OPTIMA
Mathematical Programming Society Newsletter
Steve Wright
MPS Chair's Column
September 6, 2009. I'm writing in the wake of ISMP 2009 in
Chicago, which ended last weekend. All involved in the organiza
tion were delighted with how well the meeting ran, and hope that
all attendees (over 1200 of them) found it to be a rich and valuable
experience, both professionally and personally. There are too many
highlights to recount here; suffice to say the event's success indicates
that our field of optimization is in excellent health. I have no doubt
that our research will continue to increase in depth and breadth
in the coming years and in its influence on research and under
standing in many other areas. I thank profoundly the many people
who worked so hard to make the event a success, starting with John
Birge, who chaired the organization and program committees.
John Birge, ISMP 2009 Chair (left) and
Steve Wright, MPS Chair, at the Open
Ceremony of ISMP 2009, August 23,
2009, Orchestra Hall, Chicago. Photo
by Chris Buzanis, CGPA Photography.
Many members would have noted that this ISMP followed a differ
ent framework from past meetings, which were organized at campus
locations by independent local committees. The unsuitability of cam
pus sites in Chicago pushed us toward the downtown location and
toward a more professional mode of organization involving our fel
low society INFORMS, which was able to sign contracts when MPS
was unable to do so. We found INFORMS to be a great organiza
tional partner. The benefits of high quality meeting rooms, topclass
social and ceremonial events, and the vibrant downtown location
that we enjoyed during this symposium speak for themselves. Its
success has more than justified the decision to try an alternative
model for ISMP organization this time around. This model provides
an additional option for those who are thinking about organizing
future symposia.
One novel feature of this symposium was the daily newsletter Op
tima@ISMP, which was published on each day of the meeting. I thank
editor/publisher Leah Lavelle for her great work on the newsletter,
which was popular and practical, with more than a touch of class.
The ISMP web site will stay up indefinitely. Its URL may change,
but it will be accessible through the society's web site mathprog.org.
The five issues of Optima@ISMP have been posted there, and some
photos will appear in the near future.
A major news item from ISMP was the announcement about the
2012 symposium, which will be held at TU Berlin. I thank Martin
Skutella and his colleagues at other Berlin institutions for taking on
this responsibility, and have no doubt that we'll be celebrating an
other great symposium three years from now. I'm grateful too to the
groups at London and Istanbul, who also prepared excellent bids.
Following MPS tradition, we held our major council and business
meetings of the threeyear cycle during the symposium. I welcome
the chairelect Philippe Toint and treasurerelect Juan Meza, who will
take their positions in August 2010. (Philippe serves as vicechair
until that date.) The new councilorsatlarge Jeff Linderoth, Clau
dia Sagastizibal, Martin Skutella, and Luis Vicente took office during
ISMP. The society will be in excellent hands during the new 3year
term.
The new constitution was ratified at the business meeting, a step
in our efforts to overhaul the society's legal framework. The bylaws
too have undergone major changes during the past year, to bring
them into line with governmental expectations for professional non
profit organizations, to reflect current practices, and to include new
material on such matters as ICCOPT and the mathprog.org web site.
Besides their legal function, the bylaws serve as a reference guide
for future leaders of the society on how we carry out our most
important functions, especially publications, prizes, and meetings.
During ISMP, the Committee on Stochastic Programming (COSP)
formally became a technical section of MPS. COSP was founded
originally as a standing committee of MPS, but the bylaws of the
two organizations fell out of sync over the years, a situation that has
now been remedied by this new status. We look forward to working
on projects of mutual benefit with the members of this very active
group, which includes many of our most distinguished members and
which represents a research area whose importance continues to
surge.
Our journals Mathematical Programming, Series A and B, recently
reported impact factors and article influence scores that are among
the leaders in the Applied Mathematics class. Kudos to the editorial
staff for their great work in maintaining the quality and reputation of
our publications. Mathematical Programming Computation, meanwhile,
is off to a great start, with its very first published paper (by Tobias
Achterberg) receiving this year's BealeOrchardHays Prize during
the ISMP opening ceremony. At the time of writing, our publications
are featured heavily on Springer's mathematics home page.
Contents of Issue 80 / September 2009
I Steve Wright, MPS Chair's Column
2 Santanu S. Dey and Andrea Tramontani, Recent Developments
in MultiRow Cuts
8 G6rard CornuBjols, Robert Weismantel and Laurence
Wolsey, Comments on MultiRow Cuts
10 Prizes Presented at ISMP
12 New MINLP Cybersite
12 Imprint
OPTIMA 80
After sporadic discussion over many years, a consensus emerged
clearly at the council and business meetings that the time had come
to change our society's name. Our current name has become in
creasingly oldfashioned and difficult to explain. While the alternative
term "optimization" is not fully inclusive of our activities and is also
ambiguous in certain quarters, it has much more universal recogni
tion as a title for our field of research. Possibly the most obvious
new name for the society is "Mathematical Optimization Society."
The feeling is that we should retain the current names for our jour
nals, to avoid archival confusion, but that a subtitle could be added
to indicate that they are journals of our society. Concrete proposals
will be discussed in the months ahead. As with all other issues con
cerning the society, you are welcome to make your thoughts known
to me or to other members of the society's leadership.
Finally, I mention that renewal notices for 2010 membership will
be sent in October. I urge you to renew your membership and con
tinue your participation in the society during this time of vitality and
growth.
Santanu S. Dey and Andrea Tramontani
Recent Developments in MultiRow Cuts
I Introduction
A classical way to strengthen linear programming relaxations of
mixed integer linear programs (MIP) is to add linear inequalities
known as cutting planes or cuts. Over the years, cutting planes have
proven to be indispensable tools in solving MIPs. One approach for
generating cutting planes for a given MIP is to use the facetdefining
inequalities of the convex hull of its feasible solutions. However,
since in the case of general MIPs little structure can be assumed, it is
difficult, if not impossible, to analyze the polyhedral properties of the
convex hull of the feasible solutions. Therefore, the feasible region
is typically relaxed so as to obtain a set that is more amenable to
analysis. One common relaxation often considered in the literature
is the 'single constraint relaxation'. In this scheme the cut generation
procedure may be viewed as a two step process. In the first step, all
but one of the constraints of the MIP are dropped. Alternatively, var
ious constraints are multiplied by suitable weights and then added to
obtain a single implied constraint. Then a black box for generating
a cutting plane from a single constraint (often using information on
individual variable restrictions such as nonnegativity, bounds, and in
tegrality) is invoked to derive a cut. As the single constraint defines
a relaxation of the original MIP, the resulting cut is valid for the orig
inal MIP Gomory mixed integer cuts (GMI) (Gomory [26]), mixed
integer rounding inequalities (MIR) (Nemhauser and Wolsey [34]),
and cover cuts (Balas [7], Wolsey [36]) are some examples of cuts
generated using this paradigm.
Very recently, a series of papers have focused on the possibility
of generating cuts using more than one row of the simplex tableau
(or constraints) simultaneously. Several interesting theoretical re
sults have been presented in this direction, often revisiting and re
calling other important results discovered more than 40 years ago.
The paradigm for generating cutting planes from multiple rows of
simplex tableau remains similar to that of generating cuts based on
one row. Typically, we start with a simplex tableau of the form
XB = x+ I rJxj,
jeN
x>0, (I)
xj e Z, je J,
where B (resp. N) denotes the set of basic (resp. nonbasic) variables
and the current incumbent solution (XB,XN) := (X, O) is assumed
to be integer infeasible. As before, a relaxation of (I) is generated.
However, now this new relaxation may contain more than one con
straint. Then, facetdefining inequalities (or the closely related ex
treme inequalities) for this relaxation are generated. These inequali
ties are valid for the original MIP by virtue of the fact that they are
valid for a relaxation.
The art lies in obtaining a relaxation that is both easy to ana
lyze and yet strong enough to generate potent cutting planes. In this
paper we review some of the relaxations (and their analysis) that
are closely related to the group relaxation, originally invented by
Gomory [28]. In order to construct the group relaxation, the non
negativity requirement on the basic variables XB is relaxed and new
nonnegative variables are introduced in (I). In Section 2, we review
the group relaxation, define some generic concepts used in the rest
of the paper, and summarize some recent approaches for generating
cutting planes based on multiple constraints using the classical group
relaxation.
Andersen et al. [3] and Borozan and CornuBjols [12] considered
further relaxing the group relaxation by removing the integrality re
striction on the nonbasic variables. This relaxation lead to a sig
nificantly different perspective on the multirow cutting planes. In
particular, a wonderful connection between extreme inequalities of
these relaxations and latticefree convex sets has been established via
the principle of Intersection cuts (Balas [6]). Results related to various
variants of the relaxation introduced in [3] and [12] form the bulk
of this paper and are discussed in Section 3. In Section 4, we review
some results evaluating the strength of these new classes of inequal
ities and comparing their properties with respect to split inequalities
(Cook et al. [14]). Many questions remain open and some of them
are highlighted in Section 5.
We finally note that there are numerous approaches to generat
ing cuts based on multiple rows. This review if confined only to the
recent approaches of cut generation based on the group relaxation.
2 The Group Relaxation
For the sake of simplicity we assume that the set (I) is nonempty.
The first step in the construction of the Group relaxation is the
construction of the Corner relaxation. This relaxation is obtained by
dropping the nonnegative restrictions on all the basic variables and
considering a subset of m rows of (I) associated with basic integer
constrained variables (i.e., a subset of variables xi with i e B n J),
thus obtaining
XB=XB*+ rJxj + rJyj,
jeNnJ jcN\J
S Z (2)
xj >O xj c Z, j N nJ,
yj > 0,
j N\J,
where now B denotes the set of m basic integerconstrained vari
ables corresponding to the selected rows. (Hence forth continu
ous variables are represented using the letter y to distinguish them
from integer variables represented using x.) Note that if the sim
plex tableau is nondegenerate, then the set of constraints that are
not active is exactly the set of nonnegativity constraints on the ba
sic variables. Thus, the motivation for constructing the Corner re
laxation is that the constraints that are currently active at the so
lution (XB, (x,y)N) := (X, (0,0)) are possibly more important,
while dropping the nonactive constraints may simplify the analysis of
the resulting set (2). Indeed, Gomory [27] presented necessary con
ditions for the optimal solution of the corner relaxation to be the
same as the optimal solution of the original integer program. This is
known as the Asymptotic Theorem.
September 2009
It is customary to replace x* by f, where fi = (x)i [(xF)i],
as this amounts to just translating XB by an integral vector. More
over, since XB is a free integer vector, it can be verified that if
rj, r2 e m, then in any valid strong inequality for (2) the two
variables xj, and xj2 (jl,j2 J) will have the same coefficient.
Therefore, we can replace the column rJ corresponding to an inte
ger vector xj (j e J) by a vector of its fractional components.
The group relaxation (also called the master group relaxation) is ob
tained by the addition of more variables to the corner relaxation,
i.e. we consider the set
XB f + rJxj + r rJyj,
rJ eG rJ eW
XB (3)
xj Z, je G,
XG,YW > 0 and they have finite support,
where {rylj J rn N c G and {rlj e N \ J} c W. Since any
feasible solution of (2) can be used to construct a solution of (3)
by setting the new variables to zero, the projection of (3) onto the
space of the (XB, (x,y)N) variables is a relaxation of (2).
Before proceeding we present some notation. We refer to the
set of vectors [0 1)m as Im. We use the symbols e and e to de
note addition and substraction modulo I componentwise respec
tively. Given a vector v e Rm, we let (v) be a vector belonging
to Im where the ith component of f(v) is vi(mod 1).
Gomory [28] proposed to use the columns of the integer vari
ables from a set G in (3) where G c Im, {rJ j e J nN} c G, and
the elements of G are closed under the D operations. Observe that
(3) has a new condition that XG and xw should have finite support,
i.e. only a finite number of components of XG and xw are permit
ted to be positive in any feasible solution. This condition is added
to avoid technical difficulties in the case when G or W are not finite
sets. The set of feasible solutions of (3) is denoted as R(f, G, W) in
this review.
Definition 2.1. Since all the XB variables can be written in terms of
the XG and yw variables using the first equation in (3), it is customary to
write the valid inequality in terms of the XG and yw variables. Hence, a
valid inequality for R(f, G, W) is defined as a pair of functions p : G 
R+ and Tr : W R+ such that 1,reG p (r )xj + Y)w ra r (rJ)yj > 1
is valid for all (xB,xG,yw) e R(f,G,W). Here 1(rJ) represents
the coefficient of the variable xj, which in turn is the variable corre
sponding to the column rJ in R(f, G, W). A valid inequality is called
a minimal inequality if it is not dominated by any other inequality, i.e.
(p, Tr) is a minimal inequality if there does not exist a valid inequality
(P',Tr') such that (p',Tr') # (p,Trr) and p'(u) < i(u)Vu e G
and Tr'(w) < Tr(w)Vw e W. A valid inequality is called an extreme
inequality if it cannot be written as a convex combination of two distinct
valid inequalities.
Often valid inequalities are also referred to as valid functions. These
definitions carry through to almost all the models/relaxations re
viewed in paper. Note also that extreme inequalities and facet
defining inequalities are equivalent concepts when the group G is
finite. Indeed, Gomory and Johnson [29, 30] prove the following
fundamental result: Every extreme inequality is a minimal inequality.
The advantage of constructing the group relaxation is twofold:
The polyhedral analysis of the corner relaxation presented in (2) is
'messy' since it depends on the data rJ, j e N. On the other hand,
the analysis of (3) is clean and elegant, since it contains all possible
interesting columns and is effectively 'data independent'. The sec
ond advantage is a little more subtle and it is due to the following
result by Gomory [28]: All the facetdefining inequalities of (2) can
be extracted from the extreme inequalities of (3).
We now present one representative result about minimal inequal
ities of the master group relaxation that illustrate the 'niceness' of
these structures.
Theorem 2.1 ([3 I ]). A valid inequality (p, Tr) is a minimal inequality
for R(f ,I, Rm) if and only if I1 : I [R+ and Tr : RIm R+ satisfy
the following conditions: i) (u) + (v) > (u e v) Vu, v e Im,
ii) ((u) + ((T(f) e u) = 1 Vu Im, and iii) Tr(w) =
limho(y(h)) Vw G r.
A number of families of group cuts have been discovered and proven
to be extreme based on the bedrock of Theorem 2.1 and its variants.
While many of these results are primarily for the onerow group
relaxation (i.e. the case where m 1), recently some families of
multirow cutting planes have been proven to be extreme; see Dey
and Richard [18, 19]. The results in [18, 19] present two families of
cutting planes that use onerow or multirow cuts as their building
block to construct extreme inequalities for the mrow group re
laxation. The first family corresponds to aggregation of rows. The
second family corresponds to a more intricate sequence of cut gen
eration from single and multiple rows applied in specific sequence
which ultimately leads to extreme inequalities for mrow group re
laxations. We refer the reader to [ 18, 19] for the details.
3 Corner and Group Relaxations together with
'Continuous Nonbasic Variables' Relaxation
3.1 Intersection Cuts and Latticefree Convex Sets
Before presenting the results about different relaxations that are
closely related to the group relaxation, we discuss a generic princi
ple of generating cutting planes known as intersection cutting planes,
invented by Balas [6]. We illustrate the principle of constructing in
tersection cuts for pure integer programs first: Suppose that we are
given a convex relaxation P of S, where S is the set of feasible so
lutions of a pure integer program. Let f e P \ conv(S) be a point
which we would like to separate. Let M be a convex set containing
f, such that no feasible solution of S lies in the interior of this set,
i.e., S n int(M) = 0. Then the relaxation P can be strengthened
by computing conv(P \ int(M)). The resulting inequality obtained by
this operation separates f and is called an intersection cut. Thus,
convex sets which do not contain integer points in their interior can
be used to generate intersection cuts for integer programs (and also
for MIPs as discussed in the next section). We next define these sets
formally.
Definition 3.2 ([33]). A set M c Bm is called latticefree ifint(M) n
Zm = 0. A latticefree convex set M is maximal if there exists no lattice
free convex set M' # M such that M S M'.
Maximal latticefree convex sets are very structured sets. The fol
lowing characterization is due to Lovisz [33] and Basu et al. [10].
Theorem 3.2 ([33], [10]). Let M be a maximal latticefree convex
set in Rm. Then M is either an irrational affine hyperplane in Rm or a
fulldimensional latticefree polyhedron of the form M = P + L, where P is
a polytope, L is a rational linear space, dim(M) = dim(P) + dim(L). In
the second case, M has at most 2m facets and there is an integral point
in the relative interior of each facet of M.
In two dimensions (i.e., when m = 2), fulldimensional maximal
latticefree convex sets can be classified as follows. (See Figure I.)
OPTIMA 80
0 0 0 0 0 0 0 0 0 0 0
0 0 0 a 00
o ooo oo o o o o
o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Figure I. Maximal lattice free sets
Proposition 3.1. Let M be a fulldimensional maximal latticefree con
vex set in R2. Then M is one of the following:
I. A split set {(xl,X2) I b < alX1 + a2x2 < b + 1} where al and
a2 are coprime integers and b is an integer,
2. A triangle with a least one integral point in the relative interior of
each of its edges, which in turn is either:
(a) A type I triangle: triangle with integral vertices and exactly one
integral point in the relative interior of each edge,
(b) A type 2 triangle: triangle with at least one fractional vertex
v, exactly one integral point in the relative interior of the two
edges incident to v and at least two integral points on the third
edge,
(c) A type 3 triangle: triangle with exactly three integral points on
the boundary, one in the relative interior of each edge.
3. A quadrilateral containing exactly one integral point in the relative
interior of each of its edges.
3.2 TwoRow Corner Relaxation with Continuous Nonbasic Variables
Andersen et al. [3] considered relaxing the integrality of all the non
basic variables in the tworow corner relaxation, i.e. they considered
the set R(f, 0, W):
XB f+ rJyj, XB C Z2, yj > 0 Vj, (4)
where f e Q2 \ Z2, IW = k is finite, and rJ e Q2 \ {(0,0)} for
all rJ c W. One motivation for this relaxation is the following: suc
cessful onerow cuts like MIR can be explained by considering the
effect of integrality of one free integer variable (see derivation in
Wolsey [37]) in a simple onerow system. By removing all the inte
grality requirement on the nonbasic variables, we are left with two
free integer basic variables. This model is thus expected to capture
the effect produced by two rows and two integer variables.
Nontrivial valid inequalities for R(f, 0, W) can be generated us
ing maximal latticefree convex sets containing f via intersection
cuts: Let M c R2 be a maximal latticefree convex set containing f
in its interior. Consider the set M' = (u,v) e R2 x u e M}.
Let Ro be the continuous relaxation of (4). Then we can generate a
cut for (4) by computing conv(R0 \ int(M')). It can be verified that
this is equivalent to generating the cut ,Jw T(rJ)yj > 1 where
the cut coefficients Tr(rJ) are computed as
rr A, if 3A > 0 s.t. f + rj boundary (M) (5)
r(r3J) \(5)
0, if rJ belongs to the recession cone of M.
We next show an interesting example of the set (4) and a cut of the
form (5).
Example 3.1 ([14], [3]). Consider the simple MIP
max t
s.t. (cl) t
(c2) t
(C3) t
+X + X2 < 2,
e Z2 and te R1.
By introducing nonnegative slack variables Si, s2 and s3 in constraints
(cl), (cA), and (c3) respectively, the simplex tableau corresponding to
the optimal vertex x* (2, ) of the LP relaxation of (6) reads
2
t
3
2
X1 3
2
X2 3
3
3S2
3
s2
+2S2
x G Z2, t G R1,s R3.
Cook et al. [14] have shown that a cutting plane algorithm based on split
cuts does not suffice to generate the cut needed to solve the toy problem
(6) in a finite number of iterations. If however the last two rows of(7) are
considered simultaneously (i.e., a relaxation of (7)), then the cut required
to solve the problem, namely
1 1 1
I Sl + I s2 + I s3 > 1 or equivalently t < 0,
2 2 2
can be immediately derived as an intersection cut. Andersen et al. [3]
show that the cut (8) is in fact an intersection cut arising from the lattice
free triangle with vertices vl (0,0), v2 (2,0) and v3 (0,2)
using (5).
Now we present a result modified from [3]. This result presents
necessary conditions for inequalities to be facetdefining for (4).
Theorem 3.3 ([3]). Let ,rew Tr(r)yj > 1 be a facetdefining in
equality for (4). Then Tr (r ) > 0 Vj. Let
L, ={ u e R2 1 ye F. ,s.t.
(9)
u = f + r, wryj, W T(r J)3yj < 1 .
If r (r) = 0 for some j, then Lr is the subset of a split set and the in
equality is a split inequality. If r (rJ) > 0 for all j, then Lr is a latticefree
triangle or a latticefree quadrilateral.
Constructing the inequality Y,,w Tr(rJ)yj > 1 by using a maximal
latticefree convex set M and (5), and then computing Lr using (9),
we typically obtain that L, c M where Lr may not be a maximal
latticefree convex set. Thus, by presenting a set of shapes of Lr
when Tr is facetdefining, Theorem 3.3 gives necessary conditions
for facetdefining inequalities for R(f, 0, W).
CornuBjols and Margot [15] present sufficient conditions for
facetdefining inequalities for R(f, 0, W). In order to present these
conditions we require some definitions. Let M be a maximal
latticefree triangle or quadrilateral with f in its interior. For i e
{1,..., IWI, let pi be the intersection of the ray f + Ari, A > 0
with the boundary of M, i.e., pi f + "r The point pi is called
the boundary point for ri. A boundary point pi is called active if it
is integral or if there exists another boundary point pJ such that a
strict convex combination of pi and pJ is an integral point. Given a
set of boundary points T, a point pi e T is called uniquely active if
there exists an unique point pi e T such that a strict convex com
bination of pi and pJ is an integral point. The following Reduction
Algorithm from CornuBjols and Margot [15] helps in simplifying the
characterization of facets of R(f, 0, W): i) Let T := {p ..., pk}.
ii) While there exists p e T such that p is active and p is a convex
combination of other points in T, remove p from T. iii) While there
exists a uniquely active p e T, remove p from T. iv) If T = p,q}
and the segment pq contains at least two integral points, remove
both p and q from T
The ray condition is said to holds if T = 0 at termination of the
Reduction Algorithm.
September 2009
Theorem 3.4 ([15]). Let M be a maximal latticefree convex set con
taining f in its interior. Then the inequality Yr, w Tr(ri)yi > 1 gener
ated using (5) is extreme if:
I. M is a split set, where the recession direction of M is ri for some
i G {I ,..., IW },
2. M is a maximal latticefree triangle and
(a) there exist rzl, rz2,r 3 such that the points f + ,
j e {1, 2,3} are the three corner points ofM or
(b) the ray condition holds,
3. M is a maximal latticefree quadrilateral and there exist ril,
r2, r 3, r4 such that the points p : f + ,' j
Tr(r )'
{1,2,3,4} are the four corner points of M, and there does not
exist an h c R, such that
IbJ pJ\ Ih
b bJ pj+1 1
ifj = 1,3
ifj =2,4,
where bJ is the integer point lying on the line segment pJ pi
Thus, Theorem 3.3 and Theorem 3.4 give the complete characteriza
tion of extreme qualities for (4) using the properties of latticefree
convex sets in R2.
3.3 MultiRow Master Infinite Group Relaxation with Continuous
Nonbasic Variables
Borozan and CornuBjols [12] considered relaxing the integrality of
all the nonbasic variables in the mrow semiinfinite master group
relaxation, i.e. they considered the set R(f, 0, Qm):
XB = f + Zr^J. rJyj,
XB e Zm, y > 0 and has a finite support,
where f e Qm \ Zm.
Theorem 3.5 ([12]). Any valid inequality for R(f, 0, Qm) can be
written in the form rJQ "Tr(rJ)yj > 1, where rr : Qm 
Q+ u {+oo}.A minimal valid function rr for R(f, 0, Qm) is nonnegative,
piecewise linear, positively homogeneous and convex. Furthermore the set
cl(Lr) cl{u e Qm  rr(u f) < 1} is a fulldimensional maximal
latticefree convex set containing f. Conversely, for any fulldimensional
maximal latticefree convex set M c p containing f, there exists a min
imal valid function rr for R(f, 0,Qm) such that cl(Lr) = M. When f
is in the interior ofM, this function is unique and can be computed using
(5).
Theorem 3.5 again illustrates the relationship between minimal in
equalities and latticefree convex sets. Any minimal valid inequality
for R(f, 0, Qm) arises from a maximal latticefree convex set M
containing f and viceverse. As this model has all possible columns
and is effectively data independent, the result is 'cleaner' than the
result of Theorem 3.3 and Theorem 3.4.
Theorem 3.5 has been recently generalized to the case where
W = Rm by Basu et al. [10]. One of the significant difficulties in this
generalization is proving that the minimal functions are nonnegative
(in Section 2 the inequalities were assumed to have nonnegative co
efficients. [10] does not make this assumption). We refer the readers
to [10] for the details.
Addressing the case m = 2, CornuBjols and Margot [15] de
scribed the extreme inequalities for the tworow set R(f, 0, Q2). In
particular, if f lies in the interior of a split set, a maximal latticefree
triangle or a maximal latticefree quadrilateral with no h satisfying
(10), then rr generated using (5) is an extreme inequality. CornuBjols
and Margot [ 15] also analyze the case of degenerate maximal lattice
free convex sets (i.e., sets containing f on the boundary) and show
that even these sets can generate extreme inequalities.
Hence, when considering the case of a finite problem R(f, 0, W)
(rJ e Qm for all rJ e W and IWI is finite), the following question
naturally arises: should one consider generating cuts using maximal
latticefree sets containing f on the boundary? For the case of two
rows, Cornu6jols and Margot [15] show that none of the degener
ate cases are needed to define the facets of R(f, 0, W). The same
answer has been provided by Zambelli [38] for a general number of
rows.
Theorem 3.6 ([38]). Let IWI be finite. Given a minimal valid inequality
=l ajYj > 1 for R (f 0, W) (f, rJ e Qm), there exists a latticefree
convex set M such that f lies in its interior and the inequality generated
using (5) satisfies = Trr(rJ) for all j.
(10) 3.4 Introducing Bounds on Nonbasic Variables
Andersen et al. [2] considered introducing upper bounds on the
+1 nonbasic variables for the tworow relaxation (4), i.e. they consid
ered the set
XB =f + kj= yrj,
XB G72, ye Rk, yj
where U c {1,... k} is the set of variables with no upper bound.
In general, an inequality _j= ajyj > ao may have negative coef
ficients unlike in the previous sections. Andersen et al. [2] show
that any nontrivial inequality for (12) can be written in the form
jeU oaj3Yj+jec+ ojyj+jec aoj(uj j) > 1 where C+uC
{1,... k} \ U and each of the ajs are nonnegative.
An inequality j=I jyj > ao is facetdefining for convex hull of
(12) if and only if (a, ao) is an extreme rays of the following polar
cone:
{(a, ao) e Bk+l a(j > 0 Vj e U
k
and ajyj ao > VycY' }, (13)
j=1
where Yv : {y e Rk  3xB e 2 s.t. (xB,y) is a vertex of conv
(12)}. Every extreme ray (corresponding to facetdefining inequality
of (12)) can satisfy a large number of constraints of (13) at equal
ity. For every extreme ray, Andersen et al. [2] present a nontrivial
subset of inequalities that are tight at it and which define this facet
defining inequality uniquely. Using this result, Andersen et al. [2]
are able to completely characterize facetdefining inequalities of (12)
when exactly one of the nonbasic variables has an upper bound. The
characterization is based on possible shapes of the set La (see (9)).
Specifically, it is proven that La can take all the shapes presented in
Theorem 3.3 along with pentagons. The pentagon represents a cut
that is stronger than cuts that can be obtained without the informa
tion on the upper bound.
3.5 Introducing Constraints on Basic Integer Variables
Dey and Wolsey [21], Basu et al. [II], and Fukasawa and Gun
liik [25] have considered various variants of imposing constraints
of the form AXB < b on the basic integer variables of the model
R(f, 0, W). This direction of research was previously investigated
by Johnson [32].
Interestingly, many results in Section 3.2 and Section 3.3 carry
through to this case. Observe that the nonnegativity constraints on
the basic variable were removed to obtain the group relaxation. We
OPTIMA 80
then relaxed the integrality of nonbasic variables to obtain the re
laxation R(f, 0, W) in Section 3.2. Now we are reintroducing the
nonnegativity restriction (and more general constraints) on the basic
variables.
In the case of the set R(f, 0, W), for any XB c Zm there ex
ists a y F. such that XB f + iYrJwrJyj (assuming that
conerjw {rJ} = Rm). Thus the intersection cut is generated using
maximal latticefree convex set. On the other hand, when we add
constraints AXB < b, instead of considering maximal latticefree sets
to generate intersection cuts we need to consider maximal convex
sets M such that they contain no integer point satisfying AXB < b
in their interior. Therefore, we now allow integer points satisfying
AxB > b in the interior of the convex set used to generate the
cut. Formally we make the following definition as a counterpart to
Definition 3.2.
Definition 3.3 ([21]). Let S c Zm. A convex set M c Rm is a maxi
mal Sfree convex set if int(M) n S = 0 and there exists no convex set
M' such that int(M') n S 0 and M' ? M.
It turns out that under the mild assumption of rationality of A and
b, a result very similar to Theorem 3.2 carries through. Weaker ver
sions of the following result are proven in Dey and Wolsey [21] and
Fukasawa and Gunliuk [25].
Theorem 3.7 ([I 1]). Let S be the set of integral points in some ratio
nal polyhedron in Rm such that dim(conv(S)) = m. A set M c Rm is a
maximal Sfree convex set if and only if one of the following holds:
I. M is a polyhedron such that M n conv(S) has nonempty interior,
M does not contain any point ofS in its interior and there is a point
of S in the relative interior of each of its facets. The recession cone
ofM n conv(S) is rational and it is contained in the lineality space
ofM.
2. M is a halfspace of Rm such that Mnconv(S) has empty interior
and the boundary of M is a supporting hyperplane of conv(S).
3. M is a hyperplane of Rm such that lin(M) n rec(conv(S)) is not
rational.
Using maximal Sfree convex sets, minimal inequalities can be gen
erated. We consider the case when W = Rm.
Theorem 3.8 ([21], [I I]). Let S be the set of integral points in some
rational polyhedron in Rm such that dim(conv(S)) = m. Let M be a
maximal Sfree convex set containing f in its interior. The set M {f}
can be written in the form {x I (g)T x < 1,j e {1 ... 1,}} as 0 belongs
to the interior of M {f} and M is polyhedral. Let
TrM(u) =maxljl{ (g J)Tu}.
Then re,,R rMT(rJ)yj > 1 is a minimal inequality for the set
{(xB,y) R(f, 0, Rm) IB x S}.
We note here that the inequality raejRm eM(rJ)yj > 1 may have
negative coefficients. Theorem 3.8 illustrates the use of maximal S
free convex sets to generate minimal inequalities. It is possible to
construct maximal Sfree convex set using minimal inequality (The
orem 3.9). This result together with Theorem 3.8 is a counterpart
of Theorem 3.5. Basu et al. [I I] present a very elegant proof of this
result. (See [25] for the case of m = 2.)
Theorem 3.9 ([ I ]). Let S be the set of integral points in some rational
polyhedron in Rm such that dim(conv(S)) = m. Let Tr : Rm R be a
minimal inequality for the set {(XB,y) e R(f, 0, Rm) I xBe S} of the
form YrJam nr(rJ)yj > 1. Then the set {u e Rm I r(u f) < 1} is
a maximal Sfree convex set.
Dey and Wolsey [21] also characterize extreme inequalities for
{(XB,y) e R(f,o0,R2) IXB e S}. We refer the readers to [21]
for the details.
3.6 Introducing Integral Nonbasic Variables
In Section 3.3, the integrality of nonbasic variables in the corner and
group relaxation were relaxed to obtain the set R(f, 0, W). Now
we consider reintroducing the integer nonbasic variables and study
the set R(f, Im, m). The focus is not on generating all possible
minimal inequalities as characterized by Theorem 2.1. It is instead on
generating first extreme inequalities for R(f, 0, R2) and then lifting
in the integer nonbasic variables. One motivation for this approach
is that the continuous variables get strongest possible coefficients in
such a cut.
On the one hand, exact lifting of unbounded integer variables
(or even nonnegative integer variables with upper bound greater
than I) is a difficult problem since it is a nonlinear integer pro
gram. On the other hand, a trivial valid inequality for R (f, Im, ~m)
is rJ lm rT(rJ)xj + TraRm r (rJ)yj > 1 where the function Tr
represents a valid inequality for R(f, 0, R2). The focus is to take a
middle path, i.e. generate stronger coefficients without actually solv
ing the exact lifting problem.
Dey and Wolsey [20, 23] considered the lifting of extreme in
equalities for R(f, 0, R2) corresponding to maximal latticefree
splits, triangles and quadrilaterals using the so called trivial fillin
function. This approach coincides with a coefficient strengthening
method presented by Balas and Jeroslow [8]. Given an extreme in
equality for R(f, 0, Rm), the trivial fillin function 1o : Im +,
introduced by Gomory and Johnson [30], can be defined as <'", =
infzez, {r(u + z)}. Then, the following result holds.
Theorem 3.10 ([23]). Let Tr be an extreme inequality for
R(f, 0, R2). Then (o0 is the unique lifting function in the case L, =
{u e IR2 Tr(u f) < 1) is a split set, a triangle of type I, or a tri
angle of type 2 and ((10, r) is an extreme inequality for R(f,I2, R2).
If {u e R2 I T (u f) < 1} is any other maximal latticefree convex
set, then there does not exist a unique lifting function and the trivial fillin
function is not minimal.
Conforti et al. [ 13] have considered lifting of integer variables, start
ing from the minimal inequality for {(xB,y) e R(f, 0, Rm) I xB
S}.
Theorem 3.11 ([13]). Let reaRm r(rJ)yj > 1 be a minimal in
equality for {(xB,y) e R(f, 0, R~m) IxyB S} where S is a set of
integer points in a rational polyhedron. Let dp : Rm R be function such
that (, Tr) is minimal for {(xB,x,y) e R(f, Rm, Ym) I XB S}.
Then there exists e > 0 such that Tr(u) = p(u) for all Ilull < c.
(Note that XB variables are not free and two integer variables whose
columns are equivalent modulo I may not get the same coefficient.
Hence all possible columns are considered for nonbasic integer vari
ables.) Using Theorem 3.11 I, Conforti et al. [13] extend the results
of Theorem 3.10 to higher dimensions and also present some other
classes of inequalities for the set { (xB, x, y) e R(f, Rm, Ym) I XBe
S} that have unique lifting. We refer the readers to the paper [13]
for details. Dey and Wolsey [22] have recently considered some
mixedlifting approaches combining traditional sequential lifting with
the fillin approach described above.
4 Properties of the New Cuts and Their Evaluation
4. 1 Comparing Closures
There are a number of possible ways of evaluating the quality of the
new classes of cutting planes that are derived from the extreme in
equalities of multirow relaxations. One approach is to compare the
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closures of various classes of inequalities. To this end, we require
some definitions.
Given a polyhedron of the form Q = x e Yn I azx > b, i
1,..., m. where ai > O, bi > 0 Vi 1,..., m and a scalar a > 0,
we define aQ = x e ~ I aaix > bi, i 1,..., m}. Note that
aQ D Q where a > 1. Larger the value of a, larger is the set aQ.
Given a class of cutting planes, the corresponding closure is de
fined as the set obtained by the addition of all possible inequal
ities of this class to the linear programming relaxation. Consider
first the case of two rows. Let R(f, 0, W) where rJ e Q2 for all
rj e W, f e Q2. Then the split closure S(f, 0, W), the triangle
closure T(f, 0, W), and the quadrilateral closure Q(f, 0, W) are
obtained by intersecting the continuous relaxation of R(f, 0, W)
with the cuts obtained using (5) where M is a all possible split sets,
maximal latticefree triangles, and maximal latticefree quadrilater
als respectively. Since all the facetdefining (extreme) inequalities for
R(f, 0, W) are either splits, triangles or quadrilaterals, we obtain
conv(R(f, 0, W)) = S(f, 0, W) n T(f, 0, W) n Q(f, 0, W). Basu
et al. [9] prove the following result.
Theorem 4.12 ([9]).
I. Split versus triangle and quadrilateral closures:
I(f, 0,W) c S(f, 0,W), Q(f, 0,W) c S(f, 0,W).
2. Triangle and quadrilateral closures versus conv(R(f, 0, W)):
conv(R(f, 0,W)) c T(f, 0,W) c 2conv(R(f, 0, W)),
conv(R(f 0, W)) c Q(f 0, W) c 2conv(R(f 0, W)).
3. Split closure versus conv(R(f, 0,W)): For all a > 1, there is a
choice of W and f such that S(f, 0, W) aconv(R(f, 0, W)).
Theorem 4.12 proves that the split closure can be arbitrarily bad,
while the triangle or quadrilateral closure get us within a factor of
2 of the convex hull no matter what the instance be. Therefore,
at least theoretically, if we added all possible triangle or quadri
lateral inequalities, the resulting set is quite a strong relaxation of
conv(R(f, 0, W)).
Andersen et al. [5] extended the result of [9] to the case of
more rows. Let Yr,,wTr(rJ)yj > 1 be a valid inequality. Let
d be the dimension of the linear space spanned by vectors rJ
such that Tr(rJ) = Then the splitdimension of the inequality
rJ wTr(rJ)3y > 1 is defined as dim(L,) d (see (9) for defi
nition of L,). Let C (f, 0, W) be the the intersection of all valid
inequalities for conv(R(f, OW)) with a split dimension of at most i.
Theorem 4.13 ([5]). For any a > 1, there exist f,W (f e
Qm, rJ Qm for all r e W, IWI finite) such that Cm1 (f, 0, W) P
aconv(R (f 0, W)).
It is well known that the split closure of any mixed integer set is a
polyhedron (Cook et al. [14]). It would interesting to obtain similar
results or to provide counterexample for the new class of inequali
ties. Andersen et al. [I] provide some answers in this direction. We
first require one definition. Given a facet g'x > g0 of a latticefree
convex setM, let w(g,M) := maxxMg'Tx min Mg Tx. The facet
width of M is defined as the maximum of w(g,M) over all facets
of M. Let P c Zm x RYn be a mixed integer linear set. Let M be
latticefree convex sets in mdimensions and let M' = (u,v) e
Rm x RYn I u e Mi. As discussed in Section 3.1, the linear program
ming relaxation of P (denoted P0) can be strengthened by comput
ing U(P,M) : conv(P \ int(M')). Define the w split closure of
T to be the set nfacetwidth ofM
Theorem 4.14 ([I ]). The w split closure of is a polyhedron.
4.2 Split Rank
Another approach to compare the new families of cutting planes
with split cuts is to determine the split rank of the new families
of cutting planes. As discussed in Section 4.1 the split closure is a
polyhedral relaxation of the convex hull of a mixed integer set. It
is possible to apply the split closure procedure to the the resulting
polyhedral relaxation to obtain the second split closure. Inductively,
we define the kth split closure as the split closure of the (k )th
split closure. The split rank of an inequality is defined as the smallest
integer k such the inequality is valid for the kth split closure. Thus,
the cutting plane (8) in Example 3.1 does not have a finite split rank
since it cannot be obtained in a finite number of split closure pro
cedure. As presented next, Dey and Louveaux [17] show that this
is the 'only' interesting example with this property for the tworow
case.
Theorem 4.15 ([17]). Let rw Tr (rJ)yj > 1 be a facetdefining
inequality for R(f, 0, W) (f, rJ e Q2, IW is finite). Then the split
rank of Y,Jw T(rJ)yj > 1 is finite if and only if Lr is not a maximal
latticefree triangle of type I.
For the general case of m rows, Dey [16] presents a geometric
argument to determine a lower bound on the split rank of inter
section cuts applied to a mixed integer set of the form R(f, 0, W)
(f e Qm, rJ e Q for all rJ e W, IWI finite): Given the inequality
rJ WTr (rJ)yj > 1, first a polyhedral subset of Lr called restricted
latticefree convex is constructed (under a technical assumption on
the columns W). Then it is shown that [log2 (1)1 is a lower bound on
the split rank of the intersection cut where {x1, x2,...,x1 is a sub
set of integer points on the boundary of the restricted latticefree
set such that no two points lie on the same facet of the restricted
latticefree set. We refer the readers to [16] for details.
4.3 Computational Experiments
A first computational investigation to understand the practical im
pact of multirow cuts has been conducted by Espinoza [24]. In [24],
multirow cuts are embedded in CPLEX 10.2 default branchandcut
by using CPLEX callbacks and are separated at the root node of the
branchandcut tree, after CPLEX default cutting planes. These cuts
are generated by relaxing the simplex tableau of a general MIP as
a set of the form R(f, 0, W). Espinoza [24] considered sets with
m varying from 2 up to 15, and two classes of maximal latticefree
bounded convex sets for generating cuts using (5). The first class of
maximal latticefree convex set is a special simplex, while the second
one is the cross polytope having 2' facets.
The computational experiments provided by Espinoza [24] com
pare CPLEX branchandcut enforced with multirow cuts versus
CPLEX default on a test bed of 87 MIPLIB 3.0 and MIPLIB 2003 in
stances. Despite some negative examples in which CPLEX default
yields a better dual bound at the root node and solves the problem
faster, the reported results are overall encouraging, thus showing
that the use of multirow cutting planes may hold some potential.
A geometric average speedup over CPLEX default of 31 % can be
observed for instances in which optimality is reached using the ad
ditional cuts. Finally, as reported in [24], it is worth noting that the
performance of the multirow cuts seems to improve when these
cuts are based on a larger set of rows, since the integrality gap closed
at the root node tends to increase, while a smaller number of cuts
is typically generated.
5 Open Questions
On the theoretical side, there are a large number of open ques
tions. We mention two here. Maximal latticefree convex sets in R2
OPTIMA 80
are well understood. In R3, some properties of maximal latticefree
convex sets are known (see Scarf [35], Andersen et al. [4]). How
ever, not much is known about maximal latticefree convex sets in
higher dimensions. Another promising direction of research is to un
derstand how many of the relaxations considered to construct the
set R(f, 0, W) can be revoked, while still having the possibility of
providing a complete characterization of all extreme inequalities.
The most significant challenges are probably on the practical side.
While Espinoza [24] presents encouraging result, there are numer
ous avenues for improvement. The primary difficulty is that of cut
selection. Consider the case of two rows: the number of possible
choices of rows is 0(m2) where m is the number of rows. For
each choice of two rows, there are then a number of possibilities
in term of selection of triangles or quadrilaterals. With so many
cuts, an appropriate tool for cut selection is vital for a successful
implementation. Another practical difficulty with this class of cutting
planes is that they tend to be dense. This can affect the performance
of other components of a branchandcut algorithm. At this time
various computational experiments are underway and we hope that
much progress will be made in terms of successfully exploiting these
cutting planes.
Santanu S. Dey, H. Milton Stewart School of Industrial and Systems Engineer
ing, Georgia Institute of Technology, USA. santanu.dey@isye.gatech.edu
Andrea Tramontani, DEIS, University of Bologna, Italy.
andrea.tramontani@unibo.it
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Discussion Column
G6rard CornuBjols, Robert Weismantel and Laurence Wolsey
Comments on MultiRow Cuts
Gerard: I find the recent investigations on multirow cuts fascinating.
What really got me hooked was the paper that the two of you wrote
with Kent Andersen and Quentin Louveaux [2]. Kent showed me a
preliminary draft of the paper at the Math Programming Symposium
in Rio in 2006 and I really liked the approach. I felt that it brought
a new dimension to Integer Programming. It was a big departure
from recent investigations in the theory of cutting planes, such as
the study of mixed integer cuts which can be generated from inte
grality arguments applied to a single equation. Of course there are
connections between your work and that of Egon Balas on inter
section cuts generated from convex sets in the 1970s, or of Ralph
Gomory and Ellis Johnson on the corner polyhedron around the
September 2009
same period. A great novelty of your paper was to consider a model
where integrality only appears in the basic variables. This is a very
appealing model, which preserves much of the complexity of Inte
ger Programming and at the same time is sufficiently simplified that
one can prove a lot about it. In particular, only very special convex
sets can give rise to facets in this model. The paper that you wrote
with Kent and Quentin studies the 2row case, and you showed that
only nonnegativity, split, triangles and quadrilaterals generate facets.
What an elegant result! What got you interested in this line of re
search, Laurence? What about you Robert?
Laurence: Hard to remember. However I, like G6rard, was frus
trated by the single row case in which he and others had failed to
produce anything stronger than the Gomory Mixed Integer cuts. I
was intrigued by the CookKannanSchrijver example with infinite
convergence [4], and for years the question of handling two rows
whether 0I knapsacks or other, was a problem that I kept coming
back to. I also was taken by the computational work of Kent Ander
sen, Yanjun Li and G6rard [I] in which they heuristically combined
rows so as to get cuts with small coefficients on the continuous
nonbasic variables.
So when Kent came to Core, we almost immediately decided to
look a the two row group problem. What was interesting was the
fact that the problem with just two continuous variables was equiv
alent to a two variable IP which meant there were compact ex
tended formulations for the problem with continuous nonbasics.
Also the CookKannanSchrijver example fits the two constraint
case. We soon found ourselves looking at latticefree triangles (back
to Balas intersection cuts and Ellis Johnson's gauge functions), and
managed to generate some strong valid inequalities. However the
real progress was then made by Kent, Robert and Quentin in Magde
burg.
Later when Santanu Dey came to Core, my main concern was
how to deal with integer nonbasic variables, which again led him far
beyond what was initially asked.
Robert: My point of departure was that in the mixedinteger set
ting no generally finite cutting plane algorithm is known. In particular,
the paper [4] shows that by using regular split cuts one cannot al
ways terminate in finite time, even not for problems with two integer
variables and one continuous one. The model that we introduced in
[2] allows us to generate a triangle cut that leads to finite termina
tion of a cutting plane algorithm for those low dimensional exam
ples. From my perspective, the important feature of the model is
the correspondence between facets for such models and maximally
latticepointfree polyhedra. This link has been established by now
quite well in a series of followup papers such as [5], [7] and exten
sions by you, G6rard, Michele Conforti, Giacomo Zambelli, Amitabh
Basu and further papers by Santanu Dey and Laurence or by Kent
Andersen, Christian Wagner and myself. I would like to comment
a bit further on this topic, namely the link between cutting planes
and latticepointfree polyhedra. In fact, for any polyhedron L with
no interior integer points and any polyhedron P we can generate
inequalities that are valid for the mixed integer points in P by taking
the convex hull of all points that are in P, but not in the interior
of L. In order to generate strong cuts this way we rather make L
as large as possible (Maximally latticepointfree). Every maximally
latticepointfree polyhedron is the Minkowski sum of a polytope
and a linear space whose dimension is equal to the codimension
of the polytope part. Clearly, the complexity of the cuts increases
with the dimension of this polytope part. Indeed, it was shown by
Jorg [I ] that if one allows for arbitrary disjunctions, then a finite
cutting plane algorithm for mixed integer optimization can be devel
oped. It remains, however, open to understand precisely the con
nection between the geometry of the latticepointfree polyhedron
and the event of finite termination. More precisely, suppose that
one starts with cuts arising from parallel splits. After a certain num
ber of rounds it is expected that the progress in terms of improve
ment of the objective function becomes small (tailing off). Once the
improvement in terms of the objective function is below a certain
threshold, disjunctions based on two and even higher dimensional
latticepointfree bodies should be applied. Which conditions must
such a body satisfy in order to avoid this tailing off effect? Answers
to this question will allow us to design a hierarchical scheme for
using more and more complex latticepointfree polyhedra as a cut
generation machine. Finally it remains to comment on the question
how far this approach will take us in terms of computations. I am
very optimistic and hope to share some promising experimental re
sults with all of you by the end of the year. Kent Andersen, Christian
Wagner and I are currently working on that.
G6rard Thanks for the insights into what motivated you both to
studying 2row cuts and why this area is so exciting! There are two
key relaxations in the model you proposed in [2]. One is relaxing
integrality of the nonbasic variables and the other is relaxing non
negativity of the basic variables. The latter relaxation is the classical
group relaxation of Gomory. The first is what really appealed to
me and drew me into studying this area myself. A natural question
is: What happens if one only makes one of these two relaxations?
Can one say anything interesting building on the results of [2]? Lau
rence and Santanu have looked at both strengthening of the model
in [2] and came up with two great papers [9], [10]. The first paper
lifts minimal inequalities from [2] into valid inequalities for the group
problem and shows that in several interesting cases the minimal lift
ing function is unique. This is important work. But I find the second
paper even more surprising. A development that I did not antici
pate is that much of the theory in [2] and subsequent papers goes
through even when only the first relaxation occurs. In other words,
the key to the beautiful connection to maximal latticefree convex
sets is not the group relaxation but rather relaxing integrality of the
nonbasic variables. This was nicely brought out in a paper that Lau
rence and Santanu wrote this Spring [10]. There are new difficulties
since now facets can have negative coefficients and the connection to
maximal latticefree convex sets is not quite so simple. But most of
the theory goes through. To me this means that the greatest insights
come not from the group relaxation but from the other relaxation,
namely relaxing integrality of the nonbasic variables. No doubt this
is a controversial statement. Any comments?
Laurence I like G6rard's viewpoint, though it may just be a case
of transferring difficulties elsewhere. In particular Michele, G6rard
and Giacomo [6] have just shown the generality of this model. By
augmenting the number of rows and columns, they show how to
reformulate the group relaxation as an equivalent problem with in
teger basic variables under some additional constraints and nothing
but realvalued nonnegative nonbasic variables.
There are several other intriguing questions. Santanu Dey and
Quentin Louveaux [8] have shown that for two row problems the
only valid inequalities that have infinite split rank are those arising
from the triangles with integer vertices corresponding to the Cook
KannanSchrijver example. Do maximal latticefree bodies, studied
by Kent Andersen, Christian Wagner and Robert [3] also play a spe
cial role in higher dimensions? Does this mean that in the 2row
case a cutting plane algorithm based on Gomory mixed integer cuts
solves finitely for all other objective functions?
Another wide open question concerns the links between branch
ing and cutting, see the experiments of Miroslav Karamanov and
G6rard [12] and the work of Jorg cited above. Should we be using
three way branching rather than trying to use inequalities based on
latticefree triangles, etc.?
OPTIMA 80
Finally the number of computational options for generating facets
off relaxations from two or more rows are immense, so we wait with
baited breath for the magic recipe. There's still much work here for
everyone.
References
[I] K. Andersen, G. Cornuejols and Y. Li, Reduceandsplit cuts: Improving
the performance of mixed integer Gomory cuts, Management Science 5 I
(2005) 17201732.
[2] K. Andersen, Q. Louveaux, R. Weismantel and L. Wolsey, Cutting planes
from two rows of a simplex tableau, Proceedings oflPCO XII, Ithaca, New
York (2007) 115.
[3] K. Andersen, C. Wagner and R. Weismantel, On an analysis of the strength
of mixed integer cutting planes from multiple simplex tableau rows, manuscript
(December 2008).
[4] W.J. Cook, R. Kannan, A. Schrijver, Chvital closures for mixed integer pro
gramming, Mathematical Programming 47(1990) 155174.
[5] V. Borozan, G. Cornuejols, Minimal valid inequalities for integer con
straints, (2007), Mathematics of Operations Research, to appear.
[6] M. Conforti, G. Cornuejols, G. Zambelli, A geometric perspective on lifting,
manuscript (April 2009).
[7] G. Cornuejols, F. Margot, On the facets of mixed integer programs with
two integer variables and two constraints, Mathematical Programming A 120
(2009) 429456. Conference version in LATIN 2008, 317328.
[8] S.S. Dey, Q. Louveaux, Split rank of triangle and quadrilateral inequalities,
CORE DP 2009/xx, August 2009.
[9] S.S. Dey, L.A. Wolsey, Lifting integer variables in minimal inequalities corre
sponding to latticefree triangles, IPCO 2008, Bertinoro, Italy (May 2008),
Lecture Notes in Computer Science 5035, 463475.
[10] S.S. Dey, L.A. Wolsey, Constrained Infinite Group Relaxations of MIPs,
manuscript (March 2009).
[II] M. Jorg, kdisjunctive cuts and a finite cutting plane algorithm for general mixed
integer linear programs, manuscript, arXIV:0707.3945vl, (2007).
[12] M. Karamanov and G. Cornuejols, Branching on general disjunctions, Techni
cal report, Carnegie Mellon University, (2005). URL http://integer.tepper.
cmu.edu
Prizes Presented at ISMP
George B. Dantzig Prize
Jointly awarded by the MPS and the Society of Industrial and Ap
plied Mathematics (SIAM). Committee: Yurii Nesterov, JongShi Pang
(Chair), Lex Schrijver, Eva Tardos.
The George B. Dantzig prize is awarded for original research,
which by its originality, breadth and scope, is having a major impact
on the field of mathematical programming. The winner is Professor
G6rard CornuBjols from CarnegieMellon University, for his deep
and wideranging contributions to mathematical programming, in
cluding his work on Balanced and Ideal Matrices and Perfect Graphs
and his leading role in the work on general cutting planes for mixed
integer programming over many years covering both theory and
computation.
The BealeOrchardHays Prize
Committee: Erling Andersen, Philip Gill, Jeff Linderoth, Nick Sahini
dis (chair).
This Prize is sponsored by the Society in memory of Martin Beale
and William OrchardHays, pioneers in computational mathematical
programming. The Prize is given for excellence in any aspect of com
putational mathematical programming. 'Computational mathematical
programming' includes the development of highquality mathemati
cal programming algorithms and software, the experimental evalua
tion of mathematical programming algorithms, and the development
of new methods for the empirical testing of mathematical program
ming techniques.
Selecting a prize winner this year was challenging since we re
ceived thirteen exceptional nominations. The nominated works
spanned a broad spectrum of areas, including linear programming,
semidefinite programming, nonlinear programming, integer pro
gramming, stochastic programming, and global optimization. In re
flection of the Mathematical Programming Society's international
character, these nominations originated from ten different coun
tries.
The 2009 Prize was awarded to Tobias Achterberg for his pa
per "SCIP: Solving constrained integer programs," Mathematical Pro
gramming Computation, I (2009), pp. 141, which is the first paper
to appear in the Society's new journal Mathematical Programming
Computation. This paper describes an innovative paradigm to inte
grate modeling capabilities and solution techniques from constraint
programming, mixedinteger programming, and satisfiability. These
techniques are carefully integrated after extensive computational ex
perimentation that resulted in the software SCIP, which consists of
more than 250,000 lines of code, all written by the author himself,
and which is the focus of the paper. The source code of SCIP is
available free for academic use. Compared to previous branchand
bound systems, SCIP achieves a tighter integration of the aforemen
tioned modeling and solution paradigms. In addition, it implements
several novel algorithmic constructs that have been developed in re
cent papers by the author and coworkers, including branching rule
scores, cutting plane handling, and conflict analysis. It is truly remark
able that SCIP has performed with times that are within a modest
factor of those for the best commercial codes for mixedinteger pro
grams. This fact alone is one of the best confirmations of the quality
of the work done by the author. An additional strength of SCIP is
the ease with which researchers can create branchcutandprice ap
plications by implementing "plugins" for the problemspecific rou
tines. Finally, the computational results in the paper show that the
approach developed by the author outperforms current stateof
theart techniques for proving the validity of properties on circuits
containing arithmetic.
Fulkerson Prize
Jointly awarded by the MPS and the American Mathematical Society
(AMS). Committee: William Cook (chair), Michel Goemans, Daniel
Kleitman.
Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas, The
strong perfect graph theorem, Annals of Mathematics 164 (2006) 5 1229.
Claude Berge introduced the class of perfect graphs in 1960 to
gether with a possible characterization in terms of forbidden sub
graphs. The resolution of Berge's strong perfect graph conjecture
quickly became one of the most sought after goals in graph the
ory. The pursuit of the conjecture brought together four important
elements: vertex colorings, stable sets, cliques, and clique covers.
Moreover, D. R. Fulkerson established connections between perfect
graphs and integer programming through his theory of antiblocking
polyhedra. A graph is called perfect if for every induced subgraph
H the cliquecovering number of H is equal to the cardinality of its
largest stable set. The strong perfect graph conjecture states that a
graph is perfect if and only if neither it nor its complement contains
as an induced subgraph an odd circuit having at least five edges. The
elegance and simplicity of this possible characterization led to a great
body of work in the literature, culminating in the Chudnovsky et al.
proof, announced in May 2002, just one month before Berge passed
away. The long, difficult, and creative proof by Chudnovsky et al. is
one of the great achievements in discrete mathematics.
September 2009
Thomas C. Hales, A proof of the Kepler conjecture, Annals of Mathematics
162 (2005) 10631 183.
Samuel P. Ferguson, Sphere Packings, V. Pentahedral Prisms, Discrete and
Computational Geometry 36 (2006) 167204.
In 16 II Johannes Kepler asserted that the densest packing of equal
radius spheres is obtained by the familiar cannonball arrangement.
This statement is known as the Kepler conjecture and it is a com
ponent of Hilbert's 18th problem. After four centuries, Ferguson
and Hales have now proven Kepler's assertion. The FergusonHales
proof develops deep connections between sphere packing and
mathematical programming, making heavy use of linear programming
duality and branch and bound to establish results on the density of
candidate configurations of spheres. The beautiful geometric argu
ments and innovative use of computational tools make this a land
mark result in both geometry and discrete mathematics.
Daniel A. Spielman and ShangHua Teng, Smoothed analysis of algorithms:
Why the simplex algorithm usually takes polynomial time, journal of the ACM
5 I (2004) 385463.
George Dantzig's simplex algorithm for linear programming is a
fundamental tool in applied mathematics. The work of Spielman
and Teng is an important step towards providing a theoretical un
derstanding of the algorithm's great success in practice despite
its known exponential worstcase behavior. The smoothed analy
sis introduced by the authors fits nicely between overly pessimistic
worstcase results and the averagecase theory developed in the
1980s. In smoothed analysis, the performance of an algorithm is
measured under small perturbations of arbitrary real inputs. The
SpielmanTeng proof that the simplex algorithm runs in polynomial
time under this measure combines beautiful technical results that
intersect multiple areas of discrete mathematics. Moreover, the gen
eral smoothed analysis framework is one that can be applied in many
algorithmic settings and it is now established as an important tech
nique in theoretical computer science.
Lagrange Prize
Jointly awarded by the MPS and the SIAM. Committee: Adrian Lewis
(Chair), Jorge More, Philippe Toint, Margaret Wright.
Jean B. Lasserre: A sum of squares approximation of nonnegative polynomi
als, SIAM journal on Optimization 16 (2006), 751765. (The paper was also
reproduced in the SIGEST section of SIAM Review 49 (2007), 651669.)
Lasserre has been a pioneer in the field of global polynomial opti
mization since his 2001 paper "Global optimization with polynomials
and the problem of moments", SIOPT I I, 796817. His approach
relies on certificates of positivity for polynomials on compact sets
defined by polynomial inequalities, using representations as sums of
squared polynomials a deep and powerful technique taken from
real algebraic geometry. Using semidefinite programming to recog
nize sums of squares, Lasserre constructs a hierarchy of semidefi
nite relaxations converging to a solution of the (NP hard!) global
optimization problem. The convergence is often fast or even finite
in practice, as illustrated in software packages such as GloptiPoly,
SOSTOOLS and SPARSEPOPT The duality theory for the semidefi
nite relaxations corresponds exactly to the duality between polyno
mial optimization and generalized moment problems, a relationship
explored in great generality in Lasserre's paper "A semidefinite pro
gramming approach to the generalized problem of moments", MPB
I 12 (2008), 6592.
The winning paper is a particularly striking exemplar of Lasserre's
work on polynomial optimization. He presents an elegant new proof
of a classical foundation stone for the theory: that any nonnega
tive polynomial can be approximated by sums of squares. Combining
convex duality theory and a momenttheoretic result, he constructs
approximating polynomials that are both simple and explicit. One
consequence is a satisfying simplification of his original relaxation
hierarchy. The paper is a beautiful blend of modern optimization
theory and deep classical mathematics, with striking computational
implications.
Tucker Prize
Committee: Frederic Bonnans, Fritz Eisenbrand, Sven Leyffer, Franz
Rendl (chair), Ruediger Schultz.
The Tucker Prize for an outstanding paper authored by a student
has been awarded to Mohit Singh, Microsoft Research, Cambridge,
for his thesis: Iterative Methods in Combinatorial Optimization.
Mohit Singh obtained his undergraduate degree in Computer Sci
ence and Engineering from the Indian Institute of Technology, Delhi
in 2003. He completed his Ph.D. in the Algorithms, Combinatorics
and Optimization program from Tepper School of Business, CMU
under the supervision of Prof. R. Ravi in May 2008. He is currently a
postdoctoral researcher at Microsoft Research, New England. His
research interests lie in theoretical computer science, combinatorial
optimization, and approximation algorithms. He published two or
more papers in each of the following prestigious conferences STOC,
FOCS, IPCO and APPROX.
Singh's thesis introduces a new technique for solving important
optimization problem exactly and extends it to solving their NP
hard variants obtained by introducing complicating side constraints.
The method, iterative in nature, suggests a natural recursive algo
rithm for obtaining exact solutions to problems by writing a care
fully constructed linear programming relaxation and arguing that the
relaxation always has a zero or oneelement in an optimal basic so
lution. The crux of the argument exploits the fact that the set of
tight constraints at a basic solution can be represented by a sparse
family, hence the cardinality of the support of the solution is also
sparse.
The thesis shows a large set of problems amenable to this tech
nique ranging from spanning trees in directed and undirected graphs,
minimal matroid bases and perfect matching.
While it is an impressive contribution to offer novel proofs of
several classical polyhedral results with implications to exact algo
rithms, the thesis makes an even larger contribution by showing how
the method can be extended to handle side constraints.
As an example, consider the classical Minimum Spanning Tree
problem with upper bounds on the degrees of the various nodes
in the tree. If the current fractional support of nonzero edges have
degree at most one more than the degree bound at some node, the
degree constraint for this node can be dropped since in the worst
case, even if all these edges in the fractional support were chosen in
the final solution, the degree constraint will only be violated by one.
Singh's striking result (obtained jointly with Lau) shows that if there
are no zero or onevalued edges, there will always be such a degree
constraint on some node that one can drop and continue iteratively
looking for more zero or oneedges (to delete or include) or more
constraints to relax with low violation. This settles a longstanding
conjecture for this problem due to Goemans affirmatively.
Singh's thesis carefully rounds out the various other classes of
optimization problems where such side constraints can be handled
approximately; He derives new approximation results for general
connectivity problems (such as survivable network design) with side
degree constraints, as well as for multicriteria spanning tree and ma
troid bases problems. The thesis also gives new short proofs of var
ious constrained optimization problems such as the generalized as
signment problem using the same relaxation framework.
OPTIMA 80
Singh's thesis research has already attracted follow up work in
several prestigious conferences such as STOC 2008, IPCO 2008 and
FOCS 2008, and continues to generate a flurry of research activity
around new applications of these techniques.
This is a very impressive dissertation, which by its breath and
depth qualifies the work as the winner of the 2009 A.W. Tucker
Prize.
The other two Tucker Prize finalists chosen by this year's Tucker
Prize Committee are Tobias Achterberg and Jiawang Nie.
Tobias Achterberg studied at the Technical University of Berlin
where he finished both his master and his doctoral studies. The title
of his dissertation is: Constraint Integer Programming. It was super
vised by Martin Grotschel, and finished during the summer of 2007.
He is currently with IBMCPLEX as a software developer.
In his thesis, Achterberg discusses the integration of techniques
from mixedinteger programming (MIP), constraint programming
(CP), and satisfiability (SAT) solving. All three areas deal with opti
mization or feasibility problems over integer variables, which can be
solved by tree search algorithms. Numerous industrial applications
can be modeled as MIP, CP, or SAT. In particular, MIP has drawn a lot
of academic and commercial attention.
Achterberg introduces the concept of constraint integer pro
gramming (CIP), which generalizes mixedinteger programming in
order to carry over the powerful modeling techniques from con
straint programming to mixedinteger programming. He makes use
of the entire theory of constraint and integer programming and
compares their theoretical advantages and disadvantages. He ana
lyzes the different techniques carefully by conducting practical ex
periments, and he implements all variants with outstanding program
quality and remarkable attention to detail. This results in the soft
ware SCIP, which consists of more than 250,000 lines of code. Al
though CIP is a more general problem class than MIP, the perfor
mance of SCIP on pure MIP instances is comparable to the best
commercial MIP codes.
By now, the code of Achterberg is wellknown and recognized
in the academic world. Independent researchers identified it as the
best noncommercial code for MIP, and it is used in a variety of aca
demic and industrial projects. The fact that the source code of the
software is freely available opens up new research possibilities in the
area of optimization.
Jiawang Nie did his undergraduate studies in China, finishing with
a master of science in 2000 at the Chinese Academy of Sciences.
He then moved to the University of California, Berkeley, where
he wrote a dissertation entitled: Global Optimization of Polynomial
Functions and Applications, supervised by James Demmel and Bernd
Sturmfels. The thesis was finished in the fall of 2006. Jiawang Nie is
currently assistant professor at the University of California in San
Diego.
Nie's dissertation focuses on the interplay between optimiza
tion with polynomial functions and semidefinite programming (SDP).
More precisely, he showed that quite general (and extremely diffi
cult) problems in polynomial optimization could be solved by a con
verging sequence of approximations, each efficiently computable us
ing recently developed techniques of SDP, and he formulated and
solved engineering design optimization problems using these tech
niques.
One key ingredient lies in the observation that a polynomial is
necessarily nonnegative if it has a 'sums of square' (SOS) representa
tion through other polynomials. In the dissertation he explores this
idea applied both in the context of minimizing polynomials via Sum
of Squares over the Gradient Ideal, and also through representa
tions of positive polynomials on noncompact semialgebraic sets via
KarushKuhnTucker ideals. His work improves on previous results
of Lasserre, Jibetean, Laurent and Parrilo by removing assumptions
such that the gradient variety must be finite. By exploiting the Kuhn
KarushTucker condition, he extends his results from unconstrained
to constrained optimization. These are interesting and far reaching
results, and make a significant improvement over the prior best re
sults in this area.
These theoretical results are successfully applied in various areas
of engineering, for instance shape optimization, perturbation analysis
of polynomial systems of equations, or sensor network location.
Nie currently has more than a dozen publications in top qual
ity journals on optimization, such as Mathematical Programming and
SIOPT.
Ignacio E. Grossmann and Jon Lee
New MINLP Cybersite
In a joint collaboration between Carnegie Mellon University and
the IBM TJ. Watson Research Center, researchers have launched a
Cyberlnfrastructure Collaborative site for MixedInteger Nonlinear
Programming (MINLP): www.minlp.org, that has been funded by the
National Science Foundation under Grant OCI0750826: "Open Cy
berlnfrastructure for Mixedinteger Nonlinear Programming: Col
laboration and Deployment via Virtual Environments." The core
team consists of: Larry Biegler, Ignacio E. Grossmann, Francois Mar
got and Nick Sahinidis of CMU, and Jon Lee and Andreas Wachter
of IBM. Additional collaborators include: Pietro Belotti (Lehigh Uni
versity), Pedro Castro (INETI) and Juan Ruiz (CMU).
The major goal of this site is to create a library of optimization
problems in different application areas in which one or several alter
native models are presented with the derivation of their mathemati
cal formulations. In addition, each model has one or several instances
that can serve to test various algorithms. While we are emphasizing
MINLP models, you may also wish to submit MILP and NLP mod
els that are particularly relevant to problems that also have MINLP
formulations. The site is intended to provide a mechanism for re
searchers and users to contribute towards the creation of the library
of optimization problems, and to provide a forum of discussion for
algorithm developers and application users where alternative formu
lations, as well as performance and comparison of algorithms can be
discussed. The site also provides information on various resources,
meetings and a bibliography.
We are looking for new contributions to expand the library of
test problems, and also for feedback on this site. Comments are
welcome at: minlp@andrew.cmu.edu.
Ignacio E. Grossmann, Carnegie Mellon University, grossmann@cmu.edu
Jon Lee, IBM T.J. Watson Research Center, jonlee@us.ibm.com
IMPRINT
Editor: Andrea Lodi, DEIS University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. andrea.lodi@unibo.it CoEditors: Alberto Caprara, DEIS Uni
versity of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. alberto.caprara@unibo.it Katya Scheinberg, Department of Industrial Engineering and Operations
Research, Columbia University, 500W 120th Street, New York, NY, 10027. katyascheinberg@gmail.com Founding Editor: Donald W. Hearn Published
by the Mathematical Programming Society. m Design and typesetting by Christoph Eyrich, Mehringdamm 57 / Hof 3, 10961 Berlin, Germany. optima@0x45.de
* Printed by Oktoberdruck AG, Berlin.
