRK be the relation defined by the eth inequality of
the relation Θ.
Lemma 3.27 The Θ-polar of any given set is the intersection of the Θ-polars
of the set.
52
CHAPTER 3. CONVEX GENERAL POLARITY
Lemma 3.28 Let Q ⊆ IRK . The point x ∈ IRK is in P Θ if and only if
(xW e + v e )y ≤ α − xue
is a valid inequality for Q, for all e ∈ E.
Theorem 3.29 Let Q be the non null polyhedron
Q = {y ∈ IRK | Ay ≤ b}
where A ∈ IRI×K and b ∈ IRI for I and K finite sets.
Then the Θ-polar QΘ of Q is
QΘ = {x ∈ IRJ | (∃λe ∈ IRI+ | xW e + v e = λe A, αe − xue ≥ λe b)∀e ∈ E}
3.4
Notes
Section (3.1) is based on [Griffin, Aráoz & Edmonds 82], the original reference is [Birkhoff & MacLane 65].
Section (3.1.1) is well treated in [Rockafellar 69].
Section (3.2) is also based on [Griffin, Aráoz & Edmonds 82].
Section (3.3) is based on [Aráoz, Edmonds & Griffin 83].
Special cases of β-polarity are treated by [Gomory 69] (Gomory’s Group
Problems) and [Fulkerson 70] (Blocker Pairs), it is called Reverse Polarity in
[Balas 75] and [Tind 77]. [Aráoz 74] has dealt in detail with this polarity as
well as with ω-polarity and γ-polarity. Ω-polarity extend these results.
Special cases of ω-polarity are treated by [Rockafellar 69] and, with the
name of Outer Polarity by [Balas 75].
In [Griffin 77] and in [Di Novella & Aráoz 91] the relation between the
face lattices of the Ω-polar pair P, P Ω is given.
3.5. EXERCISES
3.5
53
Exercises
Exercise 3.1 Study the polarity given by the relation xy = 0.
What are the polar and the closure of a given set?.
What are the polar pairs?.
Exercise 3.2 Consider the polarity given by the relation
xγy ≡ xy ≥ 0.
Prove that for any set Q ∈ IRJ , the polar Qγ is a convex cone.
Exercise 3.3 Extend the results of this chapter to any sets without asking
the set to be a polyhedron.
For example prove that a set is γ-closed if and only if it is a convex cone
or that an Ω-closed set is a convex set.
Exercise 3.4 Prove that:
The Ω-polar function ρ(Ω) is an order reversing bijection between the
Ω-closed polyhedra of IRJ and the Ω-closed polyhedra of IRK .
Exercise 3.5 Prove that a polyhedron is Ω-closed if and only if it is the
Ω-polar of a polyhedron.
Exercise 3.6 Prove that a polyhedron
Q ⊆ IRK
is Ω-closed if and only if Q = IRK or there exists finite sets M 6= ∅, N and
matrices SM ∈ IRM ×J , TN ∈ IRN ×J such that
Q = P (Ω; SM , TN ).
54
CHAPTER 3. CONVEX GENERAL POLARITY
Exercise 3.7 Prove that:
For any set P ⊆ IRJ , the Ω-polar of P ,
P Ω = {y ∈ IRK | xW y + xu + vy ≤ α, ∀x ∈ P }
may be expressed in terms of valid inequalities for P by
P Ω = {y ∈ IRK | x(W y + u) ≤ α − vy is a valid inequality for P }
Exercise 3.8 Prove that:
P Ω is a convex set.
Exercise 3.9 Prove that:
A polyhedron is Ω-closed if and only if it is the Ω-polar of a polyhedron.
Exercise 3.10 The Θ-polar of any given set is the intersection of the Θpolars of the set.
Exercise 3.11 Let Q ⊆ IRK . The point x ∈ IRK is in P Θ if and only if
(xW e + v e )y ≤ α − xue
is a valid inequality for Q, for all e ∈ E.
Exercise 3.12 Let Q be the non null polyhedron
Q = {y ∈ IRK | Ay ≤ b}
where A ∈ IRI×K and b ∈ IRI for I and K finite sets.
Then the Θ-polar QΘ of Q is
QΘ = {x ∈ IRJ | (∃λe ∈ IRI+ | xW e + v e = λe A, αe − xue ≥ λe b)∀e ∈ E}
3.5. EXERCISES
55
Exercise 3.13 The parametric representation in Theorem 3.29 of the Θpolar of a polyhedron defined by a linear system does not reveal much information about the structure of the Θ-polar.
Prove that when we know the vertices and extreme rays of a polyhedron
we can obtain a linear system defining the Θ-polar.
56
CHAPTER 3. CONVEX GENERAL POLARITY
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