1
A POLYHEDRAL STUDY OF INTEGER BILINEAR COVERING SETS
By
NITISH GARG
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2011
4
ACKNOWLEDGMENTS
I acknowledge my advisor, Dr. Jean-Philippe P. Richard for his immense help and
encouragement at every stage of my thesis. The frequent technical sessions with him
provided me a good insight of Operations Research and helped me to structure my
work to its present state. I would also like to thank Dr. Yongpei Guan for serving on my
committee.
I am thankful to my parents and my sisters for their constant motivation and
support.
I also take this opportunity to thank all my friends especially Navjeet Johal, Karan
Dadlani, Ameya Bhosle and Onkar Bhende for making my master’s degree into an
experience that I will cherish for many coming years.
5
TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 7
LIST OF FIGURES .......................................................................................................... 8
LIST OF ABBREVIATIONS ............................................................................................. 9
ABSTRACT ................................................................................................................... 10
CHAPTER
1 INTRODUCTION .................................................................................................... 11
1.1 Background ....................................................................................................... 11
1.2 Mathematical Programming .............................................................................. 12 1.2.1 Optimization Problem .............................................................................. 12 1.2.2 Types of Optimization Models ................................................................. 17
1.2.2.1 Convex optimization models .......................................................... 19 1.2.2.2 Nonconvex optimization models .................................................... 22
2 BRANCH-AND-BOUND ALGORITHM FOR NONCONVEX PROBLEMS .............. 26
2.1 Branch-and-Bound ............................................................................................ 28
2.2 Building Convex Underestimator ...................................................................... 32 2.3 Factorable Relaxations ..................................................................................... 34
3 CONVEX RELAXATIONS FOR BILINEAR COVERING SETS .............................. 39
3.1 Motivation ......................................................................................................... 39 3.1.1 Strassen’s Algorithm................................................................................ 39
3.1.2 Staff Scheduling Problems ...................................................................... 42
3.2 Problem Description.......................................................................................... 44
3.3 Building Convex Hull for Different Cases .......................................................... 46 3.3.1 Case 0 ..................................................................................................... 46
3.3.1.1 Problem 1 ....................................................................................... 47 3.3.1.2 Problem 2 ....................................................................................... 48 3.3.1.3 Convex hull for n=2 ........................................................................ 49
3.3.1.4 Problem 3 ....................................................................................... 50 3.3.1.5 Convex hull for case 0 .................................................................... 51
3.3.2 Case 1 ..................................................................................................... 51
3.3.2.1 Problem 4 ....................................................................................... 52 3.3.2.2 Problem 5 ....................................................................................... 53
3.3.2.3 Convex hull for n = 3 ...................................................................... 53
6
3.3.2.4 Convex hull for case 1 .................................................................... 54
3.4 Facet-Defining Inequalities and Convex Hulls .................................................. 56 3.4.1 Facet-Defining Inequality for Case 0 ....................................................... 57
3.4.2 Facet-Defining Inequality for Case 1 ....................................................... 66 3.4.3 Convex Hull Proof for Case 0 .................................................................. 71
4 SUMMARY ............................................................................................................. 84
APPENDIX: MATLAB CODES ...................................................................................... 85
LIST OF REFERENCES ............................................................................................... 94
BIOGRAPHICAL SKETCH ............................................................................................ 95
7
LIST OF TABLES
Table page 2-1 Sub-problems obtained after breaking down u1=(x-y)2z ..................................... 35
2-2 Sub-problems obtained after breaking down u2=xy2 ........................................... 37
3-1 Feasible points for x1y1+x2y2≥3 and u1=u2=3 ...................................................... 47
3-2 Convex Hull for x1y1+x2y2≥3 ............................................................................... 48
3-3 Convex Hull for x1y1+x2y2≥4 ............................................................................... 49
3-4 Convex Hull for x1y1+x2y2+x3y3≥4 ....................................................................... 51
3-5 Convex Hull for x1y1+x2y2+x3y3≥4 in case 1 ........................................................ 52
3-6 Convex Hull for x1y1+x2y2+x3y3≥5 ....................................................................... 53
3-7 Convex Hull for x1y1+x2y2+x3y3≥4 ....................................................................... 54
8
LIST OF FIGURES
Figure page 1-1 Feasible region of Problem (1-2) with only sign restrictions. .............................. 15
1-2 Feasible region of Problem (1-2) with sign restrictions and production constraint for item 1. ........................................................................................... 15
1-3 Feasible region of Problem (1-2) with sign restrictions and production constraint for items 1 and 2. ............................................................................... 16
1-4 Infeasibility resulting from the addition of constraint x1+x2≥600. ......................... 16
1-5 P1, P2, P3 are local minima and P3 is a global minima. ....................................... 17
1-6 A) Convex set B) Non-convex set C) Convex set ............................................... 18
1-7 Chord of a convex function ................................................................................. 19
2-1 Convex underestimator g(x) and convex envelope φ(x) of the function f(x) ....... 27
2-2 A) Set X B) Convex hull of X............................................................................... 27
2-3 Convex relaxation of the sub-problem u4=u32 ..................................................... 36
2-4 Convex relaxation of the sub-problem u5=y2....................................................... 37
9
LIST OF ABBREVIATIONS
BIP Binary Integer Program
BP Bilinear Program
IP Integer Program
LP Linear Program
OR Operations Research
QCQP Quadratically Constrained Quadratic Program
QP Quadratic Program
10
Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
A POLYHEDRAL STUDY OF INTEGER BILINEAR COVERING SETS
By
Nitish Garg
August 2011
Chair: Jean-Philippe P. Richard Major: Industrial and Systems Engineering
We study the polyhedral structure of an integer bilinear covering set, which
appears in the formulation of various practical problems including staff scheduling.
Starting from the analysis of specific instances, we develop a polyhedral description of
the convex hull of solution for special settings of the parameters and derive some facet-
defining inequalities.
11
CHAPTER 1 INTRODUCTION
1.1 Background
The origins of Operations Research (OR) can be found before World War II, when
a group of scientists in the British army started to research operation on the radar
technology to improve the efficiency of early warning radar systems. During the war, the
potential of the Operational Research Group was recognized and its contributions were
extended far beyond problems related to radars [4]. The group’s work was to bridge the
gap between war technologies and their practical use. Operations Research then
became synonymous to the set of mathematical techniques developed for the efficient
use of technologies and resources.
After the end of the war, OR kept growing because of two main factors [5]. The
first factor in its progress was the industrial boom that followed the war. OR caught the
eyes of many researchers because it had almost unlimited potential outside the military.
A key example is the development of simplex algorithm in 1947 by George Dantzig,
which is a popular algorithm to solve a very common family of OR problems called
Linear Programs (we describe linear programming models, later in this chapter). The
second factor that enabled the development of OR was the computer revolution, which
led to the implementation of computer codes to solve OR problems. A large
computational effort is typically required to solve an OR problem, a feature that makes
hand calculations often unpractical. Computers allowed the use of OR to spread to a
large number of practitioners and organizations.
The basic steps in the OR approach include, (i) formulating the problem in a
mathematical form that can be analyzed, and (ii) designing/using algorithms to obtain
12
solutions to the model.
Classical OR models include:
Mathematical Programs,
Stochastic Processes Models, and
Simulation Models.
In this thesis, we focus on mathematical programs.
1.2 Mathematical Programming
Mathematical programs are composed of a set of unknown variables, whose
values must be found so as to minimize/maximize a certain criterion (objective function).
Those values must be selected in such a way that specified relations (called
constraints) between variables and problem parameters are satisfied. Combination of
variables’ values satisfying all problem constraints are said to be feasible. There are
typically many ways of formulating a practical problem as a mathematical program, not
all of which are equivalent in terms of solution efficiency. Before introducing the type of
mathematical models we will be studying, we first give a more precise definition of
mathematical programs and the basic notions associated with their solution.
1.2.1 Optimization Problem
Mathematical programs (also called optimization problems) are mathematical
problems of the form:
max/min )(xf (1-1)
s.t. 0)( xgi ki ,,1
0)( xh j lj ,,1
Xx
where, nRX , )(xf , )(xgi , )(xh j are real valued functions, i.e., RRf n : ,
13
RRg n
i : , RRh n
j : . In (1-1), )(xf is called objective function while 0)( xgi and
0)( xhj are called inequality constraints and equality constraints respectively. The
objective function allows us to measure the quality of a particular set of values for the
variables x , whereas constraints define the requirements that variables x must satisfy.
To further illustrate the concept of a mathematical program and describe notion related
to its solution, we next present a simple example.
A company hires an OR analyst to maximize the revenue it gains from selling two
products that we call item 1 and item 2. The unit revenue generated by the sale of these
products is 4 and 6 respectively. The objective function of the problem is therefore
21 64)( xxxf , where 1x and 2x are the decision variables representing number of
items 1 and 2 sold respectively. Sign restrictions specify the nature of decision
variables, whether they are nonnegative or can assume both positive and negative
values. In our example, it is clear that 0, 21 xx because the number of sold items cannot
be negative.
If the company assumes that there is no restriction on the amount of these two
products it can produce and if the demand for these two products is unlimited, then the
problem will be unbounded. A problem is said to be unbounded if there are feasible
points for which the objective function takes arbitrarily large values (for a maximization
problem). So far, the model we have built is unbounded because the revenue becomes
arbitrarily large when 1x and/or 2x become arbitrarily large; see Figure 1-1 for a
depiction of the feasible region of the problem, where the feasible region is defined as
the set of values from 1x and 2x that satisfy all of the problem constraints (in our case,
the sign restrictions).
14
Typically, unbounded problems arise from issues in modeling. In the situation that
is occurring above, the analyst will realize that producing an unlimited amount of item 1
and item 2 is unlikely to respect physical production capacity in the plant and therefore
will define new constraints to represent the problem better. In particular, the analyst may
study the production plant and realize that no more than 200 units of item 1 can be
produced due to limited capacity. If so, the feasible region will be affected; see Figure 1-
2 for a graphical depiction of the updated feasible region. After studying production
capacity for item 2, the analyst might find out that production is limited to 300 units. If
so, the feasible region will be affected; see Figure 1-3 for a graphical depiction of the
updated feasible region.
After these modifications are performed, the analyst will have defined the problem,
gathered data and generated the following mathematical program:
max 21 64 xx (1-2)
s.t. 2001 x , 3002 x
0, 21 xx .
Model (1-2) is an instantiation of model (1-1) if one sets
2121 64),( xxxxf ,
200),( 1211 xxxg ,
300),( 2211 xxxg ,
and 2
RX .
The next step in the OR approach is to find a solution to model (1-2). In the case
of a maximization problem we say that a feasible solution *x is an optimal solution to (1-
1) if )()( * xfxf for all feasible solutions x in the feasible region. The value of the
objective function at an optimal solution is called optimal value, i.e., the optimal value is
15
)( *xf . For our example, it is easy to verify from Figure 1-3 that )300,200(* x and
hence the optimal value )( *xf for the problem is 260030062004 .
Now suppose that the company has committed to put 600 units of its product in
the market. To accommodate this new restriction, one would have to add the constraint
60021 xx to the previous model. The introduction of this constraint makes the model
infeasible, as there is not a single vector x that satisfies all constraints and sign
restrictions; see Figure 1-4 for a graphical illustration.
Figure 1-1. Feasible region of Problem (1-2) with only sign restrictions.
Figure 1-2. Feasible region of Problem (1-2) with sign restrictions and production
constraint for item 1.
Feasible region
x1 ≤ 200
x1
x2
Feasible region
x1
x2
16
Figure 1-3. Feasible region of Problem (1-2) with sign restrictions and production
constraint for items 1 and 2.
Figure 1-4. Infeasibility resulting from the addition of constraint x1+x2≥600.
Feasible region
x1 ≤ 200
x2 ≤ 300
x2
x1
x1 ≤ 200
x2 ≤ 300
x1 + x2 ≥ 600
x1
x2
17
The notion of optimal solution we have presented before is that of a global optimum.
Global optimum. In a maximization problem, we say that a point Dx * is globally
optimum if the objective function value at *x is greater or equal to the objective function
of any other feasible point Dx , where D is the feasible region of the problem.
Another notion is regularly used in optimization that is less restrictive about the
quality of solutions produced.
Local optimum. In a maximization problem, we say that a point Dx * is locally
optimum if the objective function value at *x is greater or equal to the objective function
of any other feasible point x in a neighborhood of *x , where D is the feasible region of
the problem.
It is easy to verify that
Global optima are always local optima,
Local optima are not necessarily global (Figure 1-5), and
A problem might have multiple local and global optima.
Figure 1-5. P1, P2, P3 are local minima and P3 is a global minima.
Typically, problems that have local optima that are not global are harder to solve
since local solution techniques can be fooled to believe that a local solution is global.
The situation is simpler in the presence of convexity, a concept that we introduce next.
1.2.2 Types of Optimization Models
Not all optimization problems are equally simple to solve. The notion of convexity
presented next, help delineate those problems that are easy to solve with current
P1 P2
P3
18
optimization techniques from those that are not. To introduce this concept, we first
introduce the following definitions.
Affine sets. A set nRC is affine if the line joining any two points in the set
C
completely lies in
C . Mathematically, we say that
C is affine if for all Cxx 21, and for
every R ,
Cxx 21 )1( .
Affine functions. A function
f : Rn Rm is affine if it is a sum of a linear function
and a constant, i.e.,
f (x) Ax b , where
A Rmn and
b Rm.
Convex sets. A set nRC is convex if the line segment joining any two points in
the set
C also lies in
C . Mathematically, we say that
C is convex if for all Cxx 21, and
for every ]1,0[ ,
Cxx 21 )1( .
We illustrate the concept of convex set in Figure 1-6.
A B C
Figure 1-6. A) Convex set B) Non-convex set C) Convex set
It is simple to verify that the intersection of two convex sets is convex but the union
of two convex sets is not necessarily convex. The concept of convexity can also be
defined for functions as we present next.
Convex functions. Let )(xf be a real valued function
f :C R, where
C is a
convex set. The function )(xf is said to be convex if the chord joining ))(,( 11 xfx and
))(,( 22 xfx for any two points
x1,x2 C is greater or equal to function )(xf for any point
19
x on the line segment joining
x1 and
x2. Mathematically, we say that )(xf is convex if
)()1()())1(( 2121 xfxfxxf
for all
(0,1) and for all
x1,x2 C. The notion of convex function is illustrated in Figure
1-7. We say that a function )(xf is concave if
f (x) is convex. In other words, we say
that function )(xf is concave, if
)()1()())1(( 2121 xfxfxxf
for all
(0,1) and for all
x1,x2 C.
The concepts of convex function and convex set are related in several ways. First
a function )(xf is convex if and only if its epigraph is a convex set, where the epigraph
of )(xf is defined as )}(,),{()( 1 xfzCxRzxfepi n . Second, if )(xf is convex
and
R, then the sub-level set of )(xf of level defined as })({ xfRx n is a
convex set.
Figure 1-7. Chord of a convex function
1.2.2.1 Convex optimization models
We define convex optimization problems to be optimization problems of the form:
min )(xf (1-3)
s.t. 0)( xgi ki ,,1
(x2, f(x2))
f(x)
(x1, f(x1))
20
0)( xhj lj ,,1
where,
)(xf is a convex real valued function, i.e., RRf n : ,
)(xgi are convex functions, and
)(xh j are affine functions.
Because the sublevel sets of convex function are convex sets, hyperplanes are
convex and intersections of convex sets are convex, it is easy to verify that the feasible
region of a convex program is a convex set.
A fundamental property of convex optimization models is that any local optimum
will also be a global optimum (for a proof, refer to [3]), whereas nonconvex optimization
problems can have many locally optimal solutions that are not global optima. An
important consequence of this result is that local search algorithms that converge to a
locally optimal solution always produce a globally optimal solution when applied to
convex problems whereas this is not always the case for nonconvex problems.
Convex optimization problems can be solved efficiently in practice and therefore
are key models in OR. Among convex problems, several subclasses have received
particular attention.
1.2.2.1.a Linear programming models
Linear programming problems are convex problems in which the objective function
and constraints are affine functions. Linear Programs (LP) are convex optimization
problems since it is clear from their definition that affine functions are convex. LPs are
usually written in the following standard form in which the only inequalities are variables’
sign restrictions:
21
min xcT (1-4)
s.t. bAx
0x .
In (1-4), Tc is a row vector of dimension n , x is a column vector of dimension n ,
A is matrix of dimension nm , b is a column vector of dimension n , and 0x requires
that each component of the column vector x is greater or equal to 0.
Generally, LP problems arising from application are not in standard form.
However, they can be converted into that form by adding slack variables, by subtracting
surplus variables, converting unrestricted variables into the form 0x . For more details
on such transformations, we refer to [8].
We mention that the example problem we discussed in the previous section is an
LP. Linear Programs are important since they can be solved very efficiently in practice.
In particular, variants of the simplex algorithm [12] and interior point methods [3] have
been implemented in commercial solution software that can nowadays be used to solve
very large instances.
1.2.2.1.b Quadratic programming models
Quadratic programming problems are optimization problems in which the objective
function is quadratic and constraints are affine functions. A quadratic program (QP) can
be expressed in the form:
min
12
xTQx cT x r (1-5)
s.t.
Gx h
Ax b
where, nnRQ , nRc , Rr ,
G Rmn , mRh ,
A Rpn, and pRb .
22
Not all QPs are convex. We describe next conditions under which they are. To this end,
we define
S
n to be the set of symmetric positive semidefinite matrices:
}0,{ nTTnnn RaXaaXXRXS
.
Proposition 1.2.2.1 Problem (1-5) is convex if nSQ .
In Problem (1-5), if quadratic inequality constraints are added, the problem
becomes known as quadratically constrained quadratic program (QCQP). A QCQP can
be expressed in the form:
min
12
xTQ0x c0
T x r0 (1-6)
s.t. ii
T
ii
T hrxcxQx 2
1 mi ,,1
Ax b
where, nn
i RQ , n
i Rc , Rri mi ,,0 , Rhi mi ,,1 ,
A Rpn and pRb .
QCQPs are convex if n
i SQ for mi ,,0 . There are various methods to solve
convex QCQPs. We refer interested readers to [3] for a description.
1.2.2.2 Nonconvex optimization models
Although convex optimization problems can be solved efficiently in theory and
practice, the situation for nonconvex optimization problems is more difficult. Entire
families of nonconvex problems can be shown to be hard to solve (under reasonable
complexity assumptions) such as integer programs, which we discuss in the next
section. However, some nonconvex problems can be readily converted into convex
optimization problem, therefore making them easy to solve. Consider for instance
min 2
2
2
1)( xxxf (1-7)
s.t. 0)1/()( 2
21 xxxg
23
0)()( 2
21 xxxh
2Rx ,
an example that is presented in [3].
It can be observed that )(xh is not affine and )(xg is not convex. However, (1-7)
can be readily converted into the following convex optimization problem:
min 2
2
2
1)( xxxf (1-8)
s.t.
g(x) x1 0
h(x) x1 x2 0
2Rx .
Problem (1-8) satisfies all the requirements of a convex optimization problem. The
sets of optimal solutions to (1-7) and (1-8) are clearly identical.
We next describe two families of nonconvex problems that are hard to solve in
theory (and also in practice).
1.2.2.2.a Integer programming models
Integer programs (IPs) are among the most challenging classes of nonconvex
optimization models and have wide practical applicability because in many practical
applications, decision variables can only assumes integers values.
Pure integer programming. A pure IP is a linear program in which all the
variables are required to take integers values. Pure IPs can be written as:
min xcT (1-9)
s.t. bAx
x Z
n .
In (1-9), Tc is a row vector of dimension n , x is a column vector of dimension n ,
A is matrix of dimension nm , b is a column vector of dimension n , and
Z
n is the set
24
of non-negative integers.
Among IPs, those in which all the variables are restricted to take the values 0 or 1
are called binary integer program (BIP). BIPs are particularly important, since they arise
in the modeling of virtually all combinatorial problems.
Mixed integer programming. In mixed IPs, some of the variables in the
optimization problem can only assume integer values. Mixed IPs can be written as:
min xcT (1-10)
s.t. bAx
xi Z
i I
xi R
i N \ I
where NI .
In (1-10), Tc is a row vector of dimension n , x is a column vector of dimension
n , A is matrix of dimension nm , b is a column vector of dimension n , and
},,1{ nN is the set of indices of the problem variables.
Over the years, many algorithms have been developed to solve IPs. Branch-and-
bound is one such an algorithm, which we are going to discuss in a subsequent part of
the thesis.
1.2.2.2.b Bilinear programming models
Another family of nonconvex programs is that of bilinear programs (BP) in which
the objective/constraint functions are bilinear. Bilinear functions are functions of the
form:
f x,y aT x bT y xTCy (1-11)
where, nRxa , mRyb , and mnRC .
25
Bilinear functions are a particular form of quadratic functions. In fact, defining
y
xz one can write (1-11) as z
C
Czzbazf
T
TTT
0
0
21,)( , from which it can be
verified that bilinear functions are typically not convex since
0
0TC
C is typically not
positive semi-definite.
A fundamental property of bilinear functions is that they become linear if anyone of
the vector x or y is assigned a particular value. In this thesis, we will study relaxation
techniques for a particular family of BPs.
BP came into existence through problems called bimatrix games. Various solution
methods have been proposed for solving BPs both locally and globally. A general
discussion regarding BPs and their solving techniques can be found in [6].In particular,
a branch-and-bound algorithm for BPs was developed by Al-Khayyal and Falk [1]. We
emphasize this particular reference, since will discuss general branch-and-bound
concepts in the following chapters of this thesis.
26
CHAPTER 2 BRANCH-AND-BOUND ALGORITHM FOR NONCONVEX PROBLEMS
We mentioned in the previous chapter that, finding global solution for nonconvex
optimization problems is difficult in general. To circumvent this difficulty, we seek to
construct good convex approximations of nonconvex problems that will allow us to
exploit the fact that convex optimization problems are easier to solve. This idea
motivates the introduction of the notion of convex relaxation.
Convex relaxation. Let
P )(min{* xfz }Xx where
f (x) : Rn R and
X Rn be an optimization problem. We say that
R )(min{* xgu }Yx is a relaxation
of
P if (i)
g(x) f (x) for all
x X and (ii)
X Y . Further, we say that
R is a convex
relaxation of
P if
R is a convex optimization problem.
Assuming that
P and
R have optimal solutions, it is easy to see that
u* z*. It
is also easy to verify that
R is unbounded if
P is unbounded and that
P is
infeasible if
R is infeasible.
Given an real valued objective function
f (x) : X R that is nonconvex, where
X Rn is a convex set, we define
g(x) to be a convex underestimator of
f (x) over
X if
g(x) is convex over
X and,
g(x) f (x) for all
x X .
We define the convex envelope
(x) of
f (x) over
X to be the pointwise
supremum of all convex underestimators of
f (x) over
X . In other words, the convex
envelope
(x) of
f (x) over
X is the largest convex function underestimating
f (x) . An
illustration of these concepts is given in Figure 2-1.
27
Figure 2-1. Convex underestimator g(x) and convex envelope φ(x) of the function f(x)
It is clear that if
P )(min{* xfz }Xx is an optimization problem where
X is
a convex set and
f (x) is a nonconvex function, then
R )(min{* xu }Xx is a
convex relaxation of
P where
(x) is the convex envelope of
f (x) over
X .
Given a set
X Rn, we define the convex hull of
X to be the smallest convex set
containing
X . An illustration of this concept is given in Figure 2-2.
A B
Figure 2-2. A) Set X B) Convex hull of X
g(x) )(x
)(xf
28
It is clear that if
P )(min{* xfz }Xx is an optimization problem where
X is a
nonconvex set and
f (x) is a convex function, then
R )(min{* xfu )}(Xconvx is a
convex relaxation of
P .
In general, given a feasible set S defined by constraints 0)( xgi for ki ,,1 , it is
difficult to find )(Sconv . However it is easily established that xS { },,1,0)( kixi
is a convex relaxation of S where )(xi is the envelope of )(xgi over )(Sconv .
Convex relaxations play an important role in solving many nonconvex optimization
problems. In the next section, we describe how convex relaxations can be combined
with divide-and-conquer principles to give rise to branch-and-bound algorithms for the
global solution of nonconvex problems.
2.1 Branch-and-Bound
Branch-and-bound (BB) is one of the most widely used methods to solve
nonconvex optimization problem to globally optimality. In this method, the initial problem
is relaxed into a convex problem that is solved. If the optimal solution of the relaxation is
also optimal for the initial problem, the process is stopped. Otherwise the problem is
divided into parts (branching) so that stronger relaxations can be built over the pieces
that hopefully have better bounds (bounding).
To illustrate the approach, we consider the following problem:
z* min 22 )10()10(),( yxyxf (2-1)
s.t. 10x
20y
0,0 yx .
29
Because this problem has a concave objective function and a polyhedral feasible
region, it is clear that an optimal solution can be found at an extreme point of the
feasible region of (2-1). It follows then by inspection that
(0,0) is an optimal solution to
the problem. However, we will ignore this to illustrate how the branch-and-bound
algorithm would solve this problem. A possible implementation of the branch-and-bound
algorithm would work as follows.
Step 0. First, the feasible region is relaxed to 0M , which is chosen to be the
convex hull of
{(0,0),(30,0),(0,30)}. This set is a simplex that contains the feasible region
D . For more information about simplices, we refer to [6].
Since the objective function of (2-1) is concave, the affine function coinciding with
),( yxf at the vertices of
M0 is the convex envelope of
f (x,y) over
M0 ; see definition of
concave function in the previous chapter. In particular, the parameters of
(x,y) ax by c can be obtained by solving the system
(0,0) 200,
(30,0) 400,
(0,30) 400 which equates the values of
(x,y) and
f (x,y) at the vertices of
M0 . We
obtain:
200c
10a
10b
and so we conclude that 2001010),( yxyx is a convex underestimator of
22 )10()10(),( yxyxf , i.e., ),(),( yxfyx 0),( Myx . Now, a lower bound to
(2-1) can be found by solving the following linear program
min 2001010),( yxyx (2-2)
s.t. DMyx 0),( .
30
The optimal value of (2-2) is 800 and is achieved at )20,10( . It follows that
z* 800 . Now, observe that the two points )0,0( and )20,10( belong to the feasible
region. Since the feasible region is convex, the segment between them, call it 0MS also
belongs to
M0 . Optimizing
f (x,y) over
S yield ),(min{0 yxf 200}),(0
MSyx ,
which is achieved at the point
x0 (0,0). Since
(0,0) is feasible with value
200, we
conclude that
z* 200 . Combining these observations, we can say that
800 z* 200. Because the gap between upper and lower bound is large, we will
subdivide the problem into smaller pieces.
Step 1. In this step, we partition
M0 into two other simplices
M1,1 conv{(0,0),(10,20),(0,30)} and
M1,2 conv{(0,0),(10,20),(30,0)}. After partitioning
M0 , we check that each one of these intersects the feasible region, i.e., DM 1,1 or
DM 2,1 . If not then it is unnecessary to investigate the corresponding simplex
further since it cannot contain the optimal solution (as it does not even contain a
feasible solution). In classical branch-and-bound terminology, we prune the
corresponding branch because we know that a solution to the actual problem cannot lie
in that branch.
In our example, both
M1,1 and
M1,2 intersect
D and so, they both must be
considered. We then must construct a convex relaxation of (2-1) over
M1,1 and
M1,2 .
)}30,0(),20,10(),0,0{(1,1 convM
)}0,30(),20,10(),0,0{(2,1 convM
)}30,0(),0,30(),0,0{(0 convM
31
Proceeding as before, we compute
1,1 ax by c and
1,2 ax by c. We calculate
the coefficients of these envelopes as explained in the previous step to obtain
1,1 30x 10y 200 and 20010102,1 yx . We now can solve convex relaxations of
(2-1) over
M1,1 and
M1,2 respectively by solving linear programs. We obtain the lower
bounds
1,1 400 achieved at
(0,20) and
1,2 300 achieved at
(10,0). Therefore, we
conclude that
z* 1 min(1,1,1,2) 400. Looking for feasible solutions in
M1,1 and
M1,2 ,
we obtain that
1,1 min f (SM1,1) 200 is achieved at )0,0( and )(min
2,12,1 MSf
200 is achieved at )0,0( , where, )}20,10(),20,0(),0,0{(1,1MS and
)}20,10(),0,10(),0,0{(2,1MS . This leads to an overall upper bound of
200),min( 2,11,11 achieved at )0,0(1 x . Because 11 , we will partition the
problem further.
Step 2. In the next iteration, we check first whether any of the previous branches
can be pruned on the basis that 1,1 or 2,1 1 . In fact, in such a situation, the best
possible solution in the branch is inferior to one already discovered in the tree and
therefore cannot be optimal. This step is one of the fundamental features of branch-and-
bound algorithms that allows entire portions of the feasible region to remain unexplored
on the basis that they cannot possibly contain an optimal solution.
In our case, the best solution found so far only has value
200 and therefore no
pruning occur at this stage. We must, therefore partition 1,1M into 1,2M and 2,2M and to
partition 1,1M into 3,2M and 4,2M so as to define
)}20,10(),20,0(),0,0{(1,2 convM ,
32
)}20,10(),20,0(),30,0{(2,2 convM ,
)}20,10(),0,10(),0,0{(3,2 convM ,
)}20,10(),0,10(),0,30{(4,2 convM .
Following the same procedure as mentioned in Step 1 to find
2 and
2, we obtain
that,
2 200 is achieved at
x2 (0,0) and
2 200. Since
2 2, we stop the
algorithm with the conclusion that )0,0(x is a globally optimal solution to (2-1).
There are many implementation details that are crucial in applying branch-and-
bound algorithm successfully, including how the feasible region is divided, how
relaxations are obtained, and in what order nodes are evaluated. Convergence of the
algorithm to an -optimal solution is not obvious but can be guaranteed under some
conditions. We refer the interested reader to [6] for a discussion of these important
questions.
2.2 Building Convex Underestimator
The above example illustrates the fact that, in order for branch-and-bound to be
effective, one needs to be able to construct convex relaxations of nonconvex problems
easily. It is also clear that such relaxations must improve as the problem gets
subdivided.
Next, we describe classical results on the derivation of convex underestimators
and convex envelopes for bilinear problems. Consider the problem
min cyxyaxyxf ),( (2-3)
s.t. Xx , Yy
where,
Rca , , RYX , .
33
Let ),( yxK underestimate the function
f (x,y) over K where K is any convex set
in domain of the function. Then it can be argued that
cyyxaxyx KK ),(),( (2-4)
where ),( yxK denotes the convex underestimator of xy over K . Now, it requires
designing a procedure to build ),( yxK .
Let },:),{( yxyxp , be a hyper-rectangle that contains the feasible
region, and let
pi {(xi,yi) :i xi i,i yi i} be such that
nn pppppp 1321 . This type of partitioning is used in the branch-and-bound
algorithm for BP developed by Al-Khayyal and Falk [1]. We seek to develop relaxations
of the terms
xi,yi over
pi .
We can safely say that:
xi i 0 (2-5)
and
yi i 0 . (2-6)
Multiplying (2-5) and (2-6) gives,
xiyi ixi iyi ii . In other words,
f1(xi,yi) ixi iyi ii underestimates
xiyi over
pi .
Similarly,
0 ii x (2-7)
and
0 ii x . (2-8)
Multiplying (2-7) and (2-8) gives iiiiiiii yxyx . In other words,
iiiiiiii yxyxf ),(2 also underestimates
xiyi over
pi .
34
Since the pointwise supremum of convex underestimators of a function also
underestimates the function, we conclude that
)},(),,(max{),( 21 iiiiip yxfyxfyx (2-9)
is a convex underestimator of
xiyi over
pi .
It can be shown that (2-9) forms the convex envelope of
xiyi over
pi . We refer to
Al-Khayyal and Falk [1] for a proof.
2.3 Factorable Relaxations
For more general functions, finding convex envelopes is typically a difficult
problem. However, several techniques have been proposed to derive convex
underestimators. We describe one such technique next that was initially proposed by G.
P. McCormick [9].
This technique applies to factorable functions that can be obtained recursively,
using sums and products of a family of univariate base functions whose convex
envelopes are known. The family of base functions can contain polynomial, exponential
or trigonometric functions. The technique proceeds by breaking down the initial function
into its constituent terms and then approximating these terms individually. The
technique can be used to obtain convex underestimators of objective functions but also
can be used to construct convex relaxations of nonconvex constraints as
˜ f (x) r is a
convex relaxation of
f (x) r whenever
˜ f (x) is a convex underestimator of
f (x) . To
illustrate the ideas behind the factorable relaxation technique, we consider the following
constraint:
f (x,y,z) (x y)2 z 3xy 2 4 (2-10)
35
where, it is known otherwise that variables belong to the unit hypercube, i.e., 10 x ,
10 y , and 10 z .
Our goal is to find a good convex underestimator
(x,y,z) of
f (x,y,z) over 3]1,0[ as
(x,y,z) 4 will provide a convex relaxation to
f (x,y,z) 4 . At first glance, it may
appear difficult to devise such a convex underestimator. For this reason, we will start
breaking down the function into components until we obtain terms that are sufficiently
simple to approximate.
To break down
f (x,y,z), we introduce two new variables
u1 and
u2 as
u1 (x y)2z (2-11)
u2 xy 2. (2-12)
Still, it is difficult to relax these terms and hence we will break them down further
by introducing new variables
u3 and
u4 as
u3 x y,
u4 u3
2,
u1 u4z. Each of these
terms is then relaxed as summarized in Table 2-1.
Table 2-1. Sub-problems obtained after breaking down
u1 (x y)2z
Sub-problem Convex relaxation
u3 x y This constraint is convex and
1 u3 1.
u4 u3
2 As shown in the Figure 2-3, the feasible region defined by this constraint is not
convex over 11 3 u . It can be
observed from the figure that
u3
2 u4 1
provides a convex relaxation of the feasible set defined by this constraint.
u1 u4z As discussed earlier in this chapter, we can use standard techniques to find a convex relaxation to this expression; see convex envelope for bilinear problems, which yields
u1 u4 ,
u1 z ,
u1 u4 z1
and
u1 0.
36
Figure 2-3. Convex relaxation of the sub-problem
u4 u3
2
A convex relaxation for the set if feasible solutions to equation
u1 (x y)2z for
3]1,0[),,( zyx can therefore be written as:
u3 x y (2-13)
u4 u3
2 (2-14)
u4 1 (2-15)
u1 u4 (2-16)
u1 z (2-17)
u1 u4 z1 (2-18)
u1 0 . (2-19)
For
u2, we proceed similarly and introduce new variable
u5 as
u5 y2,
u2 u5x .
Each of these terms is then relaxed as summarized in Table 2-2.
(u3)2
u3
37
Table 2-2. Sub-problems obtained after breaking down
u2 xy 2
Sub-problem Convex relaxation
u5 y2 As shown in Figure 2-4, the feasible region defined by this constraint is not convex over 10 y . It can be observed
from the figure that
y2 u5 y provides a
convex relaxation of the feasible set defined by this constraint.
u2 u5x Similarly to the previous case, a convex relaxation is given by
u2 u5,
u2 x ,
u2 u5 x 1 and
u2 0.
Figure 2-4. Convex relaxation of the sub-problem
u5 y2
A convex relaxation for the set of feasible solution to equation
u2 xy 2 for
2]1,0[),( yx can therefore be written as:
u5 y2 (2-20)
u5 y (2-21)
u2 u5 (2-22)
u2 x (2-23)
u2 u5 x 1 (2-24)
y2
1
y
38
u2 0, (2-25)
together with
10 x , 10 y , 10 z . (2-26)
Combining (2-13) – (2-19) and (2-20) – (2-26) with
f (x,y,z) u1 3u2 4 (2-27)
provides a convex relaxation for the set of feasible solution to constraint (2-10).
The ideas presented here can be applied general factorable functions. We refer to
Tawarmalani and Sahinidis [11] for a more detailed description. Factorable relaxations
are used in commercial global optimization software such as Baron and Lindoglobal.
39
CHAPTER 3 CONVEX RELAXATIONS FOR BILINEAR COVERING SETS
3.1 Motivation
As discussed in the previous chapters, building strong convex relaxations is key in
solving many complex nonconvex problems to global optimality. We next demonstrate
through two examples that multilinear covering constraints of the form
x i, j
j1
m
d
i1
n
(3-1)
occur in practical problems. This will be the primary motivation for our later polyhedral
studies.
3.1.1 Strassen’s Algorithm
The problem of multiplying two matrices of dimension nn is a fundamental
problem in linear algebra. When multiplying matrix
nj
niijaA
,,1
,,1
with
nj
niijbB
,,1
,,1
,
we create a new matrix
nj
niijcC
,,1
,,1
whose components ijc are obtained through the formula kj
n
k
ikij bac
1
for ni ,,1 and
nj ,,1 .
A straightforward application of the formula shows that it is possible to compute
the product of two nn matrices using 3n multiplications and )1(2 nn additions,
40
resulting in a )( 3nO algorithm. There are, however, algorithms for matrix multiplication
with running time of )( rnO where 3r . Strassen’s algorithm [10] is one such algorithm.
The fundamental idea in Strassen’s algorithm is that, when considering matrices
2221
1211
aa
aa
and
2221
1211
bb
bb,
it is possible to compute their product
2222122121221121
2212121121121111
2221
1211
babababa
babababa
cc
cc
using only 7 multiplications and 18 additions/subtractions. This can be done as follows.
First, we compute:
1T = 2211 aa 2211 bb
2T = 2221 aa 11b
3T = 11a 2212 bb
4T = 22a 1121 bb
` 5T = 1211 aa 22b
6T = 1121 aa 1211 bb
7T = 2212 aa 2221 bb .
Then, we combine these terms as follows:
11c = 7541 TTTT
41
12c = 53 TT
21c = 42 TT
22c = 6321 TTTT .
Because matrices can be block multiplied, a recursive application of the above
equations yields a )(72log
nO algorithm for matrix multiplication.
An important question is that of determining whether such rule can be obtained
systematically. This question can be answered positively as schemes to multiply two
22 matrices using only 7 multiplications corresponds to feasible solutions to a mixed
integer nonlinear problem with constraints:
7
1
0t
skhrigrstghtikt yxz 2,1,,,,, srhgki (3-2)
where,
zijt,xijt,yijt {1,0,1} 2,1, ji and 7,,1t .
In the above model, the variable ijtz represents the coefficient of tT in ijc , ijtx
represents the coefficient of ija in tT , ijty represents the coefficient of ijb in tT ,
ij 1
i j and 2,1, ji , and 0ij ji and 2,1, ji . More precisely, a feasible
solution to Problem (3-2) yields a way of computing C with the 7 multiplications as
tT =
}2,1{, ji
ijijtax
}2,1{, ji
ijijtby ,
with the following additions,
ijc =
7
1t
tijtTz .
Similarly, one could determine if there exists a way of multiplying 33 matrices
42
using 23 multiplications by determining if the following system is feasible or not:
zikt xghtyrst ighrsk 0t1
23
3,2,1,,,,, srhgki (3-3)
where,
zijt,xijt,yijt {1,0,1} 3,2,1, ji and 23,,1t .
1ij
i j , 3,2,1, ji and
ij 0
i j , 3,2,1, ji .
A feasible solution to Problem (3-2) would yield an algorithm for multiplying
3 3
matrix with 23 multiplications instead of 27. Such a feasible solution, in turn, would lead
to a )(233log
nO algorithm for matrix multiplication. A feasible solution to Problem (3-3) can
be found in [7].
The constraints of the above models involve multilinear functions. It is simple to
reformulate them using variables rstghtghrst yxu to transform (3-2) and (3-3) into a
problem containing only bilinear constraints, some of which can be relaxed to form (3-1)
3.1.2 Staff Scheduling Problems
Traditional staff scheduling models also can be formulated as bilinear integer
programs if the shifts are part of the decision variables.
To clarify the idea, consider a problem in which a call center has to ensure that at
least d operators are on duty during time slot t . Operators must be assigned to one of
the k shifts (whose characteristics must be decided) in a way that minimizes the
number of operators used. Shifts are characterized only by their start time. We assume
that shifts are required to contain a sequence of r consecutive time slots of work
followed by 1 time slot of rest, followed by r time slots of work.
To model such problem, we introduce variables tkX as
43
1tkX , if time slot t is part of shift k
0tkX , otherwise.
We introduce tkZ as
1tkZ , if shift k starts at time t
0tkZ , otherwise,
and let kY to represent the number of operators assigned to shift k .
The problem can then be formulated as:
min
K
k
kY1
s.t.
(1)
K
k
tktk dYX1
},,1{ Tt
(2)
T
t
tk rX1
2 },,1{ Kk
(3) tkkit ZX )( },,1{ Kk , },,1{ Tt , }2,,1,1,,1{ rrri
(4) tkkrt ZX 1)( },,1{ Kk , },,1{ Tt
(5)
T
t
tkZ1
1 },,1{ Kk .
In the above model, constraint (1) requires that sufficiently many operators are
present during each time period, constraint (2) imposes that operators work for r2 time
periods, constraints (3) and (4) state that the shift requires work in the period 1,,1 r
and rr 2,,1 , and rest in period r after the beginning of the shift while constraint (5)
requires that the shift has a single starting period. In model (1) - (5), we replace rt
with Trt whenever Trt .
44
In the above model, we see that the constraints that require that sufficiently many
operators are available at any point in time are bilinear and are of the form (3-1). We
also observe that they contain only integer variables.
3.2 Problem Description
As illustrated before in this chapter, constraints of the form:
n
i
ii dyx1
(3-4)
where,
xi {0,1} Ni ,
},,0{ ii uy Ni ,
ui Z Ni ,
nuuuu 321 ,
Zd ,
},,1{ nN
appear in the formulation of several practical problems. It would therefore be useful to
derive convexification procedures for them. Current convexification procedures applied
to this problem would (i) relax integrality and (ii) build convex relaxations by finding
concave overestimators of the product ii yx . This results in a double source of weakness
as (i) it can be seen that integrality makes the convex hull of feasible solutions to (3-4)
polyhedral while the convex hull of the continuous relaxation is not and (ii) the relaxation
dxg )( where g is a concave overestimator of ii yx does not yield the convex hull of
feasible solutions to (3-4). Therefore, we wish to derive expressions for the convex hull
of sets defined by (3-4).
45
We define a polyhedron as the set of feasible solutions to a finite number of
inequalities and equalities:
nRxP { }, dCxbAx (3-5)
where nmRA , mRb , npRC , and pRd . We say that P is bounded if P is
contained in a ball of finite radius. A bounded polyhedron is said to be polytope.
Because the feasible sets of integer bilinear sets are composed of a finite number of
points, the following result can be easily established using Minkowski-Weyl’s Theorem
[13].
We denote the feasible region of Problem (3-4) by B and the convex hull of B by
)(Bconv .
Proposition 3.2.1: The convex hull of the feasible solutions to Problem (3-4) is a
polytope, i.e., )(Bconv is a polytope.
As a first step in the derivation of such convex hulls, we use MATLAB to generate
the convex hulls of sample problems where the values of d and iu are fixed. To this
end, we use the MATLAB function K = convhulln(X) where },,1{ miixX
, and ix s are
the feasible solutions (vectors in n2 dimensions) of Problem (3-4). We restricted
ourselves to small values of n , d , and iu s so that the function K = convhulln(X) works
relatively fast.
During the study of the results produced by MATLAB, we found that the structure
of the convex hull is strongly influenced by the position of d in the sequence
nuuu 21 . Therefore we divide the study of the problem into different cases
depending upon position of d in the sequence. In particular, we define case k to be the
46
case where
uk1 d and
uk d .
We mention that, in case 0, every individual term
xi,yi can satisfy the right-hand-
side requirement in Inequality (3-4) by itself (without the use of any other pair of
variables). In case 1, however, the term
x1,y1 must always be used in conjunction with at
least another. Finally in case n, none of the variable pairs
xi,yi can be used without also
selecting at least another pair.
We further categorize convex hulls into two categories: specific convex hull and
generic convex hull. We call convex hulls generic, when their structure remained
unchanged for all values of n in a particular case. For example, suppose that the
convex hull is found for n = 2 in case 0. If the convex hull can be found for n = 3 in case
0 using the same method, then we say that case 0 has a generic convex hull. In our
studies, we will mainly focus on those cases for which we could identify a generic
structure for the convex hull. In the following sections, we will see that case 0 and case
1 have generic convex hulls.
3.3 Building Convex Hull for Different Cases
In this section, we use our MATLAB codes to derive convex hulls of different
instances of case 0 and case 1 and we use these examples to conjecture facet-defining
inequalities and convex hull descriptions for certain integer bilinear covering sets. In
Section 3.3.1, we consider case 0 while in Section 3.3.2, we consider case 1.
3.3.1 Case 0
In this case, we have
n
i
ii dyx1
(3-6)
47
where }1,0{ix , },,0{ ii uy , Zui , and Zd with nuuu 21 , du 1 .
3.3.1.1 Problem 1
In Problem (3-6), we set 3d and 2n to obtain:
32211 yxyx (3-7)
where
xi {0,1}, },,0{ ii uy , Zui , 3, 21 uu , and 21 uu .
To use MATLAB, we also need to choose values for 1u and 2u . For 321 uu , we
list all feasible solutions to the problem and find the convex hull using MATLAB function
K = convhulln(x). For this instance, we present all feasible solutions in Table 3-1. These
points were obtained using a code we wrote MATLAB. The code is presented in
Appendix.
Table 3-1. Feasible points for 32211 yxyx and 321 uu
1x 1y 2x 2y
1 3 0 0 1 3 1 0 1 3 0 1 1 2 1 1 1 3 1 1 1 3 0 2 1 1 1 2 1 2 1 2 1 3 1 2 1 3 0 3 0 0 1 3 1 0 1 3 0 1 1 3 1 1 1 3 0 2 1 3 1 2 1 3 0 3 1 3 1 3 1 3
Observe that, in Table 3-1, not all points are necessary to build the convex hull as
some of the points are convex combination of other points. In particular, (1,3,0,1) is a
48
convex combination of (1,3,0,0) and (1,3,0,2) and therefore could be omitted from the
list without altering the convex hull. However, computing convex hulls with these
unnecessary points does not compromise the validity of our approach.
The convex hull of feasible points of Table 3-1 generated using MATLAB, is presented
in Table 3-2.
Table 3-2. Convex Hull for 32211 yxyx
Convex hull (for restricted problem) Convex hull (for original problem)
333 21 xx 333 21 xx
33 21 xy 33 21 xy
33 21 yx 33 21 yx
321 yy 321 yy
1, 21 xx 1, 21 xx
3, 21 yy 11 uy , 22 uy
Repeating the computation for various values of 1u and 2u we conjecture the
convex hull for the original problem (3-7), presented in the right side of Table 3-2.
Intuitively, this works because the convex hull of Problem (3-4) has two kinds of
inequalities: lower bounding and upper bounding. Lower bounding inequalities do not
change as long as upper bounding inequalities do not interact with them. Limit cases
are encountered when an upper bounding inequality and a lower bounding inequality of
the convex hull meet at a vertex.
3.3.1.2 Problem 2
We consider now a case larger right-hand-side value. In particular, we set 4d
and 2n in Problem (3-6) to obtain:
42211 yxyx (3-8)
where
xi {0,1}, },,0{ ii uy , Zui , 4, 21 uu , and 21 uu .
49
To conjecture the convex hull of this problem, we apply the same procedure as
before. We first restrict 421 uu so that we can find the convex hull of the restricted
problem easily. After finding it, we replace the upper bounds on 1y and 2y with 1u and
2u respectively.
The convex hull of feasible points generated using MATLAB, is presented in Table 3-3.
Table 3-3. Convex Hull for 42211 yxyx
Convex hull (for restricted problem) Convex hull (for original problem)
444 21 xx 444 21 xx
44 21 xy 44 21 xy
44 21 yx 44 21 yx
421 yy 421 yy
1, 21 xx 1, 21 xx
4, 21 yy 11 uy , 22 uy
In the right side of the table, we present our conjecture for the convex hull when 1u
and 2u are general. From the results of Table 3-2 and Table 3-3, we next infer the
general form for the convex hull of case 0 when 2n .
3.3.1.3 Convex hull for n=2
We compare the inequalities obtained in Problem 1 and 2. Inequalities 333 21 xx
and 444 21 xx are similar, and the difference can be easily attributed to d . We
therefore infer that, ddxdx 21 is a required inequality for any value of d . We will
verify this claim later in this chapter. Second we observe that inequalities 33 21 xy
and 44 21 xy are similar and have an obvious dependence on d that can be
generalized to ddxy 21 . Similarly, inequalities 33 21 yx and 44 21 yx can be
generalized to dydx 21 . Regarding this second and third inequalities, it can be
50
argued that if anyone of them is a valid inequality then the other one is also valid,
because of symmetry. Fourth similar inequalities are 321 yy and 421 yy which
can be extended in terms of d to dyy 21 .
Hence, we conjecture that the convex hull of Problem (3-4) in case 0 when 2n
is given by:
ddxdx 21
ddxy 21
dydx 21
dyy 21
11 uy , 22 uy
11 x , 12 x .
We next seek to generalize this conjecture to situation where 3n . We do so by
considering an example first.
3.3.1.4 Problem 3
In Problem (3-6), we set 4d and 3n to obtain:
4332211 yxyxyx (3-9)
where
xi {0,1}, },,0{ ii uy , Zui , 4,, 321 uuu , and 321 uuu .
Similarly to Problem 1, we obtain the convex hull of feasible points by first
generating a list of all feasible solutions through a MATLAB code and then computing its
convex hull using K=convhulln(X). Results are presented in Table 3-4 for the case
where 4321 uuu and a conjecture for the convex hull is presented on the right side
of table when 3214 uuu .
51
Table 3-4. Convex Hull for 4332211 yxyxyx
Convex hull (for restricted problem) Convex hull (for original problem)
4444 321 xxx 4444 321 xxx
444 321 yxx 444 321 yxx
444 321 xyx 444 321 xyx
444 321 xxy 444 321 xxy
44 321 yyx 44 321 yyx
44 321 yxy 44 321 yxy
44 321 xyy 44 321 xyy
4321 yyy 4321 yyy
1,,0 321 xxx 1,,0 321 xxx
4,,0 321 yyy 110 uy , 220 uy , 330 uy
3.3.1.5 Convex hull for case 0
The results of Table 3-4 combined with those of Table 3-2 and Table 3-3, suggest
the following form for )(Bconv :
dyxdSNj
ji
Si
\
NS where },,1{ nN (3-10)
1ix Ni (3-11)
ii uy Ni (3-12)
0ix Ni for 3n (3-13)
0iy Ni for 3n (3-14)
3.3.2 Case 1
We now focus on developing a conjecture for the convex hull of integer bilinear
covering set in case 1. In this case, we consider the set of feasible solutions to
n
i
ii dyx1
(3-15)
where
xi {0,1}, },,0{ ii uy , Zui , and Zd when nuuu 21 , we assume
52
here that du 1 , du 2 .
3.3.2.1 Problem 4
In Problem (3-6), we set 4d and 3n to obtain:
4332211 yxyxyx (3-16)
where
xi {0,1}, },,0{ ii uy , Zui , 41 u , 4, 32 uu , and 321 uuu .
Similarly to Section 3.3.1.1, we will build the convex hull for Problem (3-16). We
consider two cases. First, we set 31 u and 4, 32 uu . Second, we set
u1 2 and
4, 32 uu . After building the convex hulls of these two instances, we will conjecture a
convex hull description for the general problem. Results are presented in Table 3-5.
Table 3-5. Convex Hull for 4332211 yxyxyx in case 1
Convex hull ( 31 u
and 4, 32 uu )
Convex hull (
u1 2 and
4, 32 uu )
Convex hull (for original problem)
443 321 yxx
2x1 4x2 y3 4 44 3211 yxxu
443 321 xyx
2x1 y2 4x3 4 44 3211 xyxu
43 321 yyx
2x1 y2 y3 4 43211 yyxu
44 321 yxy 44 321 yxy 44 321 yxy
44 321 xyy 44 321 xyy 44 321 xyy
4321 yyy 4321 yyy 4321 yyy
132 xx
x2 x3 1
x2 x3 1
132 yx
2x2 y3 2 )4()4( 1321 uyxu
132 xy
y2 2x3 2 )4()4( 1312 uxuy
1,,0 321 xxx 1,,0 321 xxx 1,,0 321 xxx
30 1 y 4,0 32 yy 20 1 y 4,0 32 yy 110 uy , 220 uy ,
330 uy
Comparing the inequalities, we present, in the right column a conjecture for the
convex hull description of feasible solutions to Problem (3-16).
We now examine the dependence of this convex hull on the right-hand-side value d .
53
3.3.2.2 Problem 5
In Problem (3-6), we now set
d 5 and 3n to obtain
x1y1 x2y2 x3y3 5 (3-17)
where
xi {0,1}, },,0{ ii uy , Zui ,
u1 5,
u2,u3 5, and 321 uuu .
Similarly to Section 3.3.2.1, we consider two sets of values for the bounds
321 ,, uuu . The first choice is obtained by setting
u1 4 and
u2,u3 5. The second is
obtained by setting
u1 3 and
u2,u3 5. Results are presented in Table 3-6.
Table 3-6. Convex Hull for
x1y1 x2y2 x3y3 5
Convex hull (
u1 4 and
u2,u3 5)
Convex hull (
u1 3 and
u2,u3 5)
Convex hull (for original problem)
554 321 yxx 553 321 yxx 55 3211 yxxu
554 321 xyx 553 321 xyx 55 3211 xyxu
54 321 yyx
3x1 y2 y3 5
u1x1 y2 y3 5
55 321 yxy
y1 5x2 y3 5
y1 5x2 y3 5
55 321 xyy
y1 y2 5x3 5
y1 y2 5x3 5
5321 yyy
y1 y2 y3 5
y1 y2 y3 5
132 xx 132 xx
x2 x3 1
132 yx 22 32 yx
(5 u1)x2 y3 (5 u1)
132 xy 22 32 xy
y2 (5 u1)x3 (5 u1)
1,,0 321 xxx 1,,0 321 xxx 1,,0 321 xxx
40 1 y 5,0 32 yy 30 1 y 5,0 32 yy 110 uy , 220 uy ,
330 uy
Comparing the inequalities, we present, in the right column a conjecture for the
convex hull description of feasible solutions to Problem (3-17).
3.3.2.3 Convex hull for n = 3
We are now ready to conjecture a polyhedral description of the convex hull of
integer bilinear covering sets in case 1. Comparing inequalities with similar structure for
Problems 4 and 5, we conjecture the following description of the convex hull of case 1
54
for n = 3.
Table 3-7. Convex Hull for
x1y1 x2y2 x3y3 d
Convex hull ( 4d ) Convex hull (
d 5) Convex hull (
d)
44 3211 yxxu
u1x1 5x2 y3 5
u1x1 dx2 y3 d (1)
44 3211 xyxu
u1x1 y2 5x3 5
u1x1 y2 dx3 d (2)
43211 yyxu
u1x1 y2 y3 5
u1x1 y2 y3 d (3)
44 321 yxy
y1 5x2 y3 5
y1 dx2 y3 d (4)
44 321 xyy
y1 y2 5x3 5
y1 y2 dx3 d (5)
4321 yyy
y1 y2 y3 5
y1 y2 y3 d (6)
x2 x3 1
x2 x3 1
x2 x3 1 (7)
)4()4( 1321 uyxu
(5 u1)x2 y3 (5 u1)
(d u1)x2 y3 (d u1)(8)
)4()4( 1312 uxuy
y2 (5 u1)x3 (5 u1)
y2 (d u1)x3 (d u1)(9)
1,,0 321 xxx 1,,0 321 xxx 1,,0 321 xxx (10)
110 uy ,
220 uy , 330 uy 110 uy ,
220 uy , 330 uy 110 uy , 220 uy ,
330 uy (11)
3.3.2.4 Convex hull for case 1
We next discuss each of the inequalities in Table 3-7 so as to see how they can be
generalized to problems with
n pairs of variables.
First, we observe the inequality
x i 1i 2
n
(3-18)
is always part of the description; see (7). As we discussed before,
x1y1 alone cannot
meet the right-hand-side of Problem (3-15): one of the other
xiyi has to be active. This
argument holds for any value of
n. Hence, we conjecture that the inequality (3-18) is
part of the convex hull description for any value of
n in case 1.
Second, inequalities (1), (2), (3), (4), (5), and (6) suggest that
dyxaSNj
ji
Si
i \
(3-19)
55
where, },,1{ nN , NS , },,2{ nS , and where },min{ dua ii are part of the
convex hull description. Hence, we conjecture that the family of inequalities (3-19) is
part of the description of )(Bconv for any value of
n in case 1.
To generalize (8) and (9), we write
1
\
1)( udyxudRNj
ji
Ri
(3-20)
where, },,2{ nN , NRNR \, .
Inequality (3-20) can be interpreted in light of Problem (3-6) as an application of
case 0 over variables )()( 22 nn yxyx when
x1y1 are set to their maximum values, with
the exception of inequalities based on ØR and Ø\ RN , since the cases ØR and
Ø\ RN are already handled in (3-18) and (3-19). This argument also holds true for
any value of
n in case 1.
As a result of the above discussion, we conjecture that case 1 has a generic
convex hull of the form:
x i 1i 2
n
(3-21)
dyxaSNj
ji
Si
i \
(3-22)
1
\
1)( udyxudRNj
ji
Ri
(3-23)
1ix Ni (3-24)
ii uy Ni (3-25)
0ix Ni (3-26)
0iy Ni (3-27)
56
where, },,1{ nN , },,2{ nN , NS , NS , NRNR \, , and where
},min{ dua ii .
3.4 Facet-Defining Inequalities and Convex Hulls
As described in Section 3.2, the convex hull of integer bilinear covering sets is
polyhedral. To study further the polyhedral structure of these sets, we next describe
some basic notions about polyhedra and polytopes that we will use in the remainder of
this thesis. We focus on polyhedra of the form:
nRxP { }bAx . (3-28)
Not all inequalities in the definition of a polyhedron P are necessary in describing
P . In order to differentiate the inequalities that are necessary from those that are not,
we introduce a few definitions; we refer to [3] for more details.
Affine independence. Points kxxx ,,, 21 in nR are affinely independent if the
system
01
i
k
i
ix ,
and
01
k
i
i
has 021 k as unique solution.
Next we introduce the concept of dimension of a polyhedron P . The dimension of
P ( Pdim ) is one less than the maximum number of affinely independent points in P . A
polyhedron nRP is said to be full-dimensional if nP dim . An inequality kk bxA is
said to valid for P if
57
kk bxA Px
or equivalently if
xAkmax{ kbPx } .
Given a valid inequality kk bxA for a polyhedron P , we define the face of P
induced by kk bxA to be PxFKK bA {),( }kk bxA . A valid inequality kk bxA for P is
said to be facet-defining for P if 1dimdim ),( PFKK bA . In other words, kk bxA is a
facet-defining inequality for P if there exists Pdim affinely independent points in
),( KK bAF .
Given a full-dimensional polyhedron nRP and a valid inequality kk bxA , proving
that kk bxA is a facet-defining inequality simply requires displaying n affinely
independent points in ),( KK bAF . Several variants of the idea have been proposed that are
sometimes easier to use in practice. We refer the interested reader to [2] for a
description.
3.4.1 Facet-Defining Inequality for Case 0
In our previous section, we conjectured that )(Bconv in case 0 is given by:
dyxdSNj
ji
Si
\
NS (3-29)
1ix Ni (3-30)
ii uy Ni (3-31)
0ix Ni for 3n (3-32)
0iy Ni for 3n (3-33)
where, },,1{ nN .
58
We will show next that these inequalities are facet-defining for )(Bconv . To this
end, we first show that inequalities (3-29), (3-30), (3-31), (3-32), and (3-33) are valid for
B and therefore for )(Bconv . It is easily verified that )(Bconv is full-dimensional when
2n . When 1n , the set B is trivial, i.e., dyx 11 where }1,0{1x , },,0{ 11 uy has a
convex hull that is given by
11 x
and
11 uyd .
This shows that for 1n , )(Bconv is not full-dimensional.
Proposition 3.4.1.1: Inequalities (3-30), (3-31), (3-32), and (3-33) are valid inequalities
for B
Proof: Inequalities (3-30), (3-31), (3-32), and (3-33) are valid for B since they belong to
the initial description of the problem. ■
Proposition 3.4.1.2: Inequality (3-29) is valid for B .
Proof: Consider any feasible solution ),( yx
s.t.
n
i
ii dyx1
(3-34)
where
xi {0,1}, and },,0{ ii uy . Denote },,,{ 21 nxxxx and },,,{ 21 nyyyy .
Case 1: For ØS , Inequality (3-29) reduces to
dyNj
j
.
The inequality is valid for B since
dyxyNj
jj
Nj
j
59
where the first inequality holds since
Nj
jj
Nj
j yxy and the second inequality holds
because of (3-34).
Case 2: For NS , Inequality (3-29) reduces to
ddxNi
i
or equivalently
1Ni
ix .
This inequality is clearly valid for B as, when 0Ni
ix ,
dyxNi
ii
0 .
Case 3: For ØS , Ø\ SN .
We consider two subcases.
Case 3a: Assume that the solution ),( yx of B satisfies
Si
ix 1. Then
dxdyxd i
SiSNj
ji
Si
\
,
showing that ),( yx satisfies (3-29).
Case 3b: Assume that the solution ),( yx of B satisfies
Si
ix 0 . It follows from
(3-34) that if
Si
ix 0 then
SNj
j dy\
. Then
dyyxdSNj
j
SNj
ji
Si
\\
,
showing that ),( yx satisfies (3-29). ■
60
Next, we are going to show that (3-29), (3-30), (3-31), (3-32), and (3-33) are facet-
defining inequalities for )(Bconv . In all subsequent proofs, we let Nj
jji
Ni
i yx be
any inequality that induces a face )(F in n2 dimensional space that contains the face of
)(Bconv defined by the above mentioned inequalities.
Proposition 3.4.1.3: Inequality (3-29) is facet-defining for )(Bconv .
Proof: We consider three cases:
Case 1: Assume ØS , Ø\ SN , i.e., (3-29) is of the form )(G dyxdSNj
ji
Si
\
. For
all
i S , (
xi 1,yi d) is a feasible solution that satisfies )(G at equality. Therefore, it
must belong to )(F which yields
(1)
i di
i S .
For all
i S and
j N \ S , the feasible solution (
xi 1,x j 1,yi d ) satisfies )(G at
equality and therefore must belong to the face )(F yielding
(2)
i j di ,Si SNj \ .
It follows from (1) and (2) that
j 0
j N \ S . Therefore the face )(F is defined by
Nj
jji
Si
i yx .
For all
i S and
j N \ S , the feasible solution (
x j 1,yi d,y j d ) satisfies )(G at
equality and therefore must belong to the face )(F yielding
(3) ji dd ,Si SNj \ .
For all
j N \ S , the feasible solution (
x j 1,y j d) satisfies )(G at equality and
therefore must belong to the face )(F yielding
61
(4)
d j
j N \ S or
j
d.
It follows from (3) and (4) that
i 0
i S , and so the face )(F is defined by
SNj
ji
Si
i yd
x\
.
For all
i S , the feasible solution (
xi 1,yi d) satisfies )(G at equality and therefore
must belong to the face )(F yielding
(5)
i
i S
and so the face )(F is defined by
SNj
ji
Si
yd
x\
,
which shows that the inequality defining )(F is a scalar multiple of )(G .
Case 2: Assume ØS , Inequality (3-29) reduces to )(G dyNj
j
. For all
j N ,
(
x j 1,y j d) is a feasible solution that satisfies )(G This yields
(1)
j d j
j N .
For all
i N,
j N , and
i j , the feasible solution (
xi 1,x j 1,y j d ) satisfies )(G at
equality and therefore must belong to the face )(F yielding
(2)
i j d j ,Ni jiNj , .
It follows from (1) and (2) that
i 0
i N . Therefore face )(F is defined by
Nj
jj y .
For all
j N , the feasible solution (
x j 1,y j d) satisfies )(G at equality and therefore
must belong to the face )(F yielding
62
(3)
d j
j N or
j
d
and so the face )(F is defined by
Nj
jyd
,
which shows that the inequality defining )(F is a scalar multiple of )(G .
Therefore dyNj
j
is a facet-defining inequality for )(Bconv .
Case 3: For Ø\ SN , Inequality (3-29) reduces to )(G ddxNi
i
. For all
i N ,
(
xi 1,yi d) is a feasible solution that satisfies )(G at equality and therefore must
belong to the face )(F yielding
(1)
i di
i N .
For all
i N,
j N , and
i j , the feasible solution (
xi 1,yi d,y j d ) satisfies )(G at
equality and therefore must belong to the face )(F yielding
(2)
i di d j ,Ni jiNj , .
It follows from (1) and (2) that
j 0
j N . Therefore the face )(F is defined by
Ni
ii x .
For all
i N, the feasible solution (
xi 1,yi d) satisfies )(G at equality and therefore
must belong to the face )(F yielding
(3)
i
i N
and so the face )(F is defined
Ni
ix ,
63
which shows that the inequality defining )(F is a scalar multiple of inequality )(G .
Therefore ddxNi
i
is a facet-defining inequality for )(Bconv .
Combining case 1, case 2, and case 3, we have shown that all n2 inequalities described
in (3-29) are facet-defining for )(Bconv . ■
Proposition 3.4.1.4: Inequality (3-30) is facet-defining for )(Bconv .
Proof: Consider )(G 1ix . For all Ni , ( dyx ii ,1 ) is a feasible solution that
satisfies )(G at equality and therefore must belong to the face )(F yielding
(1)
i di .
For all iNj / , the feasible solution ( 1,,1 jii xdyx ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
(2) jii d iNj / .
It follows from (1) and (2) that 0j iNj / . Therefore the face )(F is defined by
Nj
jjii yx .
For all iNj / , the feasible solution ( dydyx jii ,,1 ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
(3) jii dd iNj / .
It follows from (1) and (3) that 0j iNj / .Therefore the face )(F is defined by
iiii yx .
For any iNj / , the feasible solution ( dyxx jji ,1,1 ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
64
(4) i .
It follows from (1) and (4) that 0i , therefore the face )(F is defined by
1ix .
This shows that 1ix is facet-defining for )(Bconv . ■
Proposition 3.4.1.5: Inequality (3-31) is facet-defining for )(Bconv .
Proof: Consider )(G ii uy . For Ni , ( iii uyx ,1 ) is a feasible solution that
satisfies )(G at equality and therefore must belong to the face )(F yielding
(1) iii u .
For all iNj / , the feasible solution ( 1,,1 jiii xuyx ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
(2) jiii u iNj / .
It follows from (1) and (2) that 0j iNj / . Therefore the face )(F is defined by
Nj
jjii yx .
For all iNj / , the feasible solution ( dyuyx jiii ,,1 ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
(3) jiii du iNj / .
It follows from (1) and (3) that 0j iNj / . Therefore the face )(F is defined by
iiii yx .
For all iNj / , the feasible solution ( dyxuy jjii ,1, ) satisfies )(G at equality and
therefore must belong to the face )(F yielding
65
(4) iiu .
It follows from (1) and (4) that 0i , therefore the face )(F is defined by
i
i
yu
or equivalently
ii uy .
This shows that ii uy is facet-defining for )(Bconv . ■
Proposition 3.4.1.6: For 3n , Inequality (3-32) is facet-defining for )(Bconv .
Proof: We use the direct approach of finding n2 affinely independent points in the face
induced by 0ix . We assume without loss of generality that we consider 01 x . The
following n2 points for 3n
),( yx = ),( 22 dee
),( yx = ),( 22 deee i for ni ,,3
),( yx = ),( 22 idedee for ni ,,4,3,1
),( yx = ),( 323 dedee
),( yx = ),( 33 dee
are affinely independent and satisfy 01 x at equality. This shows that 01 x is facet-
defining for )(Bconv . ■
Proposition 3.4.1.7: For 3n , Inequality (3-33) is facet-defining for )(Bconv .
Proof: We use the direct approach of finding n2 affinely independent points in the face
induced by 0iy . We assume without loss of generality that we consider 01 y . The
following n2 points for 3n
66
),( yx = ),( 22 dee
),( yx = ),( 22 deee i for ni ,,4,3,1
),( yx = ),( 22 idedee for ni ,,3
),( yx = ),( 323 dedee
),( yx = ),( 33 dee
are affinely independent and satisfy 01 y at equality. This shows that 01 y is facet-
defining for )(Bconv . ■
In the next section, we consider the linear inequalities we developed for case 1
and sketch that they are facet-defining for )(Bconv .
3.4.2 Facet-Defining Inequality for Case 1
Previously, in this chapter, we conjectured that )(Bconv of case 1 is given by:
x i 1i 2
n
(3-35)
dyxduSNj
ji
Si
i \
},min{ (3-36)
1
\
1)( udyxudRNj
ji
Ri
(3-37)
1ix Ni (3-38)
ii uy Ni (3-39)
0ix Ni (3-40)
0iy Ni (3-41)
where, },,1{ nN , },,2{ nN , NS , NS , NRNR \, .
67
Similar to the previous section, we first prove that Inequalities (3-35), (3-36), (3-
37), (3-38), (3-39), (3-40), and (3-41) are valid. Then we proceed to prove that they are
facet-defining inequalities for )(Bconv . It is easily verified that )(Bconv is full-
dimensional when 3n . When 2n , 12 x which shows that )(Bconv is not full-
dimensional.
Proposition 3.4.2.1: Inequalities (3-38), (3-39), (3-40), and (3-41) are valid for )(Bconv .
Proof: Inequalities (3-38), (3-39), (3-40), and (3-41) are valid for )(Bconv since they
belong to the initial description of Problem (3-15). ■
Proposition 3.4.2.2: Inequality (3-35) is valid for )(Bconv .
Proof: Consider any feasible solution ),( yx . We argue that x satisfies
n
i
ix2
1.
Assume for a contradiction that
n
i
ix2
0 . Then
duyxyxn
i
ii
1
1
11 ,
a contradiction to the feasibility of ),( yx . Therefore,
n
i
ix2
1 is valid for )(Bconv . ■
Proposition 3.4.2.3: Inequality (3-36) is valid for )(Bconv .
Proof: Consider any feasible solution ),( yx of B , i.e.,
n
i
ii dyx1
where
xi {0,1},
},,0{ ii uy . It can be easily verified that
n
i
ix1
1 and
n
i
i dy1
for any ),( yx . We
consider 3 subcases:
Case 1: For ØS , Inequality (3-36) reduces to
68
dyNj
j
,
which is satisfied by any feasible solution.
Case 2: For ØS , Ø\ SN .
Case 2a: We assume first that S1 and
1\
1Si
ix . Then
dxduyxdu i
Si
i
SNj
ji
Si
i
},min{},min{\
,
which shows that ),( yx satisfies this inequality.
Case 2b: Assume next that S1 and
1\
0Si
ix . Then, it is easy to verify that for
any feasible solution
SNj
j dy\
if 01 x and 1
\
udySNj
j
if 11 x . Then
dyxuyxduSNj
j
SNj
jii
Si
\
11
\
},min{ ,
which shows that ),( yx satisfies this inequality.
Case 2c: Assume that S1 and
Si
ix 1. Then
dxdyxdu i
SiSNj
ji
Si
i \
},min{ .
This shows that ),( yx satisfies this inequality.
Case 2d: Assume that S1 and
Si
ix 0 . Then, it is easy to verify that
SNj
j dy\
. It follows that
dyyxduSNj
j
SNj
jii
Si
\\
},min{ ,
69
showing that ),( yx satisfies this inequality.
By considering the cases described above, we have shown that all feasible solutions to
B Inequality (3-36), i.e, Inequality (3-36) is valid for )(Bconv . ■
Because 11 x and 11 uy , any solution satisfying:
n
i
ii dyx1
,
also satisfies
(SP)
n
i
ii udyxdyx2
111 .
Since we know that 1uddui for all },,2{ ni , it follows from our study of case 0
that any solution to (SP) also satisfies 1
\
1)( udyxudRNj
ji
Ri
.
Proposition 3.4.2.4: Inequality (3-37) is valid for )(Bconv .
Next, we show that (3-35), (3-36), (3-37), (3-38), (3-39), (3-40), and (3-41) are
facet-defining inequalities for )(Bconv . In the proofs below we let Nj
jji
Ni
i yx
be any inequality that induces a face )(F in n2 dimensional space that is contains the
face of )(Bconv under consideration.
Proposition 3.4.2.5: Inequality (3-35) is facet-defining for )(Bconv .
Proof: Let )(G be
n
i
ix2
1. For all 1\Ni , (
xi 1,yi d) is a feasible solution that
satisfies )(G at equality and therefore must belong to the face )(F yielding
(1)
i di 1\Ni .
For all 1\Ni , Nj , and
i j , the feasible solution ( 1,,1 jii ydyx ) satisfies )(G
70
at equality and therefore must belong to the face )(F yielding
(2) jii d ,1\Ni ,Nj ji .
It follows from (1) and (2) that
j 0 Nj , and therefore the face )(F is defined by
Ni
ii x .
For },,2{ ni , the feasible solution (
xi 1,yi d) satisfies )(G at equality and therefore
must belong to the face )(F yielding
(3)
i },,2{ ni
and so the face )(F is defined by
},,2{
11
ni
ixx
.
For any },,2{ ni , the feasible solutions ( dyxx ii ,1,11 ) satisfy )(G at equality and
therefore must belong to the face )(F yielding
(4) i1 .
It follows from (3) and (4) that 01 . The face )(F is therefore defined by
1},,2{
ni
ix
,
proving that (3-35) is facet-defining inequality for )(Bconv . ■
Proposition 3.4.2.6: Inequality (3-36) is facet-defining for )(Bconv .
Proof [sketch]: In Proposition 3.4.1.3, we proved that n2 inequalities are facet-defining.
We follow the same steps to prove that 22 n inequalities described in (3-36) are facet-
defining with only one difference, which is the coefficient of 1x . Below, we discuss how
to obtain 1u as the coefficient of 1x instead of d .
71
Similar to Proposition 3.4.1.3, we reduce )(F to the form
SNj
ji
Si
i yd
x\
.
Now, assume that S1 and SNj \ , consider the feasible solution
( 1111 ,1,,1 udyxuyx jj ) that satisfies )(G at equality. This point implies that
d
ud )( 11 . ■
The proofs that Inequalities (3-38), (3-39), (3-40), and (3-41) are facet-defining for
)(Bconv are similar to those we presented in Propositions 3.4.1.4, 3.4.1.5, 3.4.1.6, and
3.4.1.7 respectively.
3.4.3 Convex Hull Proof for Case 0
We define
njuy
njx
dyx
ZZyxB
jj
j
n
j
jj
nn
,,10
,,110),(1
. (3-42)
We denote the convex hull of B by )(Bconv .
We also define
)3(,,10
)2(,,110
)1(
),(\
njuy
njx
NSdydx
RRyxS
jj
j
sj SNj
jj
nn
. (3-43)
We next prove that S provides a polyhedral description of )(Bconv .
Proposition 3.4.3.1: SBconv )(
Proof: We have shown earlier that Inequalities (1), (2), and (3) are valid for B . It follows
that SB . Since S is convex, we conclude that
72
SSconvBconv )()( .
We now show that
)(BconvS
to complete the proof.
To this end, we define the following two optimization problems
)(P *z jjjj yx min (3-44)
Byx ),(
and
)(Q *w jjjj yx min (3-45)
Syx ),( .
Observe that )(P is a mixed integer nonlinear program and )(Q is a linear program. To
show that )(BconvS , we show that ** wz for all choices of objective coefficients
),( in )(P and )(Q .
We differentiate two cases.
Case 1: Assume that there exists },,1{ ni such that 0i and 0i . In this case, it
is easy to verify that the solution
00
01*
j
j
jif
ifx
00
0*
j
jj
jif
ifuy
is an optimal solution for )(P .
It is feasible since jx j }1,0{* , juy jj },,0{* and
73
duuyxyx ii
n
j
jj
11
*
1
*
1
** .
It is optimal since is uses only variables with nonpositive objective coefficients.
Therefore, the optimal value of )(P *z is equal to
*z j
j
j
j
j u
jj
00
We next prove that ** wz .
First observe that every feasible solution satisfies
0jx (3-46)
1 jx (3-47)
0jy (3-48)
jj uy . (3-49)
Then define
jI }0j
jI }0j
jJ }0j
jJ }0j .
If we multiply (3-46) by j for all Ij , (3-47) by j for all Ij , (3-48) by j for all
Jj , and (3-48) by j for all Jj and sum the resulting inequalities, we obtain the
valid inequality for S :
*1 1
zuyx j
Jj
j
Ij
j
n
j
n
j
jjjj
. (3-50)
74
Since every feasible solution of )(Q satisfies (3-50), we obtain that
** wz .
Case 2: Assume that for all },,1{ nj either 0j or 0j . In this case, the natural
solution
00
01*
j
j
jif
ifx
00
0*
j
ji
jif
ifuy
is not directly usable since it is not feasible as
0** Nj
jj yx .
It is easy to see that it is sufficient to modify one of the above term ),( **
jj yx in order to
obtain an optimal solution to )(P . The term to modify is clearly the one for which the
change in objective value is smallest.
We define
jI }0,0 jj
jI }0,0 jj
jI }0,0 jj
jI }0,0 jj .
From our assumption in Case 2, ØI .
For Ij the value of )0,0(),( ** jj yx . Making this term feasible would require
changing it to ),1( d resulting in an increase of objective value of jjjj dd .
75
For Ij the value of ),0(),( **
jjj uyx . Making this term feasible would require
changing it to ),1( iu resulting in an increase of objective value of jjj d .
For Ij the value of )0,1(),( ** jj yx . Making this term feasible would require
changing it to ),1( d resulting in an increase of objective value of jjj dd .
Clearly, we want the smallest increase in objective value. This is obtained by finding any
index i as follows:
}{minarg ,,1
jjnj di . (3-51)
An optimal solution to the problem is therefore given by
00
01
j
j
jif
ifx
00
0
j
jj
jif
ifuy
for in \},,1{j , and
1ix
IiorIiifd
Iiifuy
j
i .
The optimal value of )(P is therefore equal to
IiorIiifd
Iiifu
z
jj
jj
iNj
iij
iNj
j
iNj
iiij
iNj
j
0\0\
0\0\
*
.
We next prove that ** wz by considering 3 subcases.
76
Case 2a: Assume that Ii . Since variables can be reordered, we may assume that
ni , },,1{ rI , },,1{ trI , },,1{ ntI and r 21 . We define
},,0{ r such that 1 n .
Because of the definition of ni in (3-51) we have that
dd njj rj ,,1 (3-52)
dnj trj ,,1 (3-53)
dd nj ntj ,,1 . (3-54)
Consider now the following valid inequalities for S
dydxydxn
tj
j
t
rj
j
r
Pj
j
P
j
j 1111
for ,,0P (3-55)
0jx (3-56)
1 jx (3-57)
0jy (3-58)
jj uy . (3-59)
For constraint (3-55) define the following weights:
1 for 0P
PP 1 for 1,,1 P
n for P ,
which are positive because of the definition of and because are sorted. Now, using
these positive weights on (3-55). We obtain
dydxyydx n
n
tj
jn
t
rj
jn
r
j
jn
j
jj
j
jjn
11111
)( . (3-60)
77
Now multiply constraints (3-58) for rj ,,1 with positive weights )( nj and add
to (3-60). Also multiply constraints (3-56) for ,,1j with positive weights
))(( jnj d and add to (3-60) to obtain
dydxyx n
n
tj
jn
t
rj
jn
r
j
jj
j
jj
1111
. (3-61)
Now multiply constraints (3-56) for rj ,,1 with positive weights j and add to (3-
61) to obtain
dydxyx n
n
tj
jn
t
rj
jn
r
j
jj
r
j
jj 1111
. (3-62)
Next multiply constraints (3-56) for trj ,,1 with positive weights )( nj d and
add to (3-62) to obtain
dyyx n
n
tj
jn
r
j
jj
t
j
jj 111
. (3-63)
Then multiply constraints (3-59) for trj ,,1 with positive weights j and add to
(3-63) to obtain
t
rj
jjn
n
tj
jn
t
j
jj
t
j
jj udyyx1111
. (3-64)
Next multiply constraints (3-58) for ntj ,,1 with positive weights )( nj and add
to (3-64) to obtain
t
rj
jjn
n
j
jj
t
j
jj udyx111
. (3-65)
Finally multiply constraints (3-57) for ntj ,,1 with positive weights j and add to
(3-65) to obtain
78
1
1111
n
tj
j
t
rj
jjnn
n
j
jj
n
j
jj udyx . (3-66)
Inequality (3-66) shows that every feasible solution to )(Q satisfies
*11
zyxn
j
jj
n
j
jj
,
i.e.,
** zw .
Case 2b: Assume that Ii . Since variables can be reordered, we may assume that
ti , },,1{ rI , },,1{ trI , },,1{ ntI and r 21 . We define
},,0{ r such that 1 t .
Because of the definition of ti in (3-51) we have that
tjj d rj ,,1 (3-67)
tj trj ,,1 (3-68)
tjd ntj ,,1 . (3-69)
Consider the following valid inequalities for S
dydxdxyn
tj
j
t
rj
j
r
Pj
j
P
j
j 1111
for ,,0P (3-70)
0jx (3-71)
1 jx (3-72)
0jy (3-73)
jj uy . (3-74)
For constraint (3-70) define the following weights:
79
1 for 0P
PP 1 for 1,,1 P
t for P ,
which are positive because of the definition of and because are sorted. Now, using
these positive weights on (3-70). We obtain
dydxdxydx t
n
tj
jt
t
rj
jt
r
j
jt
j
jjt
j
jj
11111
)( . (3-75)
Now multiply constraints (3-71) for tj ,,1 with positive weights dtj )( and
add to (3-75). Also multiply constraints (3-73) for ,,1j with positive weights
))(( jtjd and add to (3-75) to obtain
dydydx t
n
tj
jt
j
jj
t
j
jj
111
, (3-76)
or equivalently,
t
n
tj
jt
j
jj
t
j
jjd
yyx
111
(3-77)
since d is positive.
Now multiply constraints (3-73) for rj ,,1 with positive weights j and add to (3-
77) to obtain
t
n
tj
jtr
j
jj
t
j
jjd
yyx
111
. (3-78)
Next multiply constraints (3-74) for trj ,,1 with positive weights j and add to
(3-78) to obtain
80
1
1111
t
rj
jjttt
n
tj
jtt
j
jj
t
j
jj uud
yyx
. (3-79)
Multiply constraints (3-73) for ntj ,,1 with positive weights )(d
tj
and add to (3-
79) to obtain
1
111
t
rj
jjttt
n
j
jj
t
j
jj uuyx . (3-80)
Finally multiply constraints (3-72) for ntj ,,1 with positive weights j and add to
(3-80) to obtain
1
1 111
t
rj
n
tj
jjjttt
n
j
jj
n
j
jj uuyx . (3-81)
Inequality (3-81) shows that every feasible solution to )(Q satisfies
*11
zyxn
j
jj
n
j
jj
,
i.e.,
** zw .
Case 2c: Assume that Ii . Since variables can reordered, we may assume that
ri , },,1{ rI , },,1{ trI , },,1{ ntI and 121 r . We define
}1,,0{ r such that 1 ddd rr .
Because of the definition of ri in (3-51) we have that
dd rrjj rj ,,1 (3-82)
drrj trj ,,1 (3-83)
dd rrj ntj ,,1 . (3-84)
81
Consider the following valid inequalities for S
dydxydxn
tj
j
t
rj
j
r
Pj
j
P
j
j 1111
for ,,0P (3-85)
0jx (3-86)
1 jx (3-87)
0jy (3-88)
jj uy . (3-89)
For constraint (3-55) define the following weights:
1d for 0P
PP dd 1 for 1,,1 P
dd rr for P ,
which are positive because of the definition of and because are sorted. Now, using
these positive weights on (3-85). We obtain
r
j
jrr
j
jj
j
jjrr ydyddxdd111
)()(
ddyddxd rr
n
tj
jrr
t
rj
jrr )()()(11
. (3-90)
Now multiply constraints (3-88) for rj ,,1 with positive weights ))(( rrj dd
and add to (3-90). Also multiply constraints (3-86) for ,,1j with positive weights
))()(( rrjj dd and add to (3-90) to obtain
ddyddxdyddx rr
n
tj
jrr
t
rj
jrr
r
j
jj
j
jj )()()(1111
, (3-91)
or equivalently,
82
rr
n
tj
jrr
t
rj
jrr
r
j
jj
j
jj dyd
dxdyx
1111
)()( (3-92)
since d is positive.
Now multiply constraints (3-86) for rj ,,1 with positive weights j and add to (3-
92) to obtain
rr
n
tj
jrr
t
rj
jrr
r
j
jj
r
j
jj dyd
dxdyx
1111
)()( . (3-93)
Next multiply constraints (3-86) for trj ,,1 with positive weights ))(( rrj d
and add to (3-93) to obtain
rr
n
tj
jrr
r
j
jj
t
j
jj dyd
dyx
111
)(. (3-94)
Multiply constraints (3-89) for trj ,,1 with positive weights j and add to (3-94)
to obtain
t
rj
jjrr
n
tj
jrr
t
j
jj
t
j
jj udyd
dyx
1111
)(
. (3-95)
Multiply constraints (3-88) for ntj ,,1 with positive weights ))(
(d
d rrj
and
add to (3-95) to obtain
t
rj
jjrr
n
j
jj
t
j
jj udyx111
. (3-96)
Finally multiply constraints (3-87) for ntj ,,1 with positive weights j and add to
(3-96) to obtain
n
tj
j
t
rj
jjrr
n
j
jj
n
j
jj udyx1111
. (3-97)
83
Inequality (3-97) shows that every feasible solution to )(Q satisfies
*11
zyxn
j
jj
n
j
jj
,
i.e.,
** zw .
This concludes the proof. ■
Proposition 3.4.3.1 provides a proof of convex hull for integer bilinear covering
sets in case 0. We believe the polyhedral characterization for case 1 given in (3-35) –
(3-41) defines the convex hull for this case also. We do not however have a proof of this
result at this time.
84
CHAPTER 4 SUMMARY
In this thesis, we studied the polyhedral structure of an integer bilinear covering
set which appears in the formulation of several practical problems in optimization
including staff scheduling. During the study, we established that the polyhedral structure
of this covering set depends strongly on some of its parameters
We conjectured polyhedral description of the convex hull for two different cases.
We then investigated these conjectures. In particular, for these two cases, we proved
that several families of inequalities are facet-defining. For one of the cases, we also
showed that these inequalities are sufficient to describe the convex hull.
Proving that the inequalities we derived for case 1 are sufficient to describe the
convex hull in this case is a direct avenue of research in studying these covering sets.
Developing polyhedral descriptions of the convex hull for the remaining cases is another
interesting direction of research. For these other cases we believe it will typically be
hard to uncover a generic form of the convex hull but some families of valid inequalities
might be possible to describe.
85
APPENDIX MATLAB CODES
1. For Case 0, d =3, n = 2
x = [0 0 0 0]
y = [0 0 0 0]
for i = 1 : 63
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
m = 3;
else
m = 1;
end
end
y=[y;x];
end
x = [1 3 1 3]
86
for i = 1 : 63
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) >=3)
x = [x;y(i,:)]
end
end
2. For Case 0, d =4, n = 2
x = [0 0 0 0]
y = [0 0 0 0]
for i = 1 : 99
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
m = 4;
else
m = 1;
end
87
end
y=[y;x];
end
x = [1 4 1 4]
for i = 1 : 99
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) >=4)
x = [x;y(i,:)]
end
end
3. For Case 0, d =4, n = 3
x = [0 0 0 0 0 0]
y = [0 0 0 0 0 0]
for i = 1 : 999
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
88
m = 4;
else
m = 1;
end
end
y=[y;x];
end
x = [1 4 1 4 1 4]
for i = 1 : 999
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=4)
x = [x;y(i,:)]
end
end
4. For Case 1, d =4, n = 3, u1 = 3
x = [0 0 0 0 0 0]
y = [0 0 0 0 0 0]
for i = 1 : 799
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
89
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
if(j==2)
m = 3;
else
m = 4;
end
else
m = 1;
end
end
y=[y;x];
end
x = [1 3 1 4 1 4]
for i = 1 : 799
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=4)
x = [x;y(i,:)]
end
end
90
5. For Case 1, d =4, n = 3, u1 = 2
x = [0 0 0 0 0 0]
y = [0 0 0 0 0 0]
for i = 1 : 599
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
if(j==2)
m = 2;
else
m = 4;
end
else
m = 1;
end
end
y=[y;x];
end
91
x = [1 2 1 4 1 4]
for i = 1 : 599
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=4)
x = [x;y(i,:)]
end
end
6. For Case 1, d =5, n = 3, u1 = 4
x = [0 0 0 0 0 0]
y = [0 0 0 0 0 0]
for i = 1 : 1439
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
if(j==2)
m = 4;
else
92
m = 5;
end
else
m = 1;
end
end
y=[y;x];
end
x = [1 4 1 5 1 5]
for i = 1 : 1439
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=5)
x = [x;y(i,:)]
end
end
7. For Case 1, d =5, n = 3, u1 = 3
x = [0 0 0 0 0 0]
y = [0 0 0 0 0 0]
for i = 1 : 1151
x(1) = x(1) + 1
j = 1
m = rem(j,2)
while (x(j) > m)
93
x(j) = 0
x(j+1) = x(j+1) +1
j = j + 1
k = rem(j,2)
if (k == 0)
if(j==2)
m = 3;
else
m = 5;
end
else
m = 1;
end
end
y=[y;x];
end
x = [1 3 1 5 1 5]
for i = 1 : 1151
if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=5)
x = [x;y(i,:)]
end
end
94
LIST OF REFERENCES
[1] Al-Khayyal, Faiz A. and Falk, James E. (1983). Jointly Constrained Biconvex Programming. Mathematics of Operations Research 8 273-286.
[2] Balas, E. (1997). Recognizing Facet Defining Inequalities. Acta Mathematica
Vietnamica 22 7-28. [3] Boyd, Stephen and Vandenberghe, Lieven. (2004). Convex Optimization.
Cambridge University Press, New York. [4] Gadsby, G. Neville. (1965). The Army Operational Research Establishment.
Oper. Res. 16 5-18. [5] Hillier, Frederick S. and Lieberman, Gerald J. (2004). Introduction to Operations
Research, eighth edition. McGraw-Hill, Inc., New York. [6] Horst, Reiner and Tuy, Hoang. (1990). Global Optimization. Springer-Verlag,
Berlin. [7] Laderman, Julian D. (1976). A Noncommutative Algorithm for Multiplying 3 x 3
Matrices Using 23 Multiplications. American Mathematical Society 82 126-128. [8] Luenberger, David G. and Ye, Yinyu. (2007). Linear and Nonlinear Programming,
third edition. Springer Science+Business Media, LLC, New York. [9] McCormick, G. P. (1976). Computability of global solutions to factorable
nonconvex programs: Part I. Convex underestimating problems. Mathematical Programming, 10 147--175.
[10] Strassen, Volker. (1969). Gaussian Elimination is not Optimal. Numer. Math, 13
354-356. [11] Tawarmalani, M. and Sahinidis, N. V. (2002). Convexification and Global
Optimization in Continuous and Mixed-Integer Nonlinear Programming. Kluwer Academic Publishers.
[12] Winston, Wayne L. (1995). Introduction to Mathematical Programming, second
edition. Wadsworth Publishing Company, California. [13] Ziegler, Günter M. (1995). Lectures on Polytopes. Springer-Verlag, New York.
95
BIOGRAPHICAL SKETCH
Nitish Garg was born in Hisar, India in 1987. Nitish received a Bachelor of
Technology in Production and Industrial Engineering from Indian Institute of
Technology, Delhi, India in 2009.
Nitish joined the Department of Industrial and Systems Engineering at University
of Florida in August 2009. In summer 2010, Nitish participated in a research project with
Operations Research Group at London School of Economics and Political Science
(LSE) where he worked under guidance of Prof. Gautam Appa. At LSE, his work
involved devising an algorithm to compute Mutually Orthogonal Latin Squares and
implementing it with CPLEX CP Optimizer.
Since August 2010, Nitish has been working with Dr. Jean-Philippe P. Richard.
Upon completion of his master’s degree, Nitish will start working for Bloomberg L.P. as
a Financial Software Engineer. His current interests include mathematics, optimization
and algorithms.