91530.dvi MIP Guide

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c 2015 Society for Industrial and Applied Mathematics


SIAM REVIEW
Vol. 57, No. 1, pp. 3–57

Mixed Integer Linear
Programming Formulation
Techniques∗
Juan Pablo Vielma†
Abstract. A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP)
problems using standard formulation techniques. However, in some cases the resulting MIP
can be either too weak or too large to be effectively solved by state of the art solvers. In
this survey we review advanced MIP formulation techniques that result in stronger and/or
smaller formulations for a wide class of problems.
Key words. mixed integer linear programming, disjunctive programming
AMS subject classifications. 90C11, 90C10
DOI. 10.1137/130915303

1. Introduction. Throughout more than 50 years of existence, mixed integer
linear programming (MIP) theory and practice have been significantly developed, and
it is now an indispensable tool in business and engineering [68, 94, 104]. Two reasons
for the success of MIP are linear programming (LP) based solvers and the modeling
flexibility of MIP. We now have several extremely effective state-of-the-art solvers
[82, 69, 52, 171] that incorporate many advanced techniques [1, 2, 25, 23, 92, 112, 24]
and, since its early stages, MIP has been used to model a wide range of applications
[44, 45].
While in many cases constructing valid MIP formulations is relatively straightforward, some care should be taken in this construction as certain formulation attributes
can significantly reduce the effectiveness of LP-based solvers. Fortunately, constructing formulations that behave well with state-of-the-art solvers can usually be achieved
by following simple guidelines described in standard textbooks. However, more advanced techniques can often perform significantly better than textbook formulations
and are sometimes a necessity. The main objective of this survey is to summarize
the state of the art of such formulation techniques for a wide range of problems.
To keep the length of this survey under control, we concentrate on formulations for
sets of a mixed integer nature that require both integer constrained and continuous
variables. We hence purposefully place less emphasis on some related areas such as
combinatorial optimization, quadratic and polynomial 0/1 optimization, and polyhedral approximations of convex sets. These topics are certainly areas of important and
active research, so we cover them succinctly in section 12.
∗ Received by the editors April 2, 2013; accepted for publication (in revised form) July 21, 2014;
published electronically February 5, 2015.
http://www.siam.org/journals/sirev/57-1/91530.html
† Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA (jvielma@
mit.edu).

3

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JUAN PABLO VIELMA

Throughout this survey we emphasize the potential advantages of each technique.
However, we should note that given the complexities of state-of-the-art solvers, it is
hard to predict with high accuracy which formulation performs better. Some guidelines can be found in computational studies (e.g., [163, 91, 155]), but the formulation
that performs best can be strongly dependent on the specific problem structure or
data. Fortunately, there is a high correlation between certain favorable formulation
properties and good computational performance. In addition, it is often easy to construct several alternative formulations for a preliminary computational test. Finally,
we refer the reader to [78, 88, 138, 164] for complementary information on the topics
covered in this survey and their relation to other areas.
The rest of this survey is organized as follows. We begin in section 2 with a
motivating example that allows us to precisely define the idea of an MIP formulation
or model. This same example serves to illustrate one of the most important favorable
properties of an MIP formulation: the strength of its LP relaxation. Through this
section we also introduce basic MIP concepts and notation that we use in the rest of
the paper. The construction and evaluation of effective MIP formulations also requires
some basic concepts and results from polyhedral theory, which we review in section 3.
Armed with such results, in section 4, we introduce the use of auxiliary variables as
a way to construct strong formulations without incurring an excessive size. Then in
section 5 we show how these auxiliary variables can be used to construct strong formulations for a finite set of alternatives described as the union of certain polyhedra or
mixed integer sets. In section 6 we discuss how to reduce formulation size by forgoing
the use of auxiliary variables. We show how this can result in a significant loss of
strength, but also discuss cases in which this loss can be prevented. Then in section 7
we consider the use of large formulations and an LP-based technique that can be used
to reduce their size in certain cases. Sections 8 and 9 review some advanced techniques
that can be used to reduce the size of formulations and to improve the performance of
branch-and-bound-based MIP solvers. After that, in section 10 we discuss alternative
ways of combining formulations and in section 11 we consider precise geometric and
algebraic characterizations of sets that can be modeled with different classes of MIP
formulations. Finally, section 12 considers other topics related to MIP formulations.
2. Preliminaries and Motivation.
2.1. Modeling with MIP. Modeling non-convex functions has been a central topic
of MIP formulations since its early developments [119, 120, 122, 121, 85, 84, 81, 91, 89],
so our first example falls in this category. Consider the mathematical programming
problem given by
n

(2.1a)
fi (xi )
zMP := min
i=1

s.t.
(2.1b)

Ex ≤ h,

(2.1c)

0 ≤ xi ≤ u

where fi : [0, u] → Q are univariate piecewise
⎧
m1 x + c1 ,
⎪
⎪
⎪
⎪
⎨ m2 x + c2 ,
(2.2)
f (x) =
⎪
⎪
⎪
⎪
⎩
mk x + ck ,

∀i ∈ {1, . . . , n} ,

linear functions of the form
x ∈ [d0 , d1 ],
x ∈ [d1 , d2 ],
..
.
x ∈ [dk−1 , dk ],

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

5

k

for given breakpoints 0 = d0 < d1 < · · · < dk−1 < dk = u, slopes {mi }i=1 ⊆ Q, and
constants {ci }ki=1 ⊆ Q. We assume that the slopes and constants are such that the
functions are continuous, but not convex. For instance, one of these functions could
be
⎧
x + 2, x ∈ [0, 1],
⎨
−2x + 5, x ∈ [1, 2],
(2.3)
f (x) =
⎩
1, x ∈ [2, 3],
which will be our running example for this section.
Because the functions we consider are non-convex, we cannot transform (2.1) into
an equivalent LP problem. However, we can transform it into an MIP problem as
follows.
The first step in the transformation is to identify a set or a constraint that we want
to model as an MIP problem. In the case of a piecewise linear function f , an appropriate set to model is the graph of f given by gr(f ) := {(x, z) ∈ Q × Q : f (x) = z}.
Indeed, we can reformulate (2.1) by explicitly including gr(fi ) to obtain the equivalent
problem given by
(2.4a)

zMP := min

n


zi

i=1

s.t.
(2.4b)
(2.4c)
(2.4d)

Ex ≤ h,
0 ≤ xi ≤ u
∀i ∈ {1, . . . , n} ,
(xi , zi ) ∈ gr(fi ) ∀i ∈ {1, . . . , n} .

Now, as illustrated in Figure 2.1 for our running example, the graph of a univariate
continuous piecewise linear function (with bounded domain) is the finite union of line
segments, which we can easily model with MIP. For instance, for the function defined
in (2.3) we can construct the textbook MIP formulation of gr(f ) given by
(2.5a)
(2.5b)
(2.5c)
(2.5d)
(2.5e)
(2.5f)
(2.5g)
(2.5h)

0λ0 + 1λ1 + 2λ2 + 3λ3 = x,
2λ0 + 3λ1 + 1λ2 + 1λ3 = z,
λ0 + λ1 + λ2 + λ3 = 1,
λ0 ≤ y1 ,
λ1 ≤ y1 + y2 ,
λ2 ≤ y2 + y3 ,
λ3 ≤ y3 ,
y1 + y2 + y3 = 1,

(2.5i)
(2.5j)

λj ≥ 0
0 ≤ yj ≤ 1

(2.5k)

yj ∈ Z

∀j ∈ {0, . . . , 3} ,
∀j ∈ {1, 2, 3} ,
∀j ∈ {1, 2, 3} .

Formulation (2.5) illustrates how MIP formulations can use both continuous and integer variables to enforce requirements on the original variables. Formulations that use
auxiliary variables different from the original variables are usually denoted extended,
lifted, or higher-dimensional. In the case of formulation (2.5) these auxiliary variables

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JUAN PABLO VIELMA

z = f (x)
3

2

1

0
0

1

2

3

Fig. 2.1 Graph of the continuous piecewise linear function defined in (2.3).

are necessary to construct the formulation. However, as we will see in section 4, auxiliary variables can provide an advantage even when they are not strictly necessary.
For this reason we use the following definition of an MIP formulation that always
considers the possible use of auxiliary variables.
Definition 2.1. For A ∈ Qm×n , B ∈ Qm×s , D ∈ Qm×t , and b ∈ Qm consider
the set of linear inequalities and continuous and integer variables of the form
Ax + Bλ + Dy ≤ b,

(2.6a)
(2.6b)

x ∈ Qn ,

(2.6c)

λ ∈ Qs ,

(2.6d)

y ∈ Zt .

We say (2.6) is an MIP formulation for a set S ⊆ Qn if the projection of (2.6) onto
the x variables is exactly S. That is, if we have x ∈ S if and only if there exist λ ∈ Qs
and y ∈ Zt such that (x, λ, y) satisfies (2.6).
Note that generic formulation (2.6) does not explicitly consider the inclusion of
integrality requirements on the original variables x. While example (2.5) does not
require such integrality requirements, it is common for at least some of the original
variables to be integers. Formulation (2.6) implicitly considers that possibility by
allowing the inclusion of constraints of the form xi = yj . Hence, we can replace (2.6b)
with x ∈ Qn1 × Zn2 without really changing the definition of an MIP formulation.
Using Definition 2.1 we can now write a generic MIP reformulation of (2.1) by
replacing every occurrence of (xi , z) ∈ gr(fi ) with an MIP formulation of gr(fi ) to
obtain
(2.7a)

zMIP := min

n


zi

i=1

s.t.
(2.7b)

Ex ≤ h,

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

(2.7c)
(2.7d)
(2.7e)

0 ≤ xi ≤ u
 
xi
+ B i λi + Di y i ≤ bi
Ai
zi
y i ∈ Zk

7

∀i ∈ {1, . . . , n} ,
∀i ∈ {1, . . . , n} ,
∀i ∈ {1, . . . , n} ,

where Ai , B i , Di , and bi are appropriately constructed matrices and vectors, and k
is an appropriate number of integer variables. There are many choices for (2.7d), but
as long as they are valid formulations of gr(fi ), we have zMP = zMIP and we can
extract a solution of (2.1) by looking at the x variables of an optimal solution of (2.7).
However, some versions of (2.7d) can perform significantly better when solved with
an MIP solver. We study this potential difference in the next subsection.
2.2. Strength, Size, and MIP Solvers. While all state-of-the-art MIP solvers are
based on the branch-and-bound algorithm [102], they also include a large number of
advanced techniques that make it hard to predict the specific impact of an alternative
formulation. However, there are two aspects of an MIP formulation that usually have
a strong impact on both simple branch-and-bound algorithms and state-of-the-art
solvers: the size and strength of the LP relaxation, and the effect of branching on the
formulation. Instead of giving a lengthy description of the branch-and-bound algorithm and state-of-the-art solvers, we introduce the necessary concepts by considering
the first step of the solution of MIP reformulation (2.7) with a simple branch-andbound algorithm. For more details on basic branch-and-bound and state-of-the-art
solvers, we refer the reader to [22, 36, 131, 166] and [1, 2, 25, 23, 92, 112, 24], respectively.
The first step in solving MIP formulation (2.7) with a branch-and-bound algorithm is to solve the LP relaxation of (2.7) obtained by dropping all integrality
requirements. The resulting LP problem given by (2.7a)–(2.7d) is known as the root
LP relaxation and can be solved efficiently both in theory and in practice. Its optimal
n
value zLP := min { i=1 zi : (2.7b)–(2.7d)} provides a lower bound on zMIP known
as the LP relaxation bound. If the optimal solution to the LP relaxation satisfies the
dropped integrality constraints, then zMIP = zLP and this solution is also optimal
for (2.7). In contrast, if optimal solution x̄, z̄, λ̄, ȳ is such that for some i and j we
/ Z, we can eliminate this infeasibility by branching on yji . To achieve this we
have ȳji ∈
 
create two new LP problems by adding yji ≤ ȳji and yji ≥ ȳji to the LP relaxation,
respectively. These two new problems are usually denoted branches and are processed
in a similar way, which generates a binary tree known as the branch-and-bound tree.
The behavior of this first step is usually a good predictor of the performance of the
whole algorithm, so we now concentrate on the effect of an MIP formulation on this
step. We consider the effect of different formulations on subsequent steps in section 8.
To understand the effect of a formulation on the root LP relaxation we need
to understand what the LP relaxation of the formulation is modeling. Going back
to our running example, consider the LP relaxation of (2.5) given by (2.5a)–(2.5j).
Constraints (2.5a)–(2.5c) and (2.5i) show that any (x, z) that is part of a feasible
solution for this LP relaxation must be a convex combination of points (0, 2), (1, 3),
(2, 1), and (3, 1). Conversely, any (x, z) that is a convex combination of these points
is part of a feasible solution for the LP relaxation (let λ be the appropriate convex
combination multipliers, y1 = λ0 , y2 = 1−λ0 −λ3 , and y3 = λ3 ). Hence, the projection
onto the (x, z) variables of the LP Relaxation of (2.5) is equal to conv (gr (f )), the
convex hull of gr (f ). As illustrated in Figure 2.2(a), this convex hull corresponds to
the smallest convex set containing gr (f ). Now, if we used this formulation for fi in

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JUAN PABLO VIELMA

(2.1) (for simplicity imagine that all these functions are equal to f ), the LP relaxation
of (2.7) would be equivalent to
zLP := min

(2.8a)

n


φi (xi )

i=1

s.t.
Ex ≤ h,

(2.8b)

0 ≤ xi ≤ u ∀i ∈ {1, . . . , n} ,

(2.8c)

where φi : [0, u] → Q is the convex envelope or lower convex envelope [80] of fi . This
lower convex envelope is the tightest convex underestimator of f and is illustrated for
our running example in Figure 2.2(b).

z = f (x)

z = f (x)

3

3

2

2

1

1

0

0
0

1

2

3

(a) gr (f ) in red and conv (gr (f )) in light blue.

0

1

2

3

(b) conv (gr (f )) in light blue and lower convex
envelope of f in dark blue.

Fig. 2.2 Effect of the LP relaxation of formulation (2.5) of f defined in (2.3).

This reasoning suggests that, at least with respect to the LP relaxation bound,
formulation (2.5) of our example function is as strong as it can be. Indeed, the
projection onto the (x, z)-space of any formulation of f is a convex (and polyhedral)
set that must contain gr (f ) and the LP relaxation of (2.5) projects to the smallest
of such sets. By the same argument, the projection onto the x variables of the LP
relaxation of an MIP formulation of any set S ⊆ Qn must also contain conv(S)
and formulations whose LP relaxations project precisely to conv(S) yield the best
LP relaxation bounds. Jeroslow and Lowe [90, 113] denoted those formulations that
achieve this best possible LP relaxation bound as sharp. Other authors also denote
them convex hull formulations.
Definition 2.2. An MIP formulation of set S ⊆ Qn is sharp if and only if the
projection onto the x variables of its LP relaxation is exactly conv(S).
It is important to note that sharp formulations yield the best LP relaxation
bound among all MIP formulations for the sets we selected to model. However, the
LP relaxation bound can vary if we elect to model other sets. For instance, in our

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

9

example, we elected to independently model gr(fi ) for all i ∈ {1, . . . , n}. However, we
could instead have elected to model
⎧
⎫
n

⎪
⎪
⎪
⎪
⎪
⎪
fi (xi ),
z=
⎪
⎪
⎨
⎬
i=1
n
(2.9)
S = (x, z) ∈ Q × Q :
⎪
⎪
Ex ≤ h,
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
0 ≤ xi ≤ u ∀i ∈ {1, . . . , n}
to obtain a formulation that considers all possible nonconvexities at the same time.
A sharp formulation for this case would be significantly stronger as we can show that
for S defined in (2.9) we have min {z : (x, z) ∈ conv (S)} = zMP = zMIP . Because
the LP relaxation of a sharp formulation of S is equal to conv(S), we have that calculating the optimal value of (2.4) could be done by simply solving an LP problem. Of
course, unlike our original piecewise sharp formulation, constructing a sharp formulation for this joint S would normally be harder than solving (2.4). Furthermore, this
approach can result in a significantly larger formulation. Selecting which portions of
a mathematical programming problem to model independently to balance size and
final strength is a crucial and nontrivial endeavor. However, the appropriate selection
is usually ad hoc to the specific structure of the problem considered. For instance,
Croxton, Gendron, and Magnanti [42] study the case where (2.1) corresponds to a
multicommodity network flow problem with piecewise linear costs. In this setting,
they show that a convenient middle ground between constructing independent formulations of each gr (fi ) and a single formulation for the complete problem (2.9) is to
construct independent formulations for each gr
i∈Ia fi (xi ) , where Ia corresponds
to the flow variables of all commodities in a given arc a of the network. From now
on we assume that a similar analysis has already been carried out and we focus on
constructing small and strong formulations for the individual portions selected. However, in section 10 we will succinctly consider the selection of such portions and the
combination of the associated formulations.
As we have mentioned, sharp formulations are strongest in one sense. However,
if we consider the integer variables of our MIP formulation we can construct even
stronger formulations. To illustrate this, let us go back to our example and consider
an optimal solution x̄, z̄, λ̄, ȳ of the LP relaxation of (2.7) given by (2.7a)–(2.7d).
Because the root LP relaxation is usually solved by a simplex algorithm, we can expect this to be an optimal basic feasible solution. However, analyzing basic feasible
solutions of (2.7b)–(2.7d) can be quite hard, so let us analyze basic feasible solutions
of the LP relaxations of the individual formulations of gr(fi ) (i.e., (2.5a)–(2.5j)) as a
reasonable proxy. We can check that one optimal basic feasible solution of minimizing
z over (2.5a)–(2.5j) is given by λ̄2 = λ̄3 = 1/2, λ̄0 = λ̄1 = 0, ȳ1 = ȳ3 = 1/2, ȳ2 = 0,
x̄ = 2.5, and z̄ = 1. Because (2.5) is a sharp MIP formulation, it is not surprising
that this gives the same optimal value as minimizing z over (2.5). Indeed we have
that 1 = f (2.5) = φ(2.5) and x = 2.5 is a minimizer of f . However, the basic feasible
solution obtained has some of its integer variables set at fractional values. Because
a general purpose branch-and-bound solver is not aware of the specific structure of
the problem, it would have no choice but to unnecessarily branch on one of these
variables (let’s say ȳ1 ) or to run a rounding heuristic to obtain an integer feasible
solution. Hence, while sharp formulations are the strongest possible with regard to
LP relaxation bounds, they can be somewhat weak with respect to finding optimal
or good quality integer feasible solutions. Fortunately, it is possible to construct MIP
formulations that are strong from both the LP relaxation bound and integer feasibility

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JUAN PABLO VIELMA

perspectives. These formulations are those whose LP relaxations have basic feasible
solutions that automatically satisfy the integrality requirements on the y variables.
Such LP relaxations are usually denoted integral and formulations with this property
were denoted locally ideal by Padberg and Rijal [136, 137]. Ideal comes from the fact
that this is the strongest property we can expect from an MIP formulation from any
perspective. Locally serves to clarify that this property refers to a formulation for a
specifically selected portion and not to the whole mathematical programming problem
(following this convention, we should then refer to sharp formulations as locally sharp,
but we avoid it for simplicity and historical reasons). For simplicity, here we restrict
our attention to MIP formulations with LP relaxations that have at least one basic feasible solution, which is the case for all practical formulations considered in this survey.
Definition 2.3. An MIP formulation (2.6) is locally ideal if and only if its LP
relaxation has at least one basic feasible solution and all such basic feasible solutions
have integral y variables.
As expected, the following simple proposition states that a locally ideal formulation is at least as strong as a sharp formulation. We postpone the proof of this
proposition until section 3 where we introduce some useful results concerning the
feasible regions of LP problems.
Propositions 2.4. A locally ideal MIP formulation is sharp.
Finally, formulation (2.5) shows that being locally ideal can be strictly stronger
than being sharp. However, one case in which the converse of Proposition 2.4 holds is
that of traditional MIP formulations without auxiliary variables (i.e., (2.6) with s = 0
and such that for all i ∈ {1, . . . , t} constraints (2.6a) include an equality of the form
xj = yi for some j ∈ {1, . . . , n}). We again postpone the proof of this proposition
until section 3.
Propositions 2.5. For A ∈ Qm×n , b ∈ Qm , and n1 , n2 ∈ Z+ such that n =
n1 + n2 , let
(2.10a)
(2.10b)

Ax ≤ b,
x ∈ Qn1 × Zn2 ,

be an MIP formulation of S ⊆ Qn (i.e., S is precisely the feasible region of (2.10)).
If the LP relaxation of (2.10) has at least one basic feasible solution, then (2.10) is
locally ideal if and only if it is sharp.
k
3. Polyhedra. Most sets modeled with MIP are of the form S = i=1 P i , where
P i are rational polyhedra (e.g., the graph of a univariate piecewise linear function is
the union of line segments). Sets of this form usually appear as the feasible regions of
certain disjunctive programming problems [10, 11, 13]. For this reason we refer to such
sets as disjunctive constraints or disjunctive sets. While in the theory of disjunctive
programming these terms are also used to describe a slightly broader class of sets, in
this survey we concentrate on disjunctive sets that are the union of certain unbounded
rational polyhedra. To construct and evaluate MIP formulations for these and other
sets it will be convenient to use definitions and results from polyhedral theory that
we now review. We begin by considering some basic definitions and a fundamental
result that relates two natural definitions of polyhedra. We then consider the relation
between polyhedra and the feasible regions of MIP problems. After that we study
the linear transformations of polyhedra, which will be useful when analyzing the
strength of MIP formulations. Finally, we consider the smallest possible descriptions
of polyhedra to consider the real sizes of MIP formulations. We refer the reader to
[22, 131, 145, 166, 170] for omitted proofs and a more detailed treatment of polyhedra.

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

11

3.1. Definitions and the Minkowski–Weyl Theorem. Polyhedra are usually
described as the region bounded by a finite number of linear inequalities, such as
the feasible region of an LP problem. However, formulation (2.5) of the graph of
the piecewise linear function defined in (2.3) can be more easily described using the
endpoints of the line segments of this graph. These two aspects of the description of
polyhedra can be formalized through the following definition.
Definition 3.1. We say P ⊆ Qn is a rational H-polyhedron if there exist
A ∈ Qm×n and b ∈ Qm such that
(3.1)

P = {x ∈ Qn : Ax ≤ b} .

In this case, we say that the right-hand side of (3.1) is an H-representation of P .
We say P ⊆ Qn is a rational V-polyhedron if there exist finite sets V ⊆ Qn and
R ⊆ Qn such that
(3.2)

P = conv (V ) + cone (R) ,

where cone (R) is the set of all nonnegative linear combinations of elements in R and
+ denotes the Minkowski sum of sets (i.e., conv (V ) + cone (R) := {x + r : x ∈
conv (V ) , r ∈ cone (R)}). In this case, we say that the right-hand side of (3.2) is a
V-representation of P .
Both H- and V-polyhedra can also be defined in Rn . However, some results in
section 11 require the polyhedra to be rational. For this reason, from now on we
assume that, unless specified, all polyhedra considered are rational and we often refer
to them simply as polyhedra.
One of the most important results in polyhedral theory shows that the definitions
of H- and V-polyhedra are indeed equivalent (i.e., every rational polyhedron has both
an H- and a V-representation). To formalize this statement we need a few definitions
and results. We begin by considering the boundedness properties of polyhedra.
Definition 3.2. For an H- or V-polyhedron P we have the following definitions.
• Polyhedron P is a cone (or more precisely a polyhedral cone) if and only if
λx ∈ P for all λ ≥ 0 and x ∈ P .
• Polyhedron P is a polytope if and only if it is bounded.
• The recession cone of P is given by
P∞ := {d ∈ Qn : x + λd ∈ P

∀x ∈ P,

λ ≥ 0} .

The following simple proposition characterizes the recession cone of H- and Vpolyhedra.
Proposition 3.3. The recession cone of a polyhedron is always a cone. Furthermore, a nonempty polyhedron P is bounded if and only P∞ = {0}.
If P is a nonempty H-polyhedron of the form (3.1), then P∞ = {x ∈ Qn : Ax ≤ 0}.
If P is a nonempty V-polyhedron of the form (3.2), then P∞ = cone (R).
The last concept needed for the equivalence is a geometric characterization of basic
feasible solutions and certain directions of unboundedness that are not dependent on
the H-representation of a polyhedron (remember that x ∈ P is a basic feasible solution
of H-polyhedron P ⊆ Qn if and only if it satisfies n of its linear inequalities at equality
and the left-hand sides of these inequalities are linearly independent).
Definition 3.4. Let P be an H- or a V-polyhedron. Then the following hold:
• A point x ∈ P is an extreme point of P if and only if there are no x1 , x2 ∈ P
and λ ∈ (0, 1) such that x = λx1 + (1 − λ)x2 and x1 = x2 . We let ext(P )
denote the set of all extreme points of P .

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JUAN PABLO VIELMA

• A direction r ∈ P∞ \ {0} is an extreme ray of P if and only if there are no
r1 , r2 ∈ P∞ \ {0} such that r = r1 + r2 and r1 = λr2 for any λ > 0. We say
two extreme rays r and r are equivalent if and only if there exists λ > 0 such
that r = λr . We let ray(P ) denote the set of all extreme rays of P where,
for each set of equivalent extreme rays, we select exactly one representative
to be in ray(P ).
• If P has at least one extreme point, we say that P is pointed.
The definitions of extreme point and basic feasible solutions immediately coincide
for H-polyhedra and we also get an alternative characterization for extreme rays that
is analogous to basic feasible solutions.
Lemma 3.5. Let P ⊆ Qn be an H-polyhedron. Then the following hold:
• A point x ∈ P is an extreme point of P if and only it is a basic feasible
solution of P .
• A direction r ∈ P∞ \ {0} is an extreme ray of P if and only if it satisfies
n − 1 of the linear inequalities of P∞ at equality and the left-hand sides of
these inequalities are linearly independent.
Of course, as the following theorem finally shows, focusing on H-polyhedra is not
really a restriction.
Theorem 3.6 (Minkowski–Weyl). Let P ⊆ Qn such that P = ∅. Then P is a
pointed H-polyhedron if and only if P is a pointed V-polyhedron.
Furthermore, for any nonempty pointed polyhedron P we have that ext(P ) and
ray(P ) are finite and a valid V-representation of P is given by P = conv (ext(P )) +
cone (ray(P )).
The Minkowski–Weyl theorem can also be stated for nonpointed polyhedra (e.g.,
see [145, section 8.9]). However, for simplicity, from now on we assume that all polyhedra considered are pointed and nonempty. Such an assumption is naturally present
or can be easily enforced in most applications. For instance, the assumption holds
if all variables considered are nonnegative, which can be assured through standard
LP modeling tricks (e.g., through the reformulation x = u − v with u, v ≥ 0 for any
variable x with unrestricted sign).
While equivalent, both definitions of polyhedra have their advantages and, in
particular, provide alternative ways of describing sets to be modeled through MIP
formulations. We illustrate this using piecewise linear functions since modeling them
will be a running example for almost all formulations considered in this survey. The
following definition naturally extends piecewise linear functions such as the one defined
in (2.3) to the multivariate setting. In section 11.1, we will see that this definition
almost precisely describes the functions that have binary MIP formulations.
Definition 3.7. Let f : D ⊆ Qn → Q be a multivariate function. We define the
graph of f to be
gr(f ) := {(x, z) ∈ Qn × Q : x ∈ D, f (x) = z}
and the epigraph of f to be
epi(f ) := {(x, z) ∈ Qn × Q : x ∈ D, f (x) ≤ z}.
We say f is a bounded domain continuous piecewise linear function if f is con k
k
tinuous, D is bounded, and there exist mi i=1 ⊆ Qn , {ci }i=1 ⊆ Q, and rational

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

13

 k
polytopes Qi i=1 such that
D=

(3.3a)

k


Qi ,

i=1

⎧
⎪
m1 x + c1 ,
⎪
⎨
f (x) =

(3.3b)

⎪
⎪
⎩ mk x + c ,
k

x ∈ Q1 ,
..
.
x ∈ Qk .

The following proposition describes the H and V-representations of the graphs
and epigraphs of bounded domain continuous piecewise linear functions.
Proposition 3.8. Let f : D ⊆ Qn → Q be a bounded
 domain continuous
piecewise linear function for which Qi = x ∈ Qn : Ai x ≤ bi for all i ∈ {1, . . . , k}.
Then the graph of f can be described, respectively, as the union of H- and V-polyhedra
as follows:
(3.4a)

gr(f ) =

k




(x, z) ∈ Qn × Q : Ai x ≤ bi , mi x + ci = z ,

i=1

(3.4b)

gr(f ) =

k






v
.
f (v) v∈ext(Qi )

conv

i=1

Furthermore, the epigraph of f can be described, respectively, as the union of H- and
V-polyhedra with a common recession cone as follows:
(3.5a)

k



epi(f ) =
(x, z) ∈ Qn × Q : Ai x ≤ bi , mi x + ci ≤ z ,
i=1

(3.5b)

epi(f ) =

k



conv

i=1





v
0n
+ cone
,
f (v) v∈ext(Qi )
1

where in both cases the recession cone of all polyhedra considered is equal to


0n
cone
= {(x, z) ∈ Qn × Q : x = 0, z ≥ 0} .
1
3.2. Fundamental Theorem of Integer Programming and Formulation
Strength. The finite V-representation of any polyhedron guaranteed by Theorem 3.6
yields a convenient way to prove Proposition 2.4 as follows.
Propositions 2.4. A locally ideal MIP formulation is sharp.
Proof. Let (2.6) be a locally ideal MIP formulation of a set S and let Q ⊆
Qn × Qs × Qt be the polyhedron described by (2.6a). We need to show that the
projection of Q onto the x variables is contained in conv(S).
By Lemma 3.5 and locally idealness of (2.6) we have that ext (Q) ⊆ Qn × Qs × Zt .
Then, by Theorem 3.6 and through an appropriate scaling of the extreme rays of Q,


p
d
we have that there exist x̂j , ûj , ŷ j j=1 ⊆ Qn × Qs × Zt and x̃l , ũl , ỹ l l=1 ⊆
p
d


Zn × Zs × Zt such that ext (Q) = x̂j , ûj , ŷ j j=1 , ray (Q) = x̃l , ũl , ỹ l l=1 , and
Q = conv



x̂j , ûj , ŷ j

p
j=1



+ cone



x̃l , ũl , ỹ l

d 
l=1

.

14

JUAN PABLO VIELMA


Then for any (x, u, y) ∈ Q there exist λ ∈ Δp := λ ∈ Qp+ :
such that
(x, u, y) =

p


λj x̂j , ûj , ŷ j +

j=1

d


p
i=1


λi = 1 and μ ∈ Qd+

μl x̃l , ũl , ỹ l .

l=1


p
p
Let (x1 , u1 , y 1 ) := j=1 λj x̂j , ûj , ŷ j . Because points x̂j , ûj , ŷ j j=1 satisfy (2.6),
we have x̂j ∈ S for all j and hence x1 ∈ conv(S). Now, without loss of generality,
assume λ1 > 0 and let
(x2 , u2 , y 2 ) : =

p


λj x̂j , ûj , ŷ j + α

j=1

d


μl x̃l , ũl , ỹ l

l=1

= λ1 (x̄, ū, ȳ) +

p


λj x̂j , ûj , ŷ j ,

j=2

where α≥ 1 is such that (α/λ1 )

d
l=1

μl ỹ l ∈ Zt and

(x̄, ū, ȳ) := x̂1 , û1 , ŷ 1 + (α/λ1 )

d


μl x̃l , ũl , ỹ l .

l=1

Then ȳ = ŷ 1 + (α/λ1 ) l=1 μl ỹ l ∈ Zt and (x̄, ū, ȳ) satisfies (2.6) by Theorem 3.6.
Hence, x̄ ∈ S and x2 ∈ conv(S). The result then follows by noting that x = (1 −
1/α)x1 + (1/α)x2 .
To show Proposition 2.5 we need the following consequence of Theorem 3.6 known
as the Fundamental Theorem of Integer Programming. The theorem states that
the convex hull of (mixed) integer points in a rational polyhedron is also a rational
polyhedron and gives further structural guarantees on its V-representation.
Theorem 3.9. Let P ⊆ Qn be a nonempty pointed rational polyhedron and let
n1 , n2 ∈ Z+ be such that n = n1 + n2 . Then there exists a finite set V ⊆ P ∩
(Qn1 × Zn2 ) such that
d

conv (P ∩ (Qn1 × Zn2 )) = conv (V ) + cone (ray (P )) .

(3.6)

Theorem 3.9 shows that conv (P ∩ (Qn1 × Zn2 )) is the LP relaxation of a sharp
(by definition) formulation of P ∩ (Qn1 × Zn2 ). Furthermore, through Proposition 2.5
it shows that this formulation is additionally locally ideal.
Propositions 2.5. For A ∈ Qm×n , b ∈ Qm , and n1 , n2 ∈ Z+ such that n =
n1 + n2 , let
(2.10a)
(2.10b)

Ax ≤ b,
x ∈ Qn1 × Zn2 ,

be an MIP formulation of S ⊆ Qn (i.e., S is precisely the feasible region of (2.10)).
If the LP relaxation of (2.10) has at least one basic feasible solution, then (2.10) is
locally ideal if and only if it is sharp.
Proof. Locally idealness implying sharpness is direct from Propositions 2.4. For
the converse assume sharpness of (2.10) so that
Q := {x ∈ Rn : Ax ≤ b} = conv(S) = conv ({x ∈ Qn1 × Zn2 : Ax ≤ b}) .

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

15

 d
 p
Then by Theorem 3.9 there exist x̂j j=1 ⊆ P ∩ (Qn1 × Zn2 ) and x̃l l=1 ⊆ Zn such


p
that for any x ∈ ext (Q) ⊆ Q there exist λ ∈ Δp := λ ∈ Qp+ :
i=1 λi = 1 and μ ∈
Qd+ such that x = pj=1 λj x̂j + dl=1 μl x̃l . If d > 0 and μl > 0 for some l ∈ {1, . . . , d},
then, assuming without loss of generality that such l = 1, we have x = x1 /2 + x2 /2,
p
d
p
where x1 = j=1 λj x̂j + (1/2)μ1 x̃1 + l=2 μl x̃l and x2 = j=1 λj x̂j + (3/2)μ1 x̃1 +

μl x̃l are such that x1 , x2 ∈ Q and x1 = x2 . This contradicts the extremality of
p
x, so we have x = j=1 λj x̂j . If there are i, j ∈ {1, . . . , p} such that i = j, λi > 0,
and λj > 0, then, assuming without loss of generality that i = 1 and j = 2, we have
p
x = λ1 x1 / (λ1 + λ2 ) + λ2 x2 / (λ1 + λ2 ), where x1 = (λ1 + λ2 ) x̂1 + j=3 λj x̂j and
p
x2 = (λ1 + λ2 ) x̂2 + j=3 λj x̂j are such that x1 , x2 ∈ Q and x1 = x2 . This again
contradicts the extremality of x, so we may assume without loss of generality that
λ1 = 1, λi = 0 for all i ≥ 2 and μ = 0. Hence, x = x̂1 ∈ S and (2.10) is locally
ideal.
d
l=2

3.3. Linear Transformations and Projections. A convenient way to analyze the
strength of an MIP formulation is to show that its LP relaxation is the linear image
of the LP relaxation of another strong formulation. The following simple proposition
shows that the extreme points of the image LP relaxation are contained in the image
of the extreme points of the original LP relaxation. Hence, if the original formulation
is locally ideal and the linear transformation preserves integrality, then the image
formulation is also locally ideal.
Proposition 3.10. Let P ⊆ Qn be a rational polyhedron and L : Rn → Rp
be a linear transformation (i.e., L(αx + βy) = αL(x) + βL(y) for all α, β ∈ R and
x, y ∈ Rn ). Then ext (L (P )) ⊆ L (ext (P )), where L(S) := {L(x) : x ∈ S} for any
set S ⊆ Qn .
Proof. Let y ∈ ext (L (P )) and let x ∈ P be such that y = L(x). By Theorem 3.6
 d
 p
there exist x̂j j=1 ⊆ ext(P ), x̃l l=1 ⊆ ray(P ), λ ∈ Δp , and μ ∈ Qd+ such that
p

d

p

d

x = j=1 λj x̂j + l=1 μl x̃l and hence y = j=1 λj L x̂j + l=1 μl L x̃l . If d > 0
and μl > 0 for some l such that L x̃l = 0, then we contradict the extremality of y as
p
in the proof of Proposition 2.5. Then, y = j=1 λj L x̂j . If there are i, j ∈ {1, . . . , p}
i
such that i = j, λi > 0, λj > 0, and L x̂ = L x̂j , we again reach a contradiction
with the extremality of y. Hence, y = L x̂j for some j ∈ {1, . . . , p}, which concludes
the proof.
Since the projection onto a set of variables is a linear transformation, Proposition 3.10 shows that the extreme points of the projection of a polyhedron are contained
in the projection of the extreme points of the same polyhedron. Hence, because projection preserves integrality, the projection of locally ideal formulations is also locally
ideal. However, in section 5 we will show that projecting a polyhedron (or formulation) can result in a significant increment in the number of inequalities. To achieve
this we will need the following proposition that gives a more detailed description of
an H-representation of the projection of a polyhedron.
Proposition 3.11. Let A ∈ Qm×n , D ∈ Qm×p , b ∈ Qm ,

and C = μ ∈ Qm

P = {x ∈ Qn : ∃w ∈ Qp s.t. Ax + Dw ≤ b} ,

: DT μ = 0, μ ≥ 0 . Then


P = x ∈ Qn : μT Ax ≤ μT b ∀μ ∈ ray(C) .

In particular we have that P∞ = {x ∈ Qn : ∃w ∈ Qp s.t. Ax + Dw ≤ 0}.

16

JUAN PABLO VIELMA

3.4. Implied Equalities, Redundant Inequalities, and Facets. The number of
constraints of an MIP formulation is equal to the number of inequalities used in
the specific H-representation of the polyhedron associated to the LP relaxation of
that formulation. However, the size of an H-representation of a polyhedron can be
artificially inflated by adding redundant linear inequalities. Hence, to evaluate the
real size of an MIP formulation (without redundant inequalities) we need to calculate
the size of the smallest H-representation of a polyhedron. The following definition
formalizes some concepts that will allow us to describe such a smallest representation.
Definition 3.12. Let A ∈ Qm×n , b ∈ Qm , P := {x ∈ Qn : Ax ≤ b}, and ai be
the ith row of A. We say F ⊆ P is


• a face of P if and only if F = x ∈ P : al x = bl ∀l ∈ L for some L ⊆
{1, . . . , m};
• a proper face of P if and only if F is a face of P , F = ∅, and F = P ; and
• a facet of P if and only if F is a proper face of P that is maximal with respect
to inclusion.
We also say that an inequality ai x ≤ bi of P is
• an implied equality of P if and only if ai x = bi for all
 x ∈ P;

• a facet-defining inequality of P if and only if F := x ∈ P : ai x = bi is a
facet (in such case we say the inequality defines F ); and
• a redundant inequality of P for subsystem L ⊆ {1, . . . , m} with i ∈ L if and
only if


P = x ∈ Qn : al x ≤ bl ∀l ∈ L \ {i} .
Finally we say that subsystem L ⊆ {1, . . . , m} is a minimal representation of P if


P = x ∈ Qn : al x ≤ bl ∀l ∈ L
and there is no l ∈ L such that al x ≤ bl is a redundant inequality of P for L.
Note that redundancy is strongly dependent on the selected subsystem, which
can lead to the existence of multiple minimal representations when implied equalities
are present. This is illustrated in the following example.
Example 1. Let
⎞
⎛
⎤
⎡
0
0
1
⎜ 0 ⎟
⎢ 0 −1 ⎥
⎟
⎜
⎥
⎢
⎜ 1 ⎟
⎥
⎢ 1
0 ⎥
⎟
⎜
⎢
A=⎢
⎥, b = ⎜ 0 ⎟
⎟
⎜
⎢ −1 −1 ⎥
⎝ 0 ⎠
⎣ −1
1 ⎦
0
−1
0




and P = x ∈ Q2 : Ax ≤ b = x ∈ Q2 : x2 = 0, 0 ≤ x1 ≤ 1 . The faces of P are
F0 := ∅, F2 := {(0, 0)}, F3 := {(1, 0)}, and F4 := P . The facets of P are F2 and
F3 . We also have that ai x ≤ bi is an implied equality for i ∈ {1, 2}, is facet defining
for i ∈ {3, 4, 5, 6}, and is redundant for system L = {1, . . . , 5} for i ∈ {4, 5, 6}.
However, facet F2 is defined by ai x ≤ bi for any i ∈ {4, 5, 6} and at least one of these
inequalities is necessary to describe P . In fact, P has three minimal representations
given by L1 = {1, 2, 3, 4}, L2 = {1, 2, 3, 5}, and L3 = {1, 2, 3, 6}.
Constructing a minimal representation can be complicated even in the absence of
implied equalities. Fortunately, as shown by the following proposition, the concept of
facet-defining inequality and some linear algebra allows us to give a precise characterization of the number of inequalities in a minimal representation of a polyhedron.

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

17

Proposition 3.13. Let A ∈ Qm×n , b ∈ Qm , and P := {x ∈ Qn : Ax ≤ b}.
Then for any facet of F of P there exists i ∈ {1, . . . , m} such that ai x ≤ bi defines F .
Hence the number of facets of a polyhedron is always finite.
Let F ⊆ {1, . . . , m} be the set of facet-defining inequalities, f be the number
of facets of P , E ⊆ {1, . . . , m} be the set of implied equalities of P , and r =
rank [Al ]l∈E (i.e., the maximum number of linearly independent vectors in {Al }l∈E ).
Then there exist F  ⊆ F with |F  | = f and E  ⊆ E with |E  | = r such that
)
P =

x∈Q :
n

al x ≤ b l

∀l ∈ F 

al x = b l

∀l ∈ E 

*

is a minimal representation of P . In particular, every minimal representation of P
has 2r + f inequalities (or r equalities and f inequalities)
Determining what inequalities of a polyhedron are facet defining can be done using
linear algebra techniques, but this can still be a highly nontrivial endeavor. We will
present several examples of facet-defining inequalities throughout this survey, but a
detailed description of the techniques used to show that they are indeed facet defining
is beyond the scope of this survey. For more details, we refer the interested reader
to the references on polyhedral theory [22, 131, 145, 166, 170] and to [146] for a wide
range of examples, techniques, and applications from combinatorial optimization.
4. Size and Extended MIP Formulations. Strength and small size can sometimes be incompatible in MIP formulations. Fortunately, this can often be conciliated
by utilizing the power of auxiliary variables in extended formulations. We first illustrate this by showing an example where the incompatibility between strength and
small size can be resolved by using the same binary variables that are required to
construct even the simplest formulation. 
Example 2. Consider the set S := ni=1 P i where, for each i ∈ {1, . . . , n},
we have P i := {x ∈ Qn : |xi | ≤ 1, xj = 0 ∀j = i}. It is easy to check that an MIP
formulation of S is given by
(4.1a)
(4.1b)

yi − 1 ≤ xj ≤ 1 − yi
n

yi = 1,

∀i ∈ {1, . . . , n}, j = i,

i=1

(4.1c)
(4.1d)

0 ≤ yi ≤ 1

∀i ∈ {1, . . . , n},

y ∈Z .
n

Formulation (4.1) in Example 2 can be easily constructed with simple logic or
with a basic application of a well-known formulation technique (see Example 8). Unfortunately, in this case the resulting formulation is not sharp. Indeed, for n = 3
we have that xi = 2/3 for i ∈ {1, . . . , 3} and y1 = y2 = y3 = 1/3 is feasible for
the
of (4.1)
 LP 3relaxation
 given by (4.1a)–(4.1c). However, for n = 3, conv(S) =
3
x∈Q :
|x
|
≤
1
, which does not contain (2/3, 2/3, 2/3). This is illustrated
j
i=1
in Figure 4.1(a), which shows in blue the projection onto the x variables of the LP
relaxation of the formulation (4.1), and in Figure 4.1(b), which does the same for the
convex hull conv(S). Both figures show S in red.
From Figure 4.1 we can see that formulation (4.1) can be made sharp by adding
the 8 inequalities defining the diamond depicted in Figure 4.1(b). In fact, we can

18

JUAN PABLO VIELMA

(a) LP relaxation of formulation (4.1) for n = 3.

(b) Convex Hull.

Fig. 4.1 Geometric view of formulation strength.

show that the n-dimensional form of these inequalities is
n


(4.2)

ri xi ≤ 1

∀r ∈ {−1, 1}n,

i=1

and that for any n,
)
conv(S) =

x∈Q :
n

n


*
|xj | ≤ 1

= {x ∈ Qn : (4.2)} .

i=1

Then, the formulation obtained by adding (4.2) to (4.1) is automatically sharp. However, formulation (4.1)–(4.2) has two problems. First, it is not locally ideal because,
for n = 3, we have that x1 = x2 = y2 = y3 = 1/2, x3 = y1 = 0 is an extreme point of
the LP relaxation of (4.1)–(4.2). Second, the formulation is extremely large, because
the number of linear inequalities described by (4.2) is 2n . Unfortunately, each one of
these inequalities defines a different facet of conv(S) and together they form a minimal representation of conv(S). Thus removing any of them destroys the sharpness
property. Fortunately, careful use of auxiliary variables y allows constructing a much
smaller and locally ideal formulation for S.
Example 3. A polynomial-sized sharp formulation for set S in Example 2 is
given by
(4.3a)
(4.3b)

−yi ≤ xi ≤ yi
n


∀i ∈ {1, . . . , n},

yi = 1,

i=1

(4.3c)
(4.3d)

yi ≥ 0
y ∈ Zn .

∀i ∈ {1, . . . , n},

Formulation (4.3) is locally ideal and can be constructed using a well-known LP
modeling trick or by using standard MIP formulation techniques (see Examples 6

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

19

and 8). Formulation (4.3)’s only auxiliary variables are the binary variables that are
used to indicate which P i contains x. The following example shows that these binary
variables might not be enough to construct a polynomial-sized sharp formulation.
Example 4. For i ∈ {1, . . . , n}, let v i , wi ∈ Qn be defined by
)
)
n,
j
=
i,
−1, j = i,
wji =
vji =
−1, j = i,
0, j = i,
for all j ∈ {1, . . . , n}. In addition, let v 0 , w0 ∈ Qn be defined by vj0 = −wj0 = −1 for
all j ∈ {1, . . . , n}.
 n
Let S = (V × {0}) ∪ (W × {1}) ⊆ Qn+1 , where V = conv v i i=0 and W =
 i n
conv w i=0 . S and conv(S) are depicted for n = 2 in Figure 4.2, where S is
shown in red and conv(S) is shown in blue. By noting that
⎧
⎫
n
⎨
⎬

(4.4a) V × {0} = x ∈ Qn+1 : xn+1 = 0,
xj ≤ 1, −xj ≤ 1 ∀j ∈ {1, . . . , n}
⎩
⎭
j=1

and

(4.4b) W × {1} =

⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩

xn+1
x ∈ Qn+1 :
(n + 1)xi −

n

j=1

⎫
⎪
⎪
= 1, −
xj ≤ 1, ⎪
⎪
⎪
⎬
j=1
n


⎪
⎪
⎪
xj ≤ 1 ∀i ∈ {1, . . . , n}⎪
⎪
⎭

we can check that a valid formulation of S is given by
−xj ≤ 1

(4.5a)

n


(4.5b)

∀j ∈ {1, . . . , n},

xj ≤ 1 + (n − 1)(1 − y1 ),

j=1

(4.5c)
(4.5d)

(4.5e)

xn+1 = 1 − y1 ,
n

(n + 1)xi −
xj ≤ 1 + (n2 + n − 2)(1 − y2 )
−

j=1
n


∀i ∈ {1, . . . , n},

xj ≤ 1 + (n − 1)(1 − y2 ),

j=1

(4.5f)
(4.5g)
(4.5h)

xn+1 = y2 ,
y1 + y2 = 1,
y ∈ {0, 1}2.

Now for n = 3, x1 = x2 = −1, x3 = 0, and x4 = y1 = y2 = 1/2 is feasible for the LP
relaxation of (6.8), but violates x1 + x2 − x4 ≥ −2, which is a facet-defining inequality
of conv(S).
Although Figure 4.2 shows that conv(S) has few facets for small n, the following
lemma shows that the number of facets of conv(S) grows exponentially in n. Furthermore, the lemma shows that the two binary variables used by (4.5) are not enough

20

JUAN PABLO VIELMA

Fig. 4.2 Set for Example 4 for n = 2.

to yield a polynomial-sized sharp formulation even if a constant (independent of n)
number of additional auxiliary variables is used.
Lemma 4.1. Let S be the set defined in Example 4. Then the number of facets
of conv(S) grows exponentially in n. Furthermore, there is no sharp formulation of
S of the form
Ax + Bλ + Dy ≤ b, x ∈ Qn+1 , λ ∈ Qk , y ∈ Z2 ,
where A ∈ Qp(n)×(n+1) , B ∈ Qp(n)×k , D ∈ Qp(n)×2 , and b ∈ Qp(n) for some polynomial p and a constant k ∈ Z+ independent of n.
Proof. To prove the first statement we note that V and W are two n-dimensional
simplicies that are dual to each other. Then, conv(S) is the antiprism of V [28] and V
satisfies the conditions for Theorem 2.1 in [28]. Hence, by this theorem, the number
of facets of conv(S) is exactly two more than the number of proper faces of V . The
n−1
n+1
− 2 [63],
number of proper faces of an n-dimensional simplex is i=0 n+1
i+1 = 2
n+1
so we conclude that the number of facets of conv(S) is precisely 2
.
For the second statement we note that by Propositions 3.11 and 3.13 the number
of facets of the projection of the LP relaxation of the proposed formulation onto the
x variables is at most the number of extreme rays of cone
,
+
p(n)
μ ∈ Q+ : DT μ = 0, B T μ = 0 .
p(n)
, which
By Lemma 3.5, the number of extreme rays of this cone is at most p(n)−3−k
is also a polynomial. Hence, no formulation of this form can have an LP relaxation
that projects to conv(S).
Fortunately, by allowing a growing number of auxiliary variables, the following
proposition by Balas, Jeroslow, and Lowe [13, 90, 113] yields polynomial-sized formulations for a wide range of disjunctive constraints that include the set in Example 4.
We postpone the proof of this proposition to section 5.1, where we consider a slightly
more general version of this formulation.
 k
Proposition 4.2. Let P i i=1 be a finite family of polyhedra with a common


i
j
recession cone (i.e., P∞
= P∞
for all i, j), such that P i = x ∈ Qn : Ai x ≤ bi for
k
all i. Then, a locally ideal MIP formulation of S = i=1 P i is given by

(4.6a)
(4.6b)

Ai xi ≤ bi yi
k

i=1

xi = x,

∀i ∈ {1, . . . , k},

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES
k


(4.6c)

21

yi = 1,

i=1

(4.6d)

yi ≥ 0

(4.6e)

x ∈Q
i

∀i ∈ {1, . . . , k},
n

∀i ∈ {1, . . . , k},

y∈Z .
k

(4.6f)

Example 5. Using formulation (4.6) for characterization (4.4) of set S in Example 4 results in the following polynomial-sized sharp (and locally ideal) formulation
that uses a linear number of additional continuous auxiliary variables:
x1n+1 = 0,
n

x1j ≤ y1 ,

(4.7a)
(4.7b)

j=1

−x1i ≤ y1

(4.7c)
(4.7d)
(4.7e)

(4.7f)
(4.7g)
(4.7h)
(4.7i)

(n +

x2n+1
n

−
x2j
j=1
n

x2j
1)x2i −
j=1

∀i ∈ {1, . . . , n},

= y2 ,
≤ y2 ,
≤ y2

xi = x1i + x2i

∀i ∈ {1, . . . , n},
∀i ∈ {1, . . . , n + 1},

y1 + y2 = 1,
y ∈ {0, 1}2,

(4.7j)

x1 , x2 ∈ Qn+1 ,

(4.7k)

x ∈ Qn+1 .

The use of a growing number of auxiliary variables in Proposition 4.2 and similar
techniques allows the construction of polynomial-sized sharp extended formulations
for a wide range of sets. However, there are cases in which these formulations cannot
be constructed. Most examples of sets that do not have polynomial-sized extended
formulations arise from intractable combinatorial optimization problems (e.g., the
traveling salesman problem considered in [53]). However, the following recent result by
Rothvoß [140] shows that this can also happen for polynomially solvable combinatorial
optimization problems.
Theorem 4.3. Let G = (V, E) be the complete graph on |V | = n nodes where
V = {1, . . . , n} and E = {{i, j} : i, j ∈ {1, . . . , n} , i = j}. Let S be the set of
incident vectors of perfect matchings of G given by
⎫
⎧
⎬
⎨

x{i,j} = 1 ∀i ∈ {1, . . . , n} .
(4.8)
S := x ∈ {0, 1}E :
⎭
⎩
j∈{1,...,n}\{i}

If n is even, then there is no polynomial-sized sharp extended formulation for S.
The proof techniques used to show results such as Theorem 4.3 are significantly
more elaborate than those used in Lemma 4.1 and are beyond the scope of this survey.
We refer the reader interested in these techniques to [53, 140] and their references and
to [35, 95].

22

JUAN PABLO VIELMA

5. Basic Extended Formulations. Disjunctive constraints can model a wide
range of logical constraints. However, there are other aspects of MIP formulations
that are generally encountered in practice, such as feasible regions of knapsacks or
other problems with general integer variables. One class of sets that combines these
two aspects is that of the unions of mixed integer sets of the form
(5.1)

S=

k


P i ∩ (Qn1 × Zn2 ) ,

i=1




i k

where P i=1 is a finite family of polyhedra with a common recession cone. Hooker
[76] showed that such sets can be modeled through a simple extension of formulation
(4.6). Hooker also showed that this extension is sharp if the formulations of these
mixed integer sets are sharp (i.e., if P i = conv P i ∩ (Qn1 × Zn2 ) ). However, achieving this could require a large number of inequalities in the descriptions of the P i s.
Fortunately, as noted in [37], we may significantly reduce the number of inequalities
by using auxiliary variables in the description of P i . This results in the following
generalization of formulation (4.6) that also yields a locally ideal or sharp formulation
when locally ideal or sharp extended formulations of P i ∩ (Qn1 × Zn2 ) are used.
 k
Proposition 5.1. Let P i i=1 be a finite family of polyhedra with a common
i
j
recession
(i.e., P∞
= P∞
for all i, j ∈ {1, . . . , k}) and p ∈ Zk+ be such that
 cone
i
n
pi
P = x ∈ Q : ∃w ∈ Q s.t. (x, w) ∈ Qi , where


Qi = (x, w) ∈ Qn × Qpi : Ai x + Di w ≤ bi
for Ai ∈ Qmi ×n , Di ∈ Qmi ×pi , and bi ∈ Qmi for each i ∈ {1, . . . , k}. Then, for any
k
n1 , n2 ∈ Z+ such that n = n1 + n2 , an MIP formulation of S = i=1 P i ∩ (Qn1 × Zn2 )
is given by
(5.2a)
(5.2b)

Ai xi + Di wi ≤ bi yi
k


∀i ∈ {1, . . . , k},

xi = x,

i=1

(5.2c)

k


yi = 1,

i=1

(5.2d)

yi ≥ 0

(5.2e)
(5.2f)
(5.2g)
(5.2h)

∀i ∈ {1, . . . , k},

x ∈Q

n

∀i ∈ {1, . . . , k},

w ∈Q

pi

∀i ∈ {1, . . . , k},

i

i

y ∈ Zk ,
x ∈ Qn1 × Zn2 .

Furthermore, if P i = conv P i ∩ (Qn1 × Zn2 ) for all i ∈ {1, . . . , k}, then (5.2) is
sharp, and if ext Qi ⊆ Qn1 × Zn2 × Qpi for all i ∈ {1, . . . , k}, then (5.2) is locally
ideal.
Proof. For validity of (5.2), without loss of generality, assume y1 = 1 and yi = 0
i
for all i ≥ 2. Then x1 ∈ P 1 and by Proposition 3.11 we have that xi ∈ P∞
for all
i
1
i ≥ 2. Then by the common recession cone assumption we have x ∈ P∞ for all i ≥ 2
and hence x ∈ P 1 .

23

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

To prove sharpness let (x, w, y) be feasible for the LP relaxation of (5.2) and
I = {i ∈ {1, . . . , k} : yi > 0}. Then x = i∈I yi xi /yi , i∈I yi = 1, y ≥ 0, and
xi /yi , wi /yi ∈ Qi for all i ∈ I. By the assumption on P i we then have that
xi /yi ∈ conv P i ∩ (Qn1 × Zn2 ) and hence
 k

 k



i
n1
n2
i
n1
n2
x ∈ conv
conv P ∩ (Q × Z )
= conv
P ∩ (Q × Z ) = conv(S).
i=1

i=1

To prove locally idealness first note that if (x, w, y) is an extreme point of the LP
relaxation of (5.2) and y ∈ {0, 1}k , we may again assume without loss of generality
that y1 = 1 and yi = 0 for all i ≥ 2. Then by extremality of (x, w, y) we have
that xi = 0 and wi = 0 for all i ≥ 2, x = x1 , and x1 , w1 ∈ ext Q1 . Then, by
the assumption on Q1 we have x = x1 ∈ Qn1 × Zn2 and hence (x, w, y) satisfies the
integrality constraints. To finish the proof, assume for a contradiction that (x, w, y)
is an extreme point of the LP relaxation of (5.2) such that y ∈
/ {0, 1}k . Without loss
of generality we may assume that y1 , y2 ∈ (0, 1). Let ε = min{y1 , y2 , 1 − y1 , 1 − y2 } ∈
(0, 1),
⎧
⎧
⎪
⎪
⎨yi + ε, i = 1,
⎨yi − ε, i = 1,
y i = yi + ε, i = 2,
yi = yi − ε, i = 2,
⎪
⎪
⎩
⎩
yi ,
yi
otherwise,
otherwise.
xi = (y i /yi )xi , w i = (y i /yi )wi , xi = (y i /yi )xi , and w i = (y i /yi )wi for i ∈ {1, 2},
k

k

/ {1, 2}, x = i=1 xi , and x = i=1 xi . Then
xi = xi = xi and w i = w i = wi for all i ∈
(x, w, y) = (x, w, y), (x, w, y) = (1/2)(x, w, y) + (1/2)(x, w, y), and (x, w, y), (x, w, y)
are feasible for the LP relaxation of (5.2), which contradicts (x, w, y) being an extreme
point.
Formulation 5.2 can be used to construct several known formulations for piecewise
linear functions and more general disjunctive constraints. The following proposition
illustrates this by constructing a variant of (5.2) that is convenient when P i are
described through their extreme points and rays. The resulting formulation is a
straightforward extension of a formulation for disjunctive constraints introduced by
Jeroslow and Lowe [90, 113].
 k
Corollary 5.2. Let P i i=1 ⊆ Qn be a finite family of polyhedra with a common recession cone C. Then, for any n1 , n2 ∈ Z+ such that n = n1 + n2 , an MIP
k
formulation of S = i=1 P i ∩ (Qn1 × Zn2 ) is given by
(5.3a)

k




i=1 v∈ext(P i )

(5.3b)

vλiv +



rμr = x,

r∈ray(C)



∀i ∈ {1, . . . , k},

λiv = yi

v∈ext(P i )

(5.3c)

k


yi = 1,

i=1

(5.3d)

λiv ≥ 0

(5.3e)

μr ≥ 0

∀i ∈ {1, . . . , k}, v ∈ ext P i ,
∀r ∈ ray(C),

(5.3f)

y ∈ {0, 1} ,

(5.3g)

x ∈ Qn1 × Zn2 .

k

24

JUAN PABLO VIELMA

Furthermore, if P i = conv P i ∩ (Qn1 × Zn2 ) for all i ∈ {1, . . . , k}, then (5.3) is a
locally ideal formulation of S.
Proof. By Theorem 3.6 we have that P i is the projection onto the x variables
of


⎧
⎫
⎪
vλv +
rμr = x⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
r∈ray(C)
v∈ext(P i )
⎪
⎪
⎪
⎪

⎪
⎪
⎨
⎬
i
λ
=
1
ext(P )
i
n
ray(C)
v
:
Q = (x, μ, λ) ∈ Q × Q
×Q
⎪
⎪
v∈ext(P i )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
i
⎪
⎪
λ
≥
0
∀v
∈
ext
P
v
⎪
⎪
⎪
⎪
⎩
⎭
μr ≥ 0 ∀r ∈ ray (C)
for all i ∈ {1, . . . , k}. Using these extended formulations of the Qi s we have that
formulation (5.2) for S is given by


(5.4a)
vλiv +
rμir = xi
∀i ∈ {1, . . . , k},
v∈ext(P i )

(5.4b)

r∈ray(C)



λiv = yi

∀i ∈ {1, . . . , k},

v∈ext(P i )

(5.4c)

k


xi = x,

i=1

(5.4d)

k


yi = 1,

i=1

(5.4e)

λiv ≥ 0

∀i ∈ {1, . . . , k}, v ∈ ext P i ,

(5.4f)

μir ≥ 0

∀i ∈ {1, . . . , k}, r ∈ ray (C) ,

(5.4g)

yi ≥ 0

(5.4h)

∀i ∈ {1, . . . , k},

x ∈Q

n

∀i ∈ {1, . . . , k},

(5.4i)

λ ∈Q

ext(P i )

∀i ∈ {1, . . . , k},

(5.4j)

μ ∈Q

ray(C)

∀i ∈ {1, . . . , k},

i
i

i

(5.4k)

y∈Z ,

(5.4l)

x ∈ Qn1 × Zn2 .

k

We claim that the LP relaxation of (5.3) is equal to the image of the LP relaxation
of (5.4) through the linear transformation that projects out the xi and μi variables
k
and lets μ = i=1 μi . We first show that the LP relaxation of (5.3) is contained
in the image of the LP relaxation of (5.4). For this, simply note that any point in
the LP relaxation of (5.3) is the image of the point in the LP relaxation of (5.4)
i
i
obtained by letting xi =
v∈ext(P i ) vλv + (1/k)
r∈ray(C) rμr and μr = (1/k)μr
for all i ∈ {1, . . . , k} and r ∈ ray (C). The reverse containment is straightforward.
Validity then follows directly from Proposition 5.1.
For locally idealness note that by Proposition 2.5 and the assumption on P i we
have that ext P i ⊆ Qn1 × Zn2 . By noting that (x, μ, λ) ∈ ext Qi if and only
if x ∈ ext P i , μ = 0, λx = 1, and λv = 0 for v ∈ ext P i \ {x}, we have that
i
ext Qi ⊆ Qn1 ×Zn2 ×Qray(C) ×Qext(P ) . Then by Proposition 5.1 we have that (5.4)

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

25

is locally ideal and hence so is (5.3) by Proposition 3.10 and the linear transformation
described in the previous paragraph.
To distinguish extreme point/ray formulation (5.3) from inequality formulation
(5.2), we refer to the first one as the V-formulation and to the second as the Hformulation. While V-formulation (5.3) can be derived from H-formulation (5.2),
their application to specific disjunctive constraints can yield formulations with very
different structures. The following two examples illustrate this for the case n2 = 0 in
both formulations and pi = 0 for all i ∈ {1,
n. . . , k} in Proposition 5.1.
Example 6. Consider the set S = i=1 {x ∈ Qn : −1 ≤ xi ≤ 1, xj = 0 ∀j = i}
from Example 2. H-formulation (5.2) for S is given by
(5.5a)

−yi ≤ xii ≤ yi

(5.5b)
(5.5c)

n


xji

∀i, j ∈ {1, . . . , n}, i = j,

=0

yi = 1,

i=1

(5.5d)

∀i ∈ {1, . . . , n},

n


xi =

xji

∀i ∈ {1, . . . , n},

j=1

(5.5e)

y ∈ {0, 1}n.

Similarly to the proof of Proposition 5.1, we can show that the projection of (5.5) onto
the x and y variables is precisely formulation (4.3) from Example 3, which is given
by
(5.6a)
(5.6b)

−yi ≤ xi ≤ yi
n

yi = 1,

∀i ∈ {1, . . . , n},

i=1

yi ≥ 0

(5.6c)

∀i ∈ {1, . . . , n},

y ∈Z .
n

(5.6d)

If we instead use V-formulation (5.3), we obtain the alternative formulation of S given
by
(5.7a)

λi1 − λi2 = xi

∀i ∈ {1, . . . , n},

(5.7b)

λi1

∀i ∈ {1, . . . , n},

(5.7c)
(5.7d)
(5.7e)

λi2

= yi

yi
i=1
λi1 , λi2

= 1,

+
n


≥0

∀i ∈ {1, . . . , n},

y ∈ {0, 1} .
n

It is interesting to contrast formulations (5.6) and (5.7). We know that conv(S) =
n
{x ∈ Qn :
i=1 |xi | ≤ 1} and that both formulations are sharp. Hence the LP
n
relaxations of both formulations should contain lifted representations of i=1 |xi | ≤
1. H-formulation (5.6) does this using the standard trick of modeling |xi | ≤ yi as
−yi ≤ xi ≤ yi , while V-formulation (5.7) does it by the alternative trick of modeling
|xi | ≤ yi as xi = λi1 − λi2 , λi1 + λi2 =yi , and λi1 , λi2 ≥ 0. The latter trick uses the fact
that x = x+ − x− and |x| = x+ + x− , where x+ := max{x, 0} and x− := {−x, 0}.

26

JUAN PABLO VIELMA

Example 7. Let f : D ⊆ Qd → Q be a piecewise linear function defined by (3.3)
for a given finite family of polytopes {Qi }ki=1 . Formulation (5.2) for characterization
(3.4a) of gr(f ) results in the MIP formulation of gr(f ) given by
(5.8a)

z=

k


zi ,

i=1
i i

(5.8b)

z i = m x + ci ,

(5.8c)

x=

(5.8d)

Ai xi ≤ yi bi

k


xi ,

i=1

k


(5.8e)

∀i ∈ {1, . . . , k},

yi = 1,

i=1

y ∈ {0, 1}k .

(5.8f)

If we replace (5.8a) and (5.8b) in (5.8) by
(5.9)

z=

k


mi xi + ci ,

i=1

we obtain a standard MIP formulation for piecewise linear functions that is denoted
the multiple choice model in [155]. Similarly to the proof of Proposition 5.1, we can
show that the multiple choice model is the projection onto the x, xi , y, and z variables
of (5.8). If we instead use formulation (5.3) for characterization (3.4b) of gr(f ), we
obtain the formulation of gr(f ) given by
k


(5.10a)



vλiv = x,

i=1 v∈ext(Qi )

(5.10b)
(5.10c)

k




i=1

v∈ext(Qi )

f (v)λiv = z,


λiv = yi

∀i ∈ {1, . . . , k},

v∈ext(Qi )

(5.10d)

k


yi = 1,

i=1

(5.10e)
(5.10f)

λiv ≥ 0

∀i ∈ {1, . . . , k}, v ∈ ext Qi ,

y ∈ {0, 1}k ,

which is a standard MIP formulation for piecewise linear functions that is denoted the
disaggregated convex combination model in [155].
For examples of formulation (5.2) with pi > 0 and n2 > 0 we refer the reader to
[37] and [76], respectively.
6. Projected Formulations. One disadvantage of basic formulations (5.2) and
(5.4) is that they require multiple copies of the original x variables (i.e., the xi variables) or of some auxiliary variables (i.e., the λi variables). In this section we study
formulations that do not use these copies of variables and are hence much smaller.
The cost of this reduction in variables is usually a loss of strength, but it some cases

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

27

this loss can be avoided. We begin by considering the well-known Big-M approach
and an alternative formulation that combines the Big-M approach with a formulation
tailored to polyhedra with a common structure. We then consider the strength of
both classes of formulations in detail.
6.1. Traditional Big-M Formulations. One way to write MIP formulations without using copies of the original variables is to use the Big-M technique. The following
proposition proven in [164] for the bounded case and in [76] for the unbounded describes the technique.
 k
Proposition 6.1. Let P i i=1 be a finite family of polyhedra with a common


i
j
recession cone (i.e., P∞
= P∞
for all i, j), such that P i = x ∈ Qn : Ai x ≤ bi ,
where Ai ∈ Qmi ×n and bi ∈ Qmi for all i ∈ {1, . . . , k}. Furthermore, for each
i, j ∈ {1, . . . , k}, i = j, and l ∈ {1, . . . , mi }, let Mli,j ∈ Q be such that


(6.1)
Mli,j ≥ max ai,l x : Aj x ≤ bj ,
x

where a ∈ Q is the lth row of A . Then, for any n1 , n2 ∈ Z+ such that n = n1 +n2 ,
k
an MIP formulation for S = i=1 P i ∩ (Qn1 × Zn2 ) is given by
i,l

(6.2a)
(6.2b)

n

i

Ai x ≤ bi + M i − bi (1 − yi )
k


∀i ∈ {1, . . . , k},

yi = 1,

i=1

(6.2c)

yi ≥ 0

∀i ∈ {1, . . . , k},

(6.2d)
(6.2e)

y∈Z ,
x∈ Qn1 × Zn2 ,
k

where M i ∈ Qmi are such that Mli = maxj=i Mli,j for each l ∈ {1, . . . , mi }.
Proof. If all Mli,j are finite, validity of the formulation is straightforward. To
show their finiteness, assume for a contradiction that for a given i, j, and l the lefthand side of (6.1) is infinite. The
 unboundedness of this LP problem is equivalent to
j
= x ∈ Qn : Aj x ≤ 0 such that ai,l r > 0 [21, Theorem
the existence of an r ∈ P∞
i
4.13]. However, by the assumption on the recession
cones, such
 i,l
 an r is also in P∞
i
i
and hence the LP problem given by maxx a x : A x ≤ b is unbounded, which
contradicts bil being finite.
The strongest possible version of formulation (6.2) is obtained when equality holds
in (6.1), which, as illustrated in the following example, can sometimes yield sharp or
locally ideal formulations.
Example 8. Consider the set S from Example 2, which corresponds to the case
I
] ∈ Q2n×n and
Ai = [ −I
)
1, l ∈ {i, k + i} ,
bil =
0 otherwise
for all i, l ∈ {1, . . . , k}, and n2 = 0. A valid Big-M selection is to take Mli = 1 for all
i, l ∈ {1, . . . , k}. For this choice, (6.2) is equal to the nonsharp formulation (4.1). In
contrast, the strongest choice of M i given by
)
0, l ∈ {i, k + i} ,
i
Ml =
1 otherwise
yields locally ideal formulation (4.3).

28

JUAN PABLO VIELMA

Unfortunately, as the following example shows, even the strongest version of (6.2)
can fail to be sharp.
Example 9. The strongest version of formulation (6.2) for S from Example 4 is
precisely the nonsharp formulation (4.5).
For more discussion about Big-M formulations for general and specially structured
sets, we refer the reader to [18, 76, 79, 164].
6.2. Hybrid Big-M Formulations. A different class of projected formulations was
introduced by Balas, Blair, and Jeroslow [12, 26, 87] for families of polyhedra with
a common left-hand side matrix (i.e., where Ai = Aj for all i, j). Balas, Blair, and
Jeroslow showed that such formulations can have very favorable strength properties.
However, while the common left-hand side structure appears in many problems (see
Example 10), these formulations still have more limited applicability than the traditional Big-M formulation from Proposition 6.1. For this reason we here combine
the projected formulation of Balas, Blair, and Jeroslow with the traditional Big-M
formulation to introduce the following hybrid Big-M formulation that generalizes the
former and improves upon the latter. While this formulation does not require the
common left-hand side structure, it is equipped to exploit it when present.
ks
p 
Proposition 6.2. For k ∈ Zp+ , let s=1 P s,i i=1 be a finite family of polyhedra
s,i
t,j
= P∞
for all s, t, i, j) such that
with a common recession cone (i.e., P∞


P s,i = x ∈ Qn : As x ≤ bs,i ,
where As ∈ Qms ×n and bs,i ∈ Qms for each s ∈ {1, . . . , p} and i ∈ {1, . . . , ks }.
Furthermore, for each s, t ∈ {1, . . . , p}, i ∈ {1, . . . , ks }, and l ∈ {1, . . . , ms }, let
Mls,t,i ∈ Q be such that


Mls,t,i ≥ max as,l x : At x ≤ bt,i ,

(6.3)

x

where as,l ∈ Qn is the lth row of As . Then, for any n1 , n2 ∈ Z+ such that n = n1 +n2 ,
p ks
an MIP formulation for S = s=1 i=1
P s,i ∩ (Qn1 × Zn2 ) is given by
(6.4a)

As x ≤

p 
kt


M s,t,i yt,i

∀s ∈ {1, . . . , p},

t=1 i=1

(6.4b)

p 
ks


ys,i = 1,

s=1 i=1

(6.4c)

ys,i ≥ 0

∀s ∈ {1, . . . , p}, i ∈ {1, . . . , ks } ,

(6.4d)
(6.4e)

ys,i ∈ Z
x ∈ Qn1 × Zn2 .

∀s ∈ {1, . . . , p}, i ∈ {1, . . . , ks } ,

In particular, we may take M s,s,i = bs,i for all s ∈ {1, . . . , p}, i ∈ {1, . . . , ks }.
Validity of this formulation is again straightforward from finiteness of the Mls,t,i ,
which can be proven analogously to Proposition 6.1. Furthermore, the strongest
possible version of formulation (6.4) is again obtained when equality holds in (6.3).
In particular, M s,s,i = bs,i is the strongest possible coefficient unless some P s,i has
a redundant constraint. Of course, even in the case of a redundant constraint, it is
always valid and convenient to use Big-M constants such that M s,s,i ≤ bs,i . For this
reason, we assume this to be the case from now on.

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

29

Hybrid Big-M formulation (6.4) reduces to the formulation introduced by Balas,
Blair, and Jeroslow when all left-hand side matrices are identical (i.e., for p = 1), and
we take M s,s,i = bs,i for all i. An advantage of this formulation is that it can be
constructed without solving or bounding any LP problem in (6.3). In what follows
we refer to this formulation as the simple version of (6.4). An example of what can
be modeled with this simple version of the hybrid Big-M formulation is the following
class of problem that includes the SOS1 and SOS2 constraints introduced by Beale
and Tomlin [17].
 k
 k
Example 10. For given li i=1 , ui i=1 ⊆ {0, 1}n, consider the family of poly

topes P i := x ∈ Qn : lji ≤ xj ≤ uij ∀j ∈ {1, . . . , n} for i ∈ {1, . . . , k} and its union

S = ki=1 P i . For instance, if we let lji = 0 for all i, j and k = n and
)
1, j = i,
i
uj =
0 otherwise
or k = n − 1 and

)
uij =

1,
0

j ∈ {i, i + 1},
otherwise,

we have that S corresponds, respectively, to the SOS1 and SOS2 constraints introduced
by Beale and Tomlin [17]. For the cases of SOS1 and SOS2 constraints, formulation (6.4) with p = 1 reduces, respectively, to
0 ≤ xi ≤ yi
k


∀i ∈ {1, . . . , k},

yi = 1,

i=1

y ∈ {0, 1}k ,
and
0 ≤ x1 ≤ y1 ,
0 ≤ xi ≤ yi−1 + yi

∀i ∈ {2, . . . , k},

0 ≤ xk+1 ≤ yk ,
k


yi = 1,

i=1

y ∈ {0, 1}k ,
which are the standard MIP formulations for such constraints.
We end this section by showing how the simple version of hybrid big-M formulation (6.4) can be used to obtain a smaller version of V-formulation (5.3). This
results in the extension of a known formulation for piecewise linear functions (e.g.,
[45, 91, 113, 106, 165]).
 k
Corollary 6.3. Let P i i=1 ⊆ Qn be a finite family of polyhedra with a common
k
recession cone C and V := i=1 ext P i . Then, for any n1 , n2 ∈ Z+ such that
k
n = n1 + n2 , two MIP formulations of S = i=1 P i ∩ (Qn1 × Zn2 ) are given by


(6.5a)
vλv +
rμr = x,
v∈V

r∈ray(C)

30

JUAN PABLO VIELMA



(6.5b)

λv = 1,

v∈V



λv ≤

(6.5c)

∀v ∈ V,

yi

i:v∈ext(P i )
k


(6.5d)

yi = 1,

i=1

λv ≥ 0
μr ≥ 0

(6.5e)
(6.5f)

∀v ∈ V,
∀r ∈ ray(C),

(6.5g)

y ∈ {0, 1}k ,

(6.5h)

x∈ Qn1 × Zn2 ,

and
(6.6a)


v∈V

(6.6b)

vλv +



rμr = x,

r∈ray(C)



λv = 1,

v∈V

(6.6c)



λv ≥ yi

∀i ∈ {1, . . . , k},

v∈ext(P i )

(6.6d)

k


yi = 1,

i=1

(6.6e)
(6.6f)
(6.6g)
(6.6h)

λv ≥ 0
μr ≥ 0

∀v ∈ V,
∀r ∈ ray(C),

y ∈ {0, 1}k ,
x∈ Qn1 × Zn2 .

Furthermore, if P i = conv P i ∩ (Qn1 × Zn2 ) for all i ∈ {1, . . . , k}, then both formulations are sharp.
Proof. Let
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
i
Q = (x, λ, μ) ∈ Qn × QV × Qray(C) :
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩


v∈V

vλv +



⎫
rμr = x⎪
⎪
⎪
⎪
⎪
r∈ray(C)
⎪
⎪

⎪
⎪
⎪
λv = 1⎪
⎪
⎪
⎬
v∈V

λv ≤ 1 ∀v ∈ ext P i

⎪
⎪
⎪
i ⎪
⎪
λv ≤ 0 ∀v ∈ V \ ext P ⎪
⎪
⎪
⎪
⎪
⎪
λv ≥ 0 ∀v ∈ V
⎪
⎪
⎭
μr ≥ 0 ∀r ∈ ray(C)

.

k
Validity of (6.5) follows by noting that i=1 P i is the projection onto the x variables
k
k
of i=1 Qi and that (6.5) is the simple version of (6.4) for i=1 Qi . Validity of (6.6)

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

31

follows by noting that Qi can be described alternatively as
⎫
⎧


⎪
⎪
vλ
+
rμ
=
x
v
r
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
v∈V
r∈ray(C)
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
λ
=
1
v
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
v∈V
⎪
⎪
⎪
⎪

⎪
⎪
⎬
⎨
λ
≥
1
v
i
n
V
ray(C)
.
Q = (x, λ, μ) ∈ Q × Q × Q
:
v∈ext(P i )
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎪
⎪
λv ≥ 0 ∀j = i
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
j
v∈ext(P )
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
λv ≥ 0 ∀v ∈ V
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
μr ≥ 0 ∀r ∈ ray(C)
For sharpness, note that the projection onto the x variables of the LP relaxation
of both formulations is contained in conv (V ) + C. The result then follows because,
under the assumption on P i and by Theorem 3.6, we have
 k


i
n1
n2
conv(S) = conv
P ∩ (Q × Z )

= conv

= conv

= conv

k



conv P ∩ (Q
i

i=1
k


n1

×Z )
n2


P

i

i=1
k




i=1



i=1

= conv

= conv

i=1

k


k



conv ext P

i

+C


conv ext P i

+C


ext P

i

+ C = conv(V ) + C.

i=1

While (6.5) and (6.6) are equivalent, their LP relaxations are not contained in one
another. Of course this can only happen because neither formulation is locally ideal.
In fact, Lee and Wilson [106, 165] show that constraints (6.6c) are valid inequalities for
the convex hull of integer feasible points of (6.5) and hence can be used to strengthen
it. These inequalities are sometimes facet defining and are part of a larger class
of inequalities that are enough to describe the convex hull of integer feasible points
of (6.5). Unfortunately, the number of such inequalities can be exponential in k.
However, in section 7 we show how an LP-based separation of these inequalities allows
us to construct a different formulation that ends up being equivalent to (5.3). If
we specialize formulation (6.5) to piecewise linear functions, we obtain the following
formulation from [106, 165].
Example 11. Let f : D ⊆ Qd → Q be a piecewise linear function defined by (3.3)
for a given finite family of polytopes {Qi }ki=1 . Formulation (6.5) for characterization

32

JUAN PABLO VIELMA

(3.4b) of gr(f ) results in the MIP formulation of gr(f ) given by

(6.7a)
vλv = x,
v∈

k

i=1



(6.7b)
v∈

k

ext(Qi )

f (v)λv = z,

i
i=1 ext(Q )



λv ≤

(6.7c)

yi

∀v ∈

i:v∈ext(Qi )

(6.7d)

k


k


ext Qi ,

i=1

yi = 1,

i=1

(6.7e)
(6.7f)

λv ≥ 0

∀v ∈ ext Qi ,

y ∈ {0, 1}k ,

which is a standard MIP formulation for piecewise linear functions that is denoted the
convex combination model in [155].
6.3. Formulation Strength. The traditional Big-M formulation has been significantly more popular than the simple version of the hybrid Big-M formulation. One
reason for this is the somewhat limited applicability of the latter formulation. The
general version of the hybrid Big-M formulation resolves this issue as it can be used in
the same class of problems as the traditional Big-M formulation. In addition, the hybrid Big-M formulation provides an advantage over the traditional one with respect to
size, even if only partial common left-hand side structure is present in the polyhedra.
Indeed, both formulations have the same number of variables, but the hybrid formup
p
lation has 1 + s=1 ms constraints while the traditional one has 1 + s=1 ms × ks
constraints. Of course, such an advantage in size is meaningless if it is accompanied
by a significant reduction in strength. For this reason we now study the relative and
absolute strengths of these formulations. In particular, concerning the hybrid BigM formulation we show that it is always at least as strong as the traditional Big-M
formulation, that its simple version can be sharp and strictly stronger than the traditional formulation, but that even its strongest version can fail to be locally ideal
or sharp. We begin with the following proposition that shows that one case where
the hybrid and traditional formulations coincide is when S is the union of only two
polyhedra.
Proposition 6.4. If S is the union of two polyhedra whose description does not
include any redundant constraints, and equality is taken in (6.1) and (6.3), then the
LP relaxations of formulations (6.2) and (6.4) are equivalent.
Proof. For (6.2) the case of two polyhedra corresponds to k = 2, and in this case
(6.2a) is equal to
A1 x ≤ b1 + M 1,2 − b1 (1 − y1 ) ,
A2 x ≤ b2 + M 2,1 − b2 (1 − y2 ) .
For (6.3) the case of two polyhedra corresponds to p = 1 and k1 = 2 or p = 2, k1 = 1,
and k2 = 1. In the former case (6.4) is equal to
A1 x ≤ M 1,1,1 y1 + M 1,1,2 y2

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

33

and the equivalence follows from y1 + y2 = 1 by noting that for this case A1 =
A2 , M 1,2 = M 1,1,2 = b2 , and M 2,1 = M 1,1,1 = b1 , because of the nonredundancy
assumption. In the latter, (6.4) is equal to
A1 x ≤ M 1,1,1 y1 + M 1,2,1 y2 ,
A2 x ≤ M 2,1,1 y1 + M 2,2,1 y2 ,
and the equivalence follows from y1 + y2 = 1 by noting that for this case M 1,1,1 = b1 ,
M 2,2,1 = b2 , M 1,2,1 = M 1,2 , and M 2,1,1 = M 2,1 , because of the nonredundancy
assumption.
Unfortunately, as the following example shows, this equal strength can fail to
yield sharp formulations even for the case of equal left-hand side matrices and no
redundant constraints.
Example 12. For
⎛ ⎞
⎛ ⎞
⎡
⎤
1
2
1
0
1
⎜1⎟
⎜2⎟
⎢ −1
⎥
0
1
⎜ ⎟
⎜ ⎟
⎢
⎥
1
1
⎜ ⎟
⎜ ⎟
1
1 ⎥
A=⎢
⎢ 0
⎥ , b = ⎜2⎟ , b = ⎜1⎟ ,
⎝
⎝1⎠
⎣ 0 −1
⎠
⎦
2
1
0
0
0
0 −1
let P 1 = {x ∈ Q3 : Ax ≤ b1 } and P 2 = {x ∈ Q3 : Ax ≤ b2 }. The strongest versions
of formulations (6.2) and (6.4) for S = P 1 ∪ P 2 are equal to
⎛
⎞
1
⎜ 1 ⎟
⎜
⎟
1
⎟
(6.8a)
Ax ≤ b + (1 − y1 ) ⎜
⎜ −1 ⎟ ,
⎝ −1 ⎠
0
⎛
⎞
−1
⎜ −1 ⎟
⎜
⎟
⎟
Ax ≤ b2 + (1 − y2 ) ⎜
(6.8b)
⎜ 1 ⎟,
⎝ 1 ⎠
0
(6.8c)
(6.8d)

y1 + y2 = 1,
y ∈ {0, 1}2.

Formulation (6.8) is not sharp because x1 = x2 = 0, x3 = 3/2, y1 = y2 = 1/2 is
feasible for its LP relaxation and x3 ≤ 1 for any x ∈ conv(S).
The lack of sharpness of formulation (6.8) in Example 12 can be corrected by
simply adding x3 ≤ 1, as conv(S) is exactly the projection of the LP relaxation of
(6.8) onto the x variables plus this inequality. However, this correction cannot always
be done with a polynomial number of inequalities (e.g., see Lemma 4.1). Furthermore,
as the following theorem by Blair [26] shows, recognizing sharpness of even the simple
version of hybrid Big-M formulation (6.4) is hard.
Theorem 6.5. Let p = 1 and M 1,1,i = b1,i for all i ∈ {1, . . . , k} in formulation
(6.4). Deciding whether the resulting formulation is sharp is NP-hard.
Fortunately, Balas, Blair, and Jeroslow gave some practical (but not exhaustive)
necessary and sufficient conditions for sharpness of the simple version of (6.4) [12,
26, 87]. An extremely useful example of such conditions is given by the following
proposition.

34

JUAN PABLO VIELMA

Definition 6.6. For b ∈ Qm , A ∈ Qm×n , and I ⊆ {1, . . . , m} such that |I| = n
and det(AI ) = 0, we let AI (bI ) be the submatrix (subvector) of A (b) obtained by
selecting only the rows (components) indexed by I. We also let x(I, b) ∈ Qn be the
unique solution to AI x = bI . That is, x(I, b) is the basic solution associated to basis
I for polyhedron {x ∈ Qn : Ax ≤ b}.
 k
Let P i i=1 be a finite family of H-polyhedra such that for each i we have P i :=


 k
x ∈ Qn : Ax ≤ bi . We say that P i i=1 have the same shape if for all I ⊆
{1, . . . , m} such that |I| = n and det(AI ) = 0 we have either x I, bi ∈ P i for all i
/ P i for all i. In other words, polyhedra with a common structure have
or x I, bi ∈
the same shape if and only if for every basis the associated basic solution is feasible
for all polyhedra or is infeasible for all polyhedra.
k

Proposition 6.7. If p = 1, polyhedra P 1,i i=1 have the same shape, and
M 1,1,i = b1,i for all i ∈ {1, . . . , k}, then (6.4) is sharp.
One case in which these shape conditions hold are the sets in Example 10, as
the polyhedra considered are all rectangles (possibly degenerate ones). In particular,
this partially explains the success of the standard formulations for SOS1 and SOS2
constraints and gives an alternative proof of the sharpness of formulation (4.3) in
Example 3.
Unfortunately, as the following example shows, the shape condition does not
necessarily imply that formulation (6.4) is locally ideal.
Example 13. Let
⎤
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎡
1
1
1
1
1
1
1
⎜−1⎟
⎜−1⎟
⎜−1⎟
⎢−1 −1 −1 −1⎥
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎜1⎟
⎜0⎟
⎜0⎟
⎢1
0
0
0⎥
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎥
⎜1⎟
⎜1⎟
⎜0⎟
⎢0
1
0
0
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎟
⎟
⎥
⎜
⎜
⎜1⎟
⎢0
0
1
0
1
0
1
2
3
⎟
⎟
⎥
⎜
⎜
⎜ ⎟
⎢
A=⎢
⎥, b = ⎜ 0 ⎟, b = ⎜ 0 ⎟, b = ⎜ 1 ⎟,
0
0
0
1
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎜0⎟
⎜0⎟
⎜0⎟
⎢−1 0
0
0⎥
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎥
⎜0⎟
⎜0⎟
⎜0⎟
⎢ 0 −1 0
0
⎥
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎢
⎝0⎠
⎝0⎠
⎝0⎠
⎣0
0 −1 0 ⎦
0
0
0
0
0
0 −1
and P i = {x ∈ Q4 : Ax ≤ bi } for i ∈ {1, . . . , 3}. These polyhedra satisfy the shape
condition of Proposition 6.7 and hence the associated version of formulation (6.4) is
sharp. However, we can check that x3 = x4 = 1/2, x1 = x2 = 0, y1 = y3 = 1/2,
and y2 = 0 is a fractional extreme point of the LP relaxation of (6.4) and hence the
formulation is not locally ideal. Finally, note that this formulation is precisely parts
(2.5c)–(2.5k) of the standard formulation (2.5) for piecewise linear functions, we saw
in section 2.1.
Nevertheless, Proposition 6.7 can still be used to construct the following example
that shows how simple hybrid Big-M formulation (6.4) can be strictly stronger than
big-M formulation (6.2).
 k
Example 14. Let k = n − 1 and ui i=1 ⊆ {0, 1}n be such that
)
1, j ∈ {i, i + 1},
i
uj =
0 otherwise,

and let P i := x ∈ Qn : 0 ≤ xj ≤ uij


∀j ∈ {1, . . . , n} for i ∈ {1, . . . , k}. Formula-

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

tion (6.4) for S :=

n
i=1

35

P i reduces to
x1 ≤ y1 ,

(6.9a)

xi ≤ yi−1 + yi
xk+1 ≤ yk ,

(6.9b)
(6.9c)

k


(6.9d)

∀i ∈ {2, . . . , k},

yi = 1,

i=1

y ∈ {0, 1}k ,

(6.9e)

and the strongest version of Big-M formulation (6.2) for S reduces to
x1 ≤ y1 ,
xj ≤ 1 − yi

(6.10a)
(6.10b)

∀j ∈ {1, . . . , k + 1} , i ∈ {1, . . . , k} \ {j − 1, j},

xk+1 ≤ yk ,

(6.10c)

k


(6.10d)

yi = 1,

i=1

y ∈ {0, 1}k .

(6.10e)

Lemma 6.8. Hybrid Big-M formulation (6.9) is always sharp, while traditional
Big-M formulation (6.10) is not sharp for k ≥ 4.
Proof. The first formulation is sharp by Proposition 6.7.
For the lack of sharpness of the second formulation, note that x1 = xk+1 = 1/k,
xj = (k − 1)/k for j ∈ {2, . . . , k}, and yi = 1/k for i ∈ {1, . . . , k} is feasible for its LP
relaxation.
For k ≥ 4 this solution is not in conv(S) and hence this Big-M formulation is not
sharp (e.g., note that this x does not belong to the projection of the LP relaxation of
(6.9) onto the x variables).
Finally, the following proposition shows that the hybrid formulation is always at
least as strong as the traditional formulation if equivalent Big-M constants are used.
Proposition 6.9. If the Big-M constants in (6.1) and (6.3) are identical and
such that M s,s,i ≤ bs,i , then hybrid Big-M formulation (6.4) is at least as strong as
traditional Big-M formulation (6.2).
Proof. Using the notation of formulation (6.4) (i.e., accounting for the possible
common left-hand side matrices), (6.2a) from the traditional Big-M formulation is
equal to

 s,i
As x ≤ bs,i + M − bs,i (1 − ys,i ) ∀s ∈ {1, . . . , p} , i ∈ {1, . . . , ks },
s,i

where M l = maxt∈{1,...,p}, j∈{1,...,kt }, (t,j)=(s,i) Mls,t,j . Using the fact that in this case
p
ks
(6.2b) is equal to s=1 i=1
ys,i = 1, we can rewrite this equation as

s,i
M yt,i ∀s ∈ {1, . . . , p} , i ∈ {1, . . . , ks }.
As x ≤ bs,i ys,i +
t∈{1,...,p},
j∈{1,...,kt }
(t,j)=(s,i)

The result then follows from assumption M s,s,i ≤ bs,i and because, by definition,
s,i
M ≥ M s,t,i .

36

JUAN PABLO VIELMA

Proposition 6.9 and the formulation sizes suggest that, at least from a theoretical
perspective, hybrid Big-M formulation (6.4) is always preferable to traditional BigM formulation (6.2). This theoretical advantage often results in a computational
advantage [156]. However, the complexities of state-of-the-art solvers could still allow
the traditional formulation to outperform the hybrid one, particularly when the size
and strength differences are small.
7. Large Formulations. The aim of the techniques reviewed in this survey is
to construct strong, but small MIP formulations. However, there are many cases in
which all known MIP formulations (strong or weak) are large. If these formulations
are such that the number of variables is small and the number of constraints is large,
but can be separated fast, it is sometimes possible to use them in a branch-andcut procedure [131, 39, 128]. Similarly, when only the number of variables is large,
the formulations can be used in column generation or branch-and-price procedures
[15]. These procedures can be very effective and so can extensions such as branchand-cut-and-price [48] and branch-and-price for extended formulations [142]. For this
reason, when both large and small (usually extended) formulations are available it is
not always clear which is more convenient. Sometimes it is advantageous to solve the
large formulation with one of these specialized procedures [40, 50, 32], and other times
it is more convenient to solve the small extended formulation directly [31]. Exploring
these alternatives is beyond the scope of this survey, so in this section we restrict
our attention to one class of large formulations from which alternative small extended
formulations can be easily constructed. Such a construction was introduced by Martin
[115] (see also [30]) and can be summarized in the following proposition.
Proposition 7.1. Let Q := {x ∈ Qn : Ax ≤ b, Dx ≤ d} and suppose there is a
separation LP problem for R := {x ∈ Qn : Dx ≤ d}. That is, there exist F ∈ Qr×n ,
H ∈ Qr×m , c ∈ Qm , and g ∈ Qr for which the LP problem given by
(7.1a)

z(x) :=max

n


xi πi +

i=1

m


c j ρj ,

j=1

(7.1b)

F π + Hρ ≤ g,

(7.1c)

ρj ≥ 0

∀j ∈ {1, . . . , m},

is such that x ∈ R if and only if z(x) ≤ 0. Then an extended LP formulation of Q is
given by
(7.2a)

Ax ≤ b,

(7.2b)

F T w = x,

(7.2c)

H T w ≥ c,

(7.2d)

g T w ≤ 0,

(7.2e)

wj ≥ 0

∀j ∈ {1, . . . , r}.

If Q is the LP relaxation of an MIP formulation such that R has a large number
of constraints but a small separation LP problem, we can replace Dx ≤ d with (7.2b)–
(7.2e) to obtain a smaller extended formulation. Proposition 7.1 was used by Lee and
Wilson in [106, 165] to obtain an extended formulation that strengthens (6.5) and
(6.6). To describe this approach we need the following proposition from [106, 165].
For simplicity, we concentrate on formulation (6.5) for n2 = 0, which we restate in
the proposition for convenience.

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

37

 k
Proposition 7.2. Let P i i=1 ⊆ Qn be a finite family of polyhedra with a common recession cone C, ext P i be the finite sets of extreme points of polyhedron P i ,
k
ray(C) be the finite set of extreme rays of polyhedral cone C, and V := i=1 ext P i .

Then S = ki=1 P i can be modeled with formulation (6.5) given by


vλv +
rμr = x,
v∈V

r∈ray(C)



λv = 1,

v∈V



λv ≤

yi

∀v ∈ V,

i:v∈ext(P i )
k


yi = 1,

i=1

λv ≥ 0
μr ≥ 0

∀v ∈ V,
∀r ∈ ray(C),

y ∈ {0, 1}k .
While (6.5) is sharp, it is not always locally ideal. However, all facet-defining inequalities of the convex hull of solutions to (6.5) are of the form λv ≥ 0, μr ≥ 0,
or


(7.3)
yi ≤
λv
i∈I

v∈U

for some I ⊆ {1, . . . , k} and U ⊆ V . Furthermore, for a given (y, λ) a separation LP
problem for (7.3) is given by
(7.4a)

max

k


y i αi +

i=1



λ v βv ,

v∈V

(7.4b)

αi − βv ≤ 0

(7.4c)
(7.4d)

αi ≥ 0
βv ≥ 0

∀i ∈ {1, . . . , k}, v ∈ P i ,
∀i ∈ {1, . . . , k},
∀v ∈ V.

The following example shows how Propositions 7.1 and 7.2 can be used to strengthen
(6.5) to a locally ideal formulation.
Example 15. Lee and Wilson [106, 165] gave a precise characterization of sets
I and U for which (7.3) are facet defining. By adding these inequalities to (6.5) or
(6.6) we would immediately obtain a locally ideal formulation. Unfortunately, Lee
and Wilson also gave examples in which the number of facets defined by inequalities
(7.3) can grow exponentially in |V |. However, using Proposition 7.1 and the fact that
constraints (7.3) can be separated with LP problem (7.4), Lee and Wilson adapted
(6.5) to the locally ideal formulation given by


(7.5a)
vλv +
rμr = x,
v∈V

(7.5b)

r∈ray(C)



v∈V

λv = 1,

38
(7.5c)

JUAN PABLO VIELMA



wvi ≥ yi

∀i ∈ {1, . . . , k},

wvi ≤ λv

∀v ∈ V,

v∈P i

(7.5d)



i:v∈P i

(7.5e)

k


yi = 1,

i=1

(7.5f)

λv ≥ 0

∀v ∈ V,

(7.5g)

μr ≥ 0

∀r ∈ ray(C),

(7.5h)

wvi

(7.5i)

≥0

∀i ∈ {1, . . . , k}, ∀v ∈ V,

y ∈ {0, 1}k .

It is interesting to note that by eliminating variables λv and renaming wvi to λiv we
obtain formulation (5.3). In other words, as expected, if we try to strengthen (6.5) or
(6.6) to recover what we lost by eliminating the copies of λ variables of (5.3) through
Corollary 6.3, we recover (5.3).
Similar formulations for the case in which the separation problem can be solved
using dynamic programming and other extensions can be found in [96, 116]. However, note that having a generic polynomial time algorithm for the separation is not
sufficient to obtain a polynomial-sized extended formulation. For instance, if S corresponds to the incidence vectors of perfect matchings of a complete graph as defined
in (4.8), then all inequalities of conv(S) can be separated in polynomial time [146].
However, Theorem 4.3 shows that S does not have a polynomial-sized sharp extended
formulation. Finally, we note that another way to obtain small formulations from
large ones in a systematic way is to define approximate versions that are slightly
weaker, but significantly smaller [161].
8. Incremental Formulations. All formulations for disjunctive constraints we
k
have considered so far include binary variables y such that i=1 yi = 1. The problem
with such a configuration is that it can be quite detrimental to branch-and-bound
algorithms. To see this, imagine that the optimal solution to the LP relaxation at a
/Z
node of the branch-and-bound tree (e.g., the root LP relaxation) is such that y i0 ∈
for some i0 ∈ {1, . . . , k}. As described in section 2.2, a branch-and-bound-based MIP
solver will branch on yi0 by creating two new LP problems by adding yi0 ≤ yi0  to the
LP relaxation in the first branch and yi0 ≥ y i0  in the second branch. Because the LP
relaxation includes constraints 0 ≤ yi0 ≤ 1, we have that this branching is equivalent
to fixing yi0 = 0 in the first branch and yi0 = 1 in the second one. These branches
are usually denoted down-branching and up-branching, respectively, and have very
different effects on the branch-and-bound tree. The difference is that up-branching
(i.e., fixing yi0 = 1) automatically forces all other yi variables to zero, while downbranching (i.e., fixing yi0 = 0) does not imply any restriction on other yi variables.
This asymmetry results in what are usually denoted unbalanced branch-and-bound
trees, which can result in a significant slow-down of the algorithm [141].
One way to resolve this issue is to use specialized constraint branching schemes
[141]. In our particular case, an appropriate scheme is the SOS1 branching of Beale
k
and Tomlin [17] that can be described as follows. Because i=1 yi = 1 and y ≥ 0,
we have that for a fractional y there must be i0 < i1 ∈ {1, . . . , k} such that y i0 , y i1 ∈
(0, 1). We can then exclude this fractional solution by creating two new LP problems
by adding y1 = y2 = · · · = yi0 = 0 in the first branch and yi0 +1 = yi0 +2 = · · · = yk = 0

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

39

in the second one. In contrast to the traditional up-down variable branching, this
approach usually fixes to zero around half the integer variables in each branch and
hence yields much more balanced branch-and-bound trees.
An alternative to constraint branching is to use a well-known MIP formulation
trick that uses a redefinition of variables y (e.g., [27, 111, 132, 149, 33, 129, 110]).
This transformation can be formalized
straightforward
proposition.
 in the following

k
k
y
=
1
and
L
:
Q
→ Qk be the
Proposition 8.1. Let Δk := y ∈ Qk+ :
i
i=1
linear function defined as
L(y)i :=

k


yi .

j=i

Then L(Δk ) = Γk := {w ∈ Qk : 1 = w1 ≥ w2 ≥ · · · ≥ wk ≥ 0} and the inverse of L
is
)
wi − wi+1 , i < k,
−1
L (w)i :=
wk ,
otherwise.
Hence L is a one-to-one correspondence between Δk and Γk , and, in particular, L is
a one-to-one correspondence between Δk ∩ {0, 1}k and Γk ∩ {0, 1}k .
Then, in any formulation that includes the constraint y ∈ Δk ∩ {0, 1}k we can
replace such a constraint by w ∈ Γk ∩ {0, 1}k and every occurrence of yi by L−1 (w)i .
This is illustrated in the following example from [168], which shows that variable
branching on the w variables can be much more effective than variable branching on
the y variables.
Example 16. Let k be even, a ∈ Qk \ {0} be such that
a1 < · · · < a k = −1 < 0 < a k +1 = 1 < · · · < ak ,
2

2

and for S := {a1 , a2 , . . . , ak } consider the problem minx {|x| : x ∈ S}, which has an
optimal value of 1. This problem can be solved through the standard MIP formulation
given by
(8.1a)
(8.1b)

min

z,
x ≤ z,

(8.1c)

−x ≤ z,

(8.1d)

x=

k


ai y i ,

i=1

1=

(8.1e)

k


yi ,

i=1

y ∈ {0, 1}k .

(8.1f)

However, using Proposition 8.1 we can construct the alternative MIP formulation of
S given by
(8.2a)
(8.2b)

min

z,
x ≤ z,

40

JUAN PABLO VIELMA

−x ≤ z,

(8.2c)

x = a1 w1 +

(8.2d)

k


(ai − ai−1 )wi ,

i=2

wi ≥ wi+1
w1 = 1,

(8.2e)
(8.2f)

∀i ∈ {1, . . . , k − 1},

w ∈ {0, 1}k .

(8.2g)

We can show that a pure branch-and-bound algorithm requires branching in at least
k/2 of the variables of (8.1) to solve it (see [168] for more details). However, a
pure branch-and-bound algorithm can solve (8.2) branching in a single variable as
follows. The
has z = x = 0, wi = 1
 optimal solution to the LP relaxation of (8.2) 
for all i ∈ 1, . . . , k2 , w k +1 = 1/2, and wi = 0 for all i ∈ k2 + 2, . . . , k . Adding
2
w k +1 = 0 to this LP relaxation results in an optimal solution with z = 1 and x = −1,
2
while adding w k +1 = 1 results in an optimal solution with z = 1 and x = 1. Hence
2
branching on w k +1 is enough to solve the problem. Finally, it is interesting to note
2
that variable branching on w has essentially the same effect as SOS1 branching on y.
For instance, branching on w k +1 = 1 and w k +1 = 0 has the same effect as branching
2
2
on y1 = y2 = · · · = y k = 0 and y k +1 = y k +2 = · · · = yk = 0.
2
2
2
The reason for the effectiveness of branching on w comes from the fact that both
down- and up-branching on wi fix the value of many other w variables (around half
the variables depending on i). This behavior is usually denoted double contracting
[66, 67, 150] to contrast it with the behavior of branching on y (only up-branching fixes
other variables), which is denoted single contracting. The fact that double contracting
incremental formulations result in solves with fewer branch-and-bound nodes was
confirmed computationally in [155, 157].
The transformation from Proposition 8.1 can be used on any formulation that
includes y ∈ {0, 1}k . However, in some cases, an additional transformation of the
continuous variables can result in an ad-hoc incremental formulation. An example of
this is the following formulation introduced in [168] that generalizes a formulation for
piecewise linear functions introduced in [165].
 k
Proposition 8.2. Let P i i=1 ⊆ Qn be a finite family of polyhedra with a
  ri
common recession cone C and let vji j=1 := ext P i and ray(C) be the finite sets of
i
extreme points and rays of polyhedron P
k and ipolyhedral cone C, respectively. Then
a locally ideal MIP formulation of S = i=1 P is given by

v11 +

k


v1i − vri−1
wi
i

i=2

+

ri
k 

i=1 j=2

(8.3a)

+

δji vji − v1i


rμr = x,

r∈ray(C)

(8.3b)

r1

j=2

δj1 ≤ 1,

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

(8.3c)

ri


δji ≤ wi

41

∀i ∈ {2, . . . , k},

j=2

(8.3d)
(8.3e)

δri i ≥ wi+1
δji

∀i ∈ {1, . . . , k − 1},

≥0

∀i ∈ {1, . . . , k}, j ∈ {2, . . . , ri },

(8.3f)

μr ≥ 0

∀r ∈ ray(C),

(8.3g)

wi ∈ {0, 1}

∀i ∈ {2, . . . , k}.

When S = gr(f ) for a piecewise linear function f defined by (3.3) for polytopes
 i k
Q i=1 that satisfy a special ordering condition, formulation (8.3) reduces to a standard MIP formulation for piecewise linear functions that is denoted the incremental
model in [155]. For more details, examples of incremental formulations, and their
advantages, we refer the reader to [168, 20].
9. Logarithmic Formulations. Standard MIP formulations for the union of k
polyhedra use k binary variables, while incremental formulations essentially reduce
this to k − 1 variables (one of the k variables is fixed to 1). In this section we review
techniques that allow us to reduce the number of binary variables to log2 (k).
The simplest
technique for using a logarithmic number of binary variables conk
siders S = i=1 P i ⊆ Q where P i = {i − 1}. In this case, basic V-formulation (5.3)
reduces to
(9.1a)

x=

k


(i − 1)yi ,

i=1

(9.1b)

1=

k


yi ,

i=1

(9.1c)

y ∈ {0, 1}k .

We can think of formulation (9.1) as a unary encoding of x ∈ {0, 1, . . . , k − 1}. If we
instead use a binary encoding, we can obtain the formulation given by
log2 (k)



2i−1 wi ,

(9.2a)

x=

(9.2b)

x ≤ k − 1,

(9.2c)

w ∈ {0, 1}log2 (k) .

i=1

This technique appeared in the mathematical programming literature as early as [162]
and, while it is not an effective way to deal with general integer variables in MIP
[134], it has been used as the basis for effective MIP formulations of mixed integer
nonlinear programming problems (e.g., [162, 58, 72]). A similar technique is used in
[105, 103, 41] to model the constraint programming all-different requirement [75] in
problems such as graph coloring. For more general problems, MIP formulations with
a logarithmic number of binary variables were originally considered in [81, 148] and
have received significant attention recently [155, 127, 156, 107, 74, 6, 125, 159, 160].
One way to construct MIP formulations with a logarithmic number of variables
is to use the following proposition.
 k
Proposition 9.1. Let hi i=1 ⊆ {0, 1}log2 (k) be such that hi = hj for any


k
i = j. Then a locally ideal formulation of S := y ∈ {0, 1}k :
i=1 yi = 1 is given

42

JUAN PABLO VIELMA

by
(9.3a)

k


yi = 1,

i=1

(9.3b)

k


hi yi = w,

i=1

w ∈ {0, 1}log2 (k) ,
yi ≥ 0

(9.3c)
(9.3d)

∀i ∈ {1, . . . , k}.

Recently, formulations that are essentially identical to this one were independently
proposed in [6, 74, 107, 159, 160]. However, the basic idea behind formulation (9.3)
has in fact been part of the mathematical programming folklore for a long time. For
 k
instance, Glover [58] attributes (9.3) for a specific choice of hi i=1 to Sommer [148].
The following proposition from [74, 6] shows how to use (9.3) to construct formulations for simple disjunctive constraints.
 k
Proposition 9.2. Let hi i=1 ⊆ {0, 1}log2 (k) be such that hi = hj for any
i = j. Then a locally ideal formulation for S = {a1 , a2 , . . . , ak } is given by
(9.4a)

k


yi = 1,

i=1

(9.4b)

k


hi yi = w,

i=1

(9.4c)

k


ai yi = x,

i=1

(9.4d)

w ∈ {0, 1}log2 (k) ,

(9.4e)

yi ≥ 0

∀i ∈ {1, . . . , k}.

If log2 (k) ∈ Z and there exists (c0 , c) ∈ Q × Qlog2 (k) such that
log2 (k)

(9.5)

ai = c0 +



hij cj

∀i ∈ {1, . . . , k} ,

j=1

then formulation (9.4) can be reduced to the locally ideal formulation given by
log2 (k)

(9.6a)

c0 +



cj wj = x,

j=1

(9.6b)

w ∈ {0, 1}log2 (k) .

An advantage of (9.6) over (9.4) is the elimination of y variables. However, unlike
(9.4), (9.6) requires log2 (k) ∈ Z. For the case log2 (k) ∈
/ Z, formulation (9.6) remains
 k
valid only if we add the constraint w ∈ hi i=1 . If hi corresponds to the binary
log (k)

expansion of i − 1, this constraint can be enforced through i=1 2
2i−1 wi ≤ k − 1.
The resulting formulation is a generalization of (9.2) and, unfortunately, it can fail

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

43

 k
to be locally ideal. Stronger methods to implement w ∈ hi i=1 can be found in
[9, 41, 130].
One way to use Proposition 9.1 to construct MIP formulations for more general
disjunctive constraints is to combine it with formulation (5.3). The resulting formulation was introduced in [155] for piecewise linear functions and its extension to general
polyhedra was introduced in [168].
 k
Proposition 9.3. Let P i i=1 ⊆ Qn be a finite family of polyhedra with a
common recession cone C and let ext P i and ray(C) be the finite sets of extreme
points and rays of polyhedron P i and polyhedral cone C, respectively. Furthermore,
 k
let hi i=1 ⊆ {0, 1}log2 (k) be such that hi = hj for any i = j. Then a locally ideal
k
MIP formulation of S = i=1 P i is given by
(9.7a)
k




vλiv +

i=1 v∈ext(P i )

(9.7b)

(9.7c)



rμr = x,

r∈ray(C)
k




i=1

v∈ext(P i )

k




λiv = 1,
hi λiv = w,

i=1 v∈ext(P i )

λiv ≥ 0
μr ≥ 0

(9.7d)
(9.7e)

∀i ∈ {1, . . . , k}, v ∈ ext P i ,
∀r ∈ ray(C),

w ∈ {0, 1}log2 (k) .

(9.7f)

The following proposition shows an entirely different technique that was introduced in [81].
 k
Proposition 9.4. Let P i i=1 be a finite family of polyhedra with a common


i
j
recession cone (i.e., P∞
= P∞
for all i, j), such that P i = x ∈ Qn : Ai x ≤ bi with
 k
Ai ∈ Qmi ×n and bi ∈ Qmi for all i ∈ {1, . . . , k}. Also, let hi i=1 ⊆ {0, 1}log2 (k) be
k
such that hi = hj for any i = j. An MIP formulation for S = i=1 P i is given by
⎛
⎞
log2 (k)



(9.8a)
hij −
wj +
wj ⎠ ∀i ∈ {1, . . . , k},
Ai x ≤ bi + M i ⎝
j=1

(9.8b)

j:hij =1

j:hij =0

w ∈ {0, 1}log2 (k) ,

for sufficiently large M i ∈ Qmi .
Formulation (9.8) does not require any auxiliary variables besides w. However,
because it is an adaptation of Big-M formulation (6.2), it is not necessarily locally
ideal or sharp.
Other logarithmic formulations for specific constraints were introduced in [127,
125, 159, 160]. Further comparisons between logarithmic and incremental formulations can be found in [168].
10. Combining Formulations and Propositional Logic. Another way to keep
the sizes of MIP formulations controlled is to construct them for different parts of

44

JUAN PABLO VIELMA

a mathematical programming problem independently and then combine them. This
combination of formulations, which was denoted model linkage by Jeroslow and Lowe
[91], can reduce the strength of the final formulation, but can also result in a significant
reduction in size.
The
-r simplest way to combine formulations is to intersect them. For instance, if
S = i=1 S i , we can obtain an MIP formulation of S by intersecting MIP formulations
for S i . However, alternative formulations can be obtained by considering the specific
structure of parts S i . One example of this is
(10.1)

S=

kj
r 
.

P i,j ,

j=1 i=1

where P i,j are given polytopes. In this case, (10.1) can also be written using the
single disjunctive constraint,
(10.2)



S=
s∈

r

j=1 {1,...,kj }

r
.

P sj ,j .

j=1

Formulating this single combined constraint directly results in stronger, but larger
MIP formulations than when combining formulations for each constraint. However,
using the rules of propositional calculus we can construct other variations between
extreme cases (10.1) and (10.2) [11, 86]. Systematically constructing such variations
can lead to formulations that effectively balance size and strength (e.g., [139, 143,
144]). Furthermore, in some cases formulations based on (10.1) do not incur any loss
of strength [159, 160].
The reverse transformation from (10.2) to (10.1) is not always evident and sometimes requires the addition of auxiliary variables. Examples of this include the formulations for joint probabilistic constraints introduced in [101, 114, 157]. In this case,
auxiliary binary variables that keep track of violated constraints are used to combine
formulations for single row or k-row probabilistic constraints into a formulation for
the full row joint probabilistic constraint.
For more information on propositional calculus, logic, and other techniques from
constraint programming with MIP formulations, we refer the reader to [78, 88, 164,
76, 79, 77].
11. MIP Representability. A systematic study of which sets can be modeled as
MIP problems began with the work of Meyer [118, 119, 120, 122, 121] and Ibaraki
[81], and the first precise characterization of these sets was given by Jeroslow and
Lowe [90, 113]. For general MIP formulations with unbounded integer variables this
characterization is quite technical, so Jeroslow and Lowe [90, 113] also introduced a
much simpler characterization for MIP formulations with binary or bounded integer
variables. This characterization was later extended by Hooker [76] to consider a
restricted use of unbounded integer variables that nevertheless allows for the modeling
of most sets that are used in practice, while preserving most of the simplicity of
Jeroslow and Lowe’s characterization. The first simple characterization to correspond
precisely to the sets modeled by basic formulations (4.6) was denoted bounded MIP
representability by Jeroslow and Lowe.
Definition 11.1. S ⊆ Qn is bounded MIP representable if and only if it has an
MIP formulation of the form (2.6), where y is additionally constrained to be a binary
vector.

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

45

Jeroslow and Lowe showed that bounded MIP representable sets are precisely the
union of a finite number of polyhedra with a common recession cone.
Proposition 11.2. A set S ⊆ Qn is bounded MIP representable if and only if
 k
i
j
= P∞
for all i, j and
there exists a finite family of polyhedra P i i=1 such that P∞
(11.1)

S=

k


P i,

i=1

or, equivalently, if and only if
(11.2)

S=C+

k


P i,

i=1

 k
i
= {0} for all i) and C is a
where P i i=1 is a finite family of polytopes (i.e., P∞
polyhedral cone.
Proof. The fact that a set of the form (11.1) is bounded MIP representable follows
from Proposition 4.2.
For the converse, let (2.6) be an MIP formulation of set S such that y is bounded.
Then the feasible region of this MIP formulation is a finite union of polyhedra (one
for each possible value of y) with the same recession cone. The result then follows by
projecting this union onto the x variables.
Of course, through unary or binary expansion, bounded MIP representability is
equivalent to asking for y to be bounded in (2.6). However, we can extend the class
of MIP representable sets by allowing unbounded integer variables. If we restrict
these unbounded integer variables to be part of the original variables x, we obtain
the second simple characterization introduced by Hooker. Because this representation
does not allow unbounded integer auxiliary variables, we denote it “projected MIP
representability.”
Definition 11.3. S ⊆ Qn is projected MIP representable if and only if it has
an MIP formulation of the form
(11.3a)

Ax + Bλ + Dy ≤ b,

(11.3b)
(11.3c)

x ∈ Qn1 × Zn2 ,
λ ∈ Qs ,

(11.3d)

y ∈ {0, 1}t.

Note that (11.3) is a special version of (2.6) in which the linear inequalities are
such that every integer auxiliary variable is either bounded or identical to one of the
original variables. Projected MIP representability certainly subsumes bounded MIP
representability, but as shown by Hooker [76] it can model a much wider class of sets.
Proposition 11.4. A set S ⊆ Rn is projected MIP representable if and only if
 k
i
j
= P∞
for all i, j and
there exists a finite family of polyhedra P i i=1 such that P∞
(11.4)

S=

k


P i ∩ (Qn1 × Zn2 ) .

i=1

Proof. The fact that a set of the form (11.4) is projected MIP representable
follows from Proposition 5.1.

46

JUAN PABLO VIELMA

The converse is analogous to Proposition 11.2.
Projected MIP representable sets are then precisely those sets that can be formulated by (5.2) and (5.3). While projected MIP representability covers most sets that
are used in practice, it does not characterize all sets that can be modeled with MIP
formulations. We now consider the most general version of the MIP representation
theorem introduced by Jeroslow and Lowe [90, 113], for which we need the following
definition.
Definition 11.5. A finitely generated integral monoid is a set M ⊆ Zn such

 d
 d
i
d
that there exists ri i=1 ⊆ Zn for which M =
i=1 μi r : μ ∈ Z+ . We say that
 d
M is generated by ri i=1 .
General MIP representability is obtained by simply replacing cone C by a finitely
generated integral monoid M in bounded MIP characterization (11.2).
Theorem 11.6. A set S ⊆ Qn is MIP representable if and only if
(11.5)

S=M+

k


Pi

i=1




i k

for a finite family of polytopes P i=1 and a finitely generated monoid M .
Proof. For necessity assume S has a MIP formulation of the form (2.6) and let
Q ⊆ Qn ×Qs ×Qt be the polyhedron described by
 (2.6a). BypTheorem 3.6 and because
Q is a rational polyhedron, there exist points x̂j , ûj , ŷ j j=1 ⊆ Qn × Qs × Qt and

d
scaled rays x̃l , ũl , ỹ l l=1 ⊆ Zn × Zs × Zt such that for any (x, u, y) feasible for (2.6)
there exist λ ∈ Δp and μ ∈ Qd+ such that
(x, u, y) =

p


λj x̂j , ûj , ŷ j +

j=1

d


μl x̃l , ũl , ỹ l .

l=1

Now, for such λ and μ define
(x∗ , u∗ , y ∗ ) :=

p


λj x̂j , ûj , ŷ j +

j=1

d


(μl − μl ) x̃l , ũl , ỹ l

l=1

and
(x∞ , u∞ , y ∞ ) :=

(11.6)

d


μl  x̃l , ũl , ỹ l .

l=1

Then, (x∞ , u∞ , y ∞ ) ∈ Zn × Zs × Zt and (x∗ , u∗ , y ∗ ) belongs to bounded set
⎫
⎧
p
d
⎬
⎨

λj x̂j , ûj , ŷ j +
μl x̃l , ũl , ỹ l : λ ∈ Δp , μ ∈ [0, 1]d .
S0 :=
⎭
⎩
j=1

l=1

Also, because y = y ∗ + y ∞ and y ∈ Zt we have y ∗ ∈ Zt . Hence (x∗ , u∗ , y ∗ ) ∈
S0 ∩ Qn × Qs × Zt , which is a finite union of polytopes (one for each possible value
 k
of y ∗ in S0 ). Let P i i=1 be the projection of such polytopes onto the x variables,

so that x∗ ∈ ki=1 P i . The result then follows from x = x∗ + x∞ and the fact that
 d
(11.6) implies x∞ belongs to the integral monoid generated by x̃l l=1 .

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

47

For sufficiency, let S be of the form (11.5), let (4.6) be an MIP formulation for
 i d
i
n
i=1 P , and let M be generated by r i=1 ⊆ Z . Then an MIP formulation for S
is given by
k

Ai xi ≤ bi yi

(11.7a)
(11.7b)

k

i=1

xi +

d


∀i ∈ {1, . . . , k},

zl rl = x,

l=1
k


(11.7c)

yi = 1,

i=1

xi ∈ Qn

(11.7d)

∀i ∈ {1, . . . , k},

(11.7e)

y ∈ {0, 1} ,

(11.7f)

z ∈ Zd+ .

k

By comparing (11.2) and (11.5), we see that general MIP representability replaces
the continuous recession directions of polyhedral cone C with the discrete recession
directions of monoid M . The fact that this replacement gives further modeling power
stems from the following lemma, which shows that continuous recession directions
can be obtained from discrete recession directions and a polytope through Minkowski
addition of sets.

 d
i
d
Lemma 11.7. Let C =
i=1 μi r : μ ∈ Q+ be a polyhedral cone generated by
 d

 i d
i
d
is a polytope
r i=1 ⊆ Zn . Then C = M + P , where P :=
i=1 λi r : λ ∈ [0, 1]

 d
i
d
is
a
finitely
generated
integral
monoid.
and M :=
μ
r
:
μ
∈
Z
+
i=1 i
Proof. The result follows from noting that
d

i=1

μi r i =

d

i=1

μi  r i +

d


(μi − μi ) ri .

i=1

Using this lemma we have that bounded MIP characterization (11.2) is equivalent
to
(11.8)

S = M + Q0 +

k


Qi

i=1

 k
for a finite family of polytopes Qi i=0 and a finitely generated integral monoid M .
k
We then obtain general MIP characterization (11.5) by noting that Q0 + i=1 Qi =
k
i
and that P i = Q0 + Qi is a polytope for every i. We could carry
i=1 Q0 + Q
out the reverse transformation if we could factor out a common polyhedron Q0 from
polytopes P i from bounded MIP characterization (11.2). However, as the following
example from [90, 113] shows, this factorization cannot always be done and general
characterization (11.5) includes more cases than bounded characterization (11.1)–
(11.2) and than projected MIP characterization (11.4).
Example 17.
Let S = ({0} ∪ [2, ∞)) ∩ Z. S is a finitely generated integral
monoid and a general MIP formulation for S is given by
x − 2y1 − 3y2 = 0,
y ∈ Z2+ .

48

JUAN PABLO VIELMA

However, S does not satisfy bounded characterization (11.1)–(11.2) or projected MIP
characterization (11.4).
Bounded, projected, and general MIP representability can sometimes be hard to
recognize (e.g., see Example 20). One way to check the potential for MIP representability is through the following necessary conditions proven in [90, 113].
Proposition 11.8. If S ⊆ Qn is MIP representable, then S is closed and conv(S)
is a polyhedron.
Unfortunately, the following example from [153] shows that these conditions are
not sufficient for MIP representability.
Example 18. Let


max
xi , xj ≥ 0 ∀j ∈ {1, . . . , n}
S = x ∈ Qn : xn =
i∈{1,...,n−1}

=

n−1


{x ∈ Qn : xn = xi ,

xn ≥ xj ≥ 0

∀j ∈ {1, . . . , n}} .

i=1



n−1
We have that conv(S) = x ∈ Qn : xn ≤ i=1 xi , xn ≥ xj ≥ 0 ∀j ∈ {1, . . . , n}
is a polyhedron, but S is not MIP representable.
We now illustrate some of these representability concepts by considering MIP
formulations of piecewise linear functions. However, we first introduce the following
example, which shows that rationality is crucial to the presented representability
results. For more information concerning MIP representability for nonpolyhedral or
nonrational polyhedral sets, we refer the reader to [49, 73].
19.
with irrational data given by S =
√ Consider the MIP formulation

 Example
x ∈ R2 : x2 − 2x1 ≤ 0, x2 ≥ 0, x ∈ Z2 . We have that S is closed and the closure
of the convex hull of S is the nonrational polyhedron given by the LP relaxation of
this formulation. However, conv(S) is not closed. Furthermore, we have that S is an
integral monoid (it is composed by integer points and is closed under addition), but
by Lemma 3 in [83] we have that S is not finitely generated.
11.1. MIP Representability of Functions. Functions that have MIP formulations include most piecewise linear functions. The study and use of MIP formulations
for piecewise linear functions has been and still is a very prolific area of research with
a wide range of applications. While it is beyond the scope of this paper to include
a detailed survey of the literature on such formulations, we have included several
MIP formulations of piecewise linear functions and we now cover some issues concerning representability of piecewise linear functions. For more details we refer the
reader to [155] and the references therein. Some recent work concerning piecewise
linear functions that are not referenced in [155] include [151, 51, 57, 124, 62, 123,
126, 127, 43, 156, 153]. Finally, we note that there is also a vast literature on incorporating piecewise linear functions into optimization models without using MIP
[17, 54, 55, 56, 38, 99, 158, 47, 169, 46]
Functions with bounded MIP representable graphs and epigraphs include most
piecewise linear functions. In particular, for continuous multivariate functions with
bounded domain we can give the following precise characterization.
Proposition 11.9. Let f : D ⊆ Qd → Q be a continuous function.
1. If gr(f ) is bounded MIP representable, then epi(f ) is bounded MIP representable
2. gr(f ) is bounded MIP representable if and only if there exist {mi }ki=1 ⊆ Qn ,

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

49

{ci }ki=1 ⊆ Q, and a finite family of polytopes {Qi }ki=1 such that
D=

(11.9a)

k


Qi ,

i=1

⎧
1
⎪
⎪ m x + c1 ,
⎨
f (x) =

(11.9b)

⎪
⎪
⎩ mk x + c ,
k

x ∈ Q1 ,
..
.
x ∈ Qk .

Proof. For item 1 we first note that epi(f ) = C+ + gr(f ), where C+ := {(0, z) ∈
Qn ×Q : z ≥ 0}, and that for f with bounded domain gr(f ) is bounded. Now, if gr(f )
k
is bounded MIP representable, we have by Proposition 11.2 that gr(f ) = i=1 P i for


a finite family of polytopes {P i }ki=1 . Then, epi(f ) = C+ + ki=1 P i = ki=1 C+ + P i
k

and C+ + P i i=1 is a finite family of polyhedra with common recession cone C+ .
Hence, by Proposition 11.2 epi(f ) is bounded MIP representable.
For the “if” part of item 2 let P i := {(x, z) ∈ Qn × Q : x ∈ Qi , z = mi x + ci } for

each i ∈ {1, . . . , k}. Then {P i }ki=1 is a finite family of polytopes and gr(f ) = ki=1 P i ;
hence by Proposition 11.2 gr(f ) is bounded MIP representable. For the only if part,
k
if gr(f ) is bounded MIP representable, then, by Proposition 11.2, gr(f ) = i=1 P i
for a finite family of polytopes {P i }ki=1 . Let Qi be the projection onto the x variables
of polytope P i and let {v j }pj=1 be the extreme points of polytope Qi . We claim that
there exists a solution mi , ci ∈ Qn × Q to the system mi v j + ci = f (v j ) for all
j ∈ {1, . . . , p}. If such a solution exists we have that P i = {(x, z) ∈ Qn × Q : x ∈
Qi , mi x + ci = z}, which gives the desired result. To show the claim first assume
without loss of generality that v d+1 = 0. After that, assume again /without
loss of
0p
generality that {v j }dj=1 are linearly independent where 1 < d = rank v j j=1 (if d =


0, then Qi = v d+1 = {0} and the claim is straightforward). Let mi ∈ Qn and ci ∈ Q
be such that ci = f v d+1 = f (0) and mi v j = f (v j ) − ci for all j ∈ {1, . . . , d}, which
exist by the linear independence assumption. To arrive at a contradiction assume
without loss of generality that mi v d+2 + ci > f v d+1 . Because of the assumption on
{v j }dj=1 , there exist λ ∈ Qd such that v d+2 = di=1 λi v i . Furthermore, without loss
of generality we may assume that λ1 ≤ λi for all i ∈ {1, . . . , d}. Let
ṽ(δ) := δ

d+1

i=1

1
vi
d+1


+ (1 − δ)v d+2 =

d


λ̃i (δ)v i ,

i=1

1
+ (1 − δ)λi for all i ∈ {1, . . . , d}. If λ1 ≥ 0, let δ1 := 0, and if
where λ̃i (δ) := δ d+1
λ1 < 0, let

λ1
 ∈ (0, 1).
δ1 := 
1
λ1 − d+1
If

d
i=1

λi ≤ 1, let δ2 := 0, and if
δ2 :=

d
i=1

λi > 1, let

d
i=1
d
i=1

λi − 1

λi −

d
d+1

∈ (0, 1).

50

JUAN PABLO VIELMA

Then, for δ ∗ = max {δ1 , δ2 } we have i=1 λ̃i (δ ∗ ) ≤ 1 and λ̃i (δ ∗ ) ≥ 0 for all i ∈
k
i
{1, . . . , d}. Hence, ṽ (δ ∗ ) ∈ conv {v j }d+1
j=1 . Now, because gr(f ) =
i=1 P , we
i
i
i
∗
have v , f (v ) ∈ P for all i ∈ {1, . . . , d + 2}. Then, by the definition of ṽ (δ ) and
d+1
1
convexity of P i , we have ṽ (δ ∗ ) , j=1 δ d+1
f (v i )+(1−δ)f (v d+2 ) ∈ P i . Furthermore,
i
by the definition of m , ci , the assumption on f (v d+2 ), and the fact that 1 − δ ∗ > 0,
we have mi ṽ (δ ∗ ) + ci > f (ṽ (δ ∗ )). However, because ṽ (δ ∗ ) ∈ conv {v j }d+1
j=1 and
i
i
∗
∗
due to the definition of m , ci we also have m ṽ (δ ) + ci = f (ṽ (δ )), which is a
contradiction.
Conditions (11.9) imply that a function with a bounded MIP representable graph
is automatically continuous. However, if we only require the epigraph of a function to
be bounded MIP representable, we can also model some discontinuous functions and
some functions with unbounded domains. This is illustrated in the following example,
which also shows that to recognize bounded MIP representability it is sometimes
necessary to have some redundancy in characterization (11.1) (i.e., the interiors of
some polyhedra must intersect to comply with the common recession cone condition).
Example 20. Let f : [0, ∞) → Q be defined as
d

)
f (x) =

−x + 2,
x − 1,

x ∈ [0, 1),
x ∈ [1, ∞).

This function is depicted in black Figure 11.1(a), where its epigraph is depicted in
red. The function is discontinuous at 1 so its graph is not closed and hence cannot
be MIP representable by Proposition 11.8. However, epi(f ) = P 1 ∪ P 2 where P 1 :=
{(x, z) : x + z ≥ 2, z − x ≥ 0, x ≥ 0} and P 2 := {(x, z) : z − x ≥ −1, x ≥ 1} are
polyhedra with a common recession cone. This is illustrated in Figure 11.1(b), where
P 1 is depicted in green and P 2 is depicted in blue. Proposition 11.2 then implies that
epi(f ) is bounded MIP representable.
Some continuous functions with unbounded domains also have bounded MIP
representable graphs, but the necessary conditions for this to hold are more restrictive.
For more details concerning functions with unbounded domains, we refer the reader
to [84, 113, 120], and for discontinuous functions to [155] and the references therein.
12. Other Topics.
12.1. Combinatorial Optimization and Approximation of Convex Sets. An
important part of LP or polyhedral methods for combinatorial optimization can be
interpreted as the construction of sharp or locally ideal formulations for such problems [146]. The use of extended formulations for such constructions is a mature and
yet extremely active area of research. For instance, in 1991 Yannakakis [167] raised
the question of the nonexistence of a polynomial-sized sharp extended formulation
for the traveling salesman problem (TSP) and for perfect matchings. This nonexistence was only proven recently for the TSP by Fiorini et al. [53] and for matching by
Rothvoß [140]. In addition, the construction of extended formulations for combinatorial optimization is strongly related to approximation questions in convex geometry
(e.g., [98]). For example, the recent paper by Kaibel and Pashkovich [97] develops
a technique for constructing extended formulations that generalizes and unifies approaches for the polyhedral approximation of convex sets [19] and for formulations of
combinatorial optimization problems [61].

51

MIXED INTEGER LINEAR PROGRAMMING FORMULATION TECHNIQUES

3

3

2

2

1

1

0

0
0

1

2

3

0

(a) Epigraph of function from Example 20

1

2

3

(b) Epigraph as union of polyhedra.

Fig. 11.1 Discontinuous Function with Unbounded Domain.

For more information on extended formulations in combinatorial optimization we
refer the reader to [35, 154, 95] and for more on polyhedral approximations of convex
sets we refer the reader to [14, 16, 152] and [93, Chapters 8 and 17].
12.2. Linearization of Products. Another important MIP formulation technique
that we omitted because of space is the linearization of products of binary or bounded
integer variables and the products of these variables with bounded continuous variables. These linearizations are some of the oldest MIP formulation techniques [58,
7, 100, 133, 59, 162, 60, 70, 8, 117, 72, 135] and continue to be a very active area
of research [5, 3, 71, 64, 65, 4]. They are an important tool for nonconvex mixed
integer nonlinear programming [29] and have been extensively studied in the context of mathematical programming reformulations [109, 108]. In particular, many of
these linearized formulations can be obtained through the reformulation-linearization
technique of Sherali and Adams [147].
Acknowledgment. The author would like to thank two anonymous referees and
one associate editor for their helpful feedback.
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