Title: Optima
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Language: English
Creator: Mathematical Programming Society, University of Florida
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Place of Publication: Gainesville, Fla.
Publication Date: January 2006
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SMathematical Programming Society Newsleer
I Mathematical Programming Society Newsletter


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Local Branching:

Basics and Extensions

Matteo Fischetti*
Andrea Lodit

*DEI, University ofPadova
Via Gradenigo 6/A, 35100 Padova, Italy

DEIS, University A'f,',i,.,',:
Viale R.,S.,,.'., 2, 40136 Bologna, Italy

tIBM T.. Watson Research Center
P.O. Box 218, Yorktown Heights, NY10598

E-mail: matteo.fischetti@unipd.it

July 5, 2005

The availability of effective exact or heuristic
solution methods for general Mixed-
Integer Programs (MIPs) is of paramount
importance for practical applications. In
the present paper we investigate the use
of a generic MIP solver as a black-box
"tactical" tool to explore effectively suitable
solution subspaces defined and controlled
at a "strategic" level by a simple external
branching framework. The procedure is
in the spirit of well-known local search
metaheuristics, but the neighborhoods
are obtained through the introduction
in the MIP model of completely general
linear inequalities called local branching
cuts. The new solution strategy is exact in
nature, though it is designed to improve
the heuristic behavior of the MIP solver
at hand. It alternates high-level strategic
branchings to define the solution
neighborhoods and low-level tactical
branchings to explore them. The result
is a completely general scheme aimed at
favoring early updatings of the incumbent
solution, hence producing high-quality
solutions at early stages of the computation.

1 Introduction
Mixed-Integer linear Programming
(MIP) plays a central role in modeling
difficult-to-solve (NP-hard) combinatorial
problems. However, the exact solution
of the resulting models often cannot be
carried out for the problem sizes of interest
in real-world applications; hence, one is
interested in effective heuristic methods.
Although several heuristic and metaheuristic
frameworks have been proposed in the
literature for specific classes of problems,
only a few papers deal with general-purpose
MIP heuristics, including [1], [8], [9],
[11], [12], [14] and [3] among others.

Exact MIP solvers are nowadays very
sophisticated tools designed to hopefully
deliver, within acceptable computing time,
a provable optimal solution of the input
MIP model, or at least a heuristic solution
with a practically-acceptable error. In fact,
what matters in many practical cases is the
possibility of finding reasonable solutions
as early as possible during the computation.
In this respect, the "heuristic behavior"
of the MIP solver plays a very important
role: An aggressive solution strategy that

improves the incumbent solution at very
early stages of the computation is strongly
preferred to a strategy designed for finding
good solutions only at the late steps of the
computation (that for difficult problems
will unlikely be reached within the time
limit). Many commercial MIP solvers
allow the user to have a certain control
on their heuristic behavior through a set
of parameters affecting the visit of the
branching tree, the frequency of application
of the internal heuristics, the fact of
emphasizing the solution integrality rather
than its optimality, etc. Unfortunately,
in some hard cases a general-purpose
MIP solver may prove not adequate even
after a clever tuning, and one tends to
quit the MIP framework and to design
ad-hoc heuristics for the specific problem
at hand, thus losing the advantage of
working in a generic framework.

Recently, Fischetti and Lodi [5] investigated
the use of a general-purpose MIP solver
as a black-box "tactical" tool to explore
effective suitable solution subspaces defined
and controlled at a "strategic" level by a
simple external branching framework.
The procedure is in the spirit of well-
known local search metaheuristics, but
the neighborhoods are obtained through
the introduction in the MIP model of
(invalid) linear inequalities called local
branching cuts. This allows one to work
within a perfectly general MIP framework,
and to take advantage of the impressive
research and implementation effort
devoted to the design of MIP solvers.

A main point: Soft vs. hard variable fixing.
A commonly used, and often effective,
heuristic scheme fitting into the framework
described in the introduction is based on
the so-called (hard) variable fixing or diving
idea, which can be described as follows.
We assume to have an exact or heuristic
black-box solver for the problem at hand.
The solver is first applied to the input data,
but its parameters are set so as to quickly
abort execution and return a 'p....b-l,
infeasible) "target solution" x. This solution
is defined, e.g., as the solution of the root-
node Linear Programming (LP) relaxation,
possibly after a clever rounding of some of
its fractional variables, or as any heuristic
solution of the problem. Solution xis then
analyzed, and some of its nonzero variables

0SP T I MA 2



are heuristically rounded up to the nearest
integer (if non-integer) and then fixed to
this value. The method is then iteratively re-
applied on the restricted problem resulting
from fixing: The black-box solver is called
again, a new target solution is found,
some of its variables are fixed, and so on.
In this way the problem size reduces after
each fixing, hence the black-box solver
can concentrate on smaller and smaller
core problems" with increasing chances
of solving them to proven optimality.

A critical issue in variable-fixing methods
is of course related to the choice of the
variables to be fixed at each step. As a
matter of fact, for difficult problems high-
quality solutions are only found after several
rounds of fixing. On the other hand, wrong
choices at early fixing levels are typically
very difficult to detect, even when bounds
on the optimal solution value are computed
before each fixing: In the hard cases, the
bound typically remains almost unchanged
for several fixings, and increases suddenly
after an apparently-innocent late fixing.
Therefore, one has to embed in the scheme
backtracking mechanisms to recover
from bad fixings, a very difficult task.

The question is then how to fix a relevant
number of variables without losing the
possibility of finding good feasible solutions.
To better illustrate this point, suppose
one is given a heuristic 0-1 solution x ofa
pure 0-1 MIP model with n variables, and
wants to concentrate on a core subproblem
resulting from fixing to 1 at least 90% (say)
of its nonzero variables. How should one
choose the actual variables to be fixed? Put
in these terms, the question lends itself to a
simple answer: Just add to the MIP model
a linear softfixing constraint of the form

YJX, 0.9 J
J J (1)

and apply the black-box solver to the
resulting MIP model. In this way one
avoids a too-rigid fixing of the variables
in favor of a more flexible condition
defining a suitable neighborhood of the
current target solution, to be explored by
the black-box solver itself. In the example,
the underlying hypothesis is that the 10%
of slack left in the right-hand side of (1)
drives the black-box solver as effectively
as fixing a large number of variables, but

with a much larger degree of freedom-
hence better solutions can be found.

Good results using a hard fixing policy
have been reported recently by Danna, Le
Pape and Rothberg [3]. Their method called
Relaxation Induced Neighborhood Search
hard fixes in some nodes of a branch-and-
cut tree each integer variable xi that assumes
an integer value in the current continuous
relaxation, which is in turn coincident
with its value in the incumbent solution.
An eventually smaller additional MIP is
then defined and heuristically explored in
the attempt of improving the incumbent
solution. The method appears in the arsenal
of ILOG-Cplex since the recent version 9.0.

2 Local branching as a heuristic for
The soft fixing mechanism outlined in
the previous section leads naturally to the
general framework exploited in [5] to find
good approximate solutions for hard MIPs.
This framework, which is exact in nature, is
designed to improve the heuristic behavior
of the MIP solver at hand. It alternates
high-level strategic branchings to define
solution neighborhoods and low-level
tactical branchings (performed within the
MIP solver) to explore them. The result can
then be viewed as a two-level branching
strategy aimed at favoring early updatings
of the incumbent solution, hence producing
improved solutions at early stages of the
computation. More precisely, we consider a
generic MIP with 0-1 variables of the form:


Here, the
is partition
B 0 is t
while the
( inrlpY t

neighborhood WV( x, k) of x as the set of
the feasible solutions of (P) satisfying the
additional local branching constraint:

A(x,T):= (1- x) + Y x JS J SB\S
where the two terms in left-hand side count
the number of binary variables flipping
their value (with respect to x) either from
1 to 0 or from 0 to 1, respectively.

In the relevant case in which the cardinality
of the binary support of any feasible
solution of (P) is a constant, this constraint
can more conveniently be written in
its equivalent "asymmetric" form:
(1 -x ) < k' (= k / 2) (8)
The above definition is consistent, e.g.,
with the classical k'-OPT neighborhood
for the Traveling Salesman Problem (TSP),
where constraint (8) allows one to replace
at most k'edges of the reference tour x.

As its name suggests, the local branching
constraint can be used as a branching
criterion within an enumerative scheme
for (P). Indeed, given the incumbent
solution x, the solution space associated
with the current branching node can be
partitioned by means of the disjunction:

A(x,T) k

(left branch)

A(x,T) 2 k +1 (right branch) (9)
A 1 1 1 1 I .

As to me neignDornooa-size parameter
k, it should be chosen as the largest value
producing a left-branch subproblem which
Ax > b (3) is likely to be much easier to solve than
the one associated with its father. The
x e {0,1} V 0 (4) idea is that the neighborhood L( x, k)
> 0, integer Vj e G (5) corresponding to the left branch must be
"sufficiently small" to be optimized within
x, 2 0 j E C (6) short computing time, but still "large
variable index set N. = 1, . ., n} enough" to likely contain better solutions
tned into (B, G, Q, where than x. According to our computational
:he index set of the 0-1 variables, experience, the choice of k is seldom a
possibly empty sets q and problem by itself, in that values of k in range
he Oen e intreOr anl the [10, 20] proved effective in most cases.

continuous variables, respectively.

Given a feasible reference solution xof
(P), let S: = {j eS: = = 1} denote the
binary support ofx. For a given positive
integer parameter k, we define the k-OPT

A first implementation of the local
branching idea is illustrated in Figure
1, where the triangles marked by the
letter "T" (for Tactical) correspond to
the branching subtrees to be explored

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through a standard "tactical" branching
criterion such as, branching on fractional
variables-i.e., they represent the application
of the black-box exact MIP solver.

In the figure, we assume to have a starting
incumbent solution x 1 at the root node
1. The left-branch node 2 corresponds
to the optimization within the k-OPT
neighborhood N( x' k), which is
performed through a tactical branching
scheme converging (hopefully in short
computing time) to an optimal solution in
the neighborhood, say x2. This solution
then becomes the new incumbent solution.
The scheme is then re-applied to the right-
branch node 3, where the exploration of
V( x 2, k) \ NW(x k) at node 4 produces a
new incumbent solution 3. Node 5 is then
addressed, which corresponds to the initial
problem (P) amended by the two additional
constraints A(x, x ) > k + 1 and A(x, x2) >
k + 1. In the example, the left-branch node
6 produces a subproblem that contains no
improving solution. In this situation the
addition of the branching constraint
A(x, x 3) > k + 1 leads to the right-branch
node 7, which is explored by tactical
branching. Note that the fractional LP
solution of node 1 is not necessarily cut off
in both son nodes 2 and 3, as is always the
case when applying standard branching on
variables. The same holds for nodes 3 and
5. In fact, the local branching philosophy
is quite different from the standard one:
We do not want to force the value of a
fractional variable, but we rather instruct
the solution method to explore first some

A(z,~t)< k

initial solution 21

_A(z, +k+l

A(a'.i) 16Xe

no improved solution
Figure 1: The basic local branching scheme.

promising regions of the solution space. The
advantage of the local-branching scheme
is an early (and more frequent) update of
the incumbent solution. In other words,
we quickly find better and better solutions
until we reach a point where local branching
cannot be applied any more (node 7, in the
example); hence, we have to resort to tactical
branching to conclude the enumeration.

This behavior is illustrated in Figure 2,
where we solved MIP instance tr24-15
[15] by means of three codes: The
commercial solver ILOG-Cplex 7.0 in
the two versions emphasizing the solution
optimality or feasibility, respectively,
and the local branching scheme where
ILOG-Cplex 7.0 (optimality version) is
used to explore the "T-triangle" subtrees,
and the local branching constraints are
of type (7) with k = 18. Apart from the
emphasizing setting, all the three codes
were run with the same parameters. As
to the initial reference solution x 1 needed
in the local branching framework, it was
obtained as the first feasible solution
found by ILOG-Cplex 7.0 (optimality
version)-the corresponding computing time
is included in the local-branching running
time. The test was performed on Digital
Alpha Ultimate Workstation 533 MHz.

According to the figure, the performance
of the local branching scheme is quite
satisfactory in that it is able to improve the
initial solution several times in the early
part of the computation. As a matter of fact,
the local-branching incumbent solution is
significantly better than that of the two


136650 -




other codes during almost the entire run.
As to optimality convergence, the local
branching method concludes its run after
1,878 CPU seconds, whereas ILOG-Cplex
7.0 in its optimization version converges
to optimality within 3,827 CPU seconds
(the feasibility version is unable to prove
optimality within a time limit of 6,000
CPU seconds). Note, however, that the
enhanced convergence behavior of the local
branching scheme in proving optimality
cannot be guaranteed in all cases. Indeed,
the framework described in Figure 1 has
been specialized to obtain good approximate
solutions for hard MIPs by using effective
ideas from the metaheuristic area such as
a time limit on the exploration of each
node and sophisticated diversification
strategies. For space limit, we do not include
the final description of the framework
(see [5] for details), but an example of
the execution on the hard MIP instance
B1C1S1 [15] is reported in Figure 3.
Complete results are given in [5].

3 Local branching extensions
The main idea discussed in Section 1
opens many interesting fields of application
in which the basic local branching
idea can extend its range of use.

Tighter integration within the MIP solver.
There are two ways of exploiting local
branching constraints. The first uses
the local branching constraints as a
"strategic" branching rule within an exact
solution method, to be alternated with a
more standard "tactical" branching rule.
This approach, described in Section 2,

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 2: Solving MIP instance tr24-15 (solution value vs. CPU seconds).

0 --T I M A -



uses the MIP solver as a black-box for
performing the tactical branchings. This
is remarkably simple to implement, but
has the disadvantage of wasting part
of the computational effort devoted,
to the exploration of the nodes where
no improved solution could be found
within the node time limit. Therefore, a
more integrated (and flexible) framework
where the two branching rules work
in tight cooperation is expected to
produce an enhanced performance.

Local search by branch-and-cut.
A second way of using the local branching
constraints is within a genuine heuristic
framework akin to Tabu Search (TS) or
Variable Neighborhood Search (VNS). As a
matter of fact, all the main ingredients of
these metaheuristics (defining the current
solution neighborhood, dealing with tabu
solutions or moves, imposing a proper
diversification, etc.) can easily be modeled
in terms of linear cuts to be dynamically
inserted and removed from the model.
This naturally leads to a completely general
(and hopefully powerful) TS or VNS
framework for MIPs possibly designed to
take into account the structure of specific
combinatorial problems. Very promising
results in this direction have been obtained
by Fischetti, Polo and Scantamburlo [7].

Working with infeasible reference
As stated, the local branching framework
requires a starting (feasible) reference
solution x 1, which we assume is provided
by the MIP black-box solver. However, for
28000 -

2855000 -
2S50 ------ I --

difficult MIPs (such as, hard set partitioning
models) even the definition of this solution
may require an excessive computing
time. In this case, one should consider
the possibility of working with infeasible
reference solutions. It is then advisable
to adopt the widely-used mechanism in
metaheuristic frameworks consisting in
modifying the MIP model by adding slack
variables to (some of) the constraints, while
penalizing them in the objective function.
A sophisticated version of this simple idea
has been considered by Fischetti and Lodi
[6] in conjunction with a recently proposed
algorithm to find an initial solution for
MIPs called Feasibility Pump [4].

Dealing with general-integer variables.
Local branching is based on the assumption
that the MIP model contains a relevant set
of 0-1 variables to be used in the definition
of the "distance function" A(x, 3).
According to our computational experience,
this definition is effective even in case of
MIPs involving general integer variables,
in that the 0-1 variables (which are often
associated with big-M terms) are likely to
be largely responsible for the difficulty of
the model. However, it may be the case that
the MIP model at hand does not involve
any 0-1 variable, or that the 0-1 variables do
not play a relevant role in the model-hence
the introduction of the local branching
constraint does not help the MIP solver. In
this situation, one is interested in modified
local branching constraints including the
general-integer variables in the definition
of the distance function A(x, x). To this
end, suppose MIP model (P) involves the

bounds j< x< < uj for the integer variables
x, (j I: = Bu G). Then a
suitable local branching constraint
can be defined as follows:

A,(x,Ex):= P (Uj--X)+

X ^j,-i)+
X p(u -x>-)+

Jt < < a,

where weights K are defined, e.g., as
R = 1/(uj 1j) for allj e I, while the
variation terms x and x. require the
introduction into the MIP model of
additional constraints of the form:

x, x+ x- x, x 0j >0, x 0,

Vj I: < < uj.

Use of local branching constraints
within special-purpose codes.
As outlined in the introduction, there is no
need of using local branching constraints
within a general-purpose MIP solvers. In
fact, these constraints can be effectively
integrated within special-purpose (black-
box) codes, both exact and heuristic,
designed for specific problems so as to
enhance their heuristic capability. Obviously
the only requirement is that the code is
able to take into account linear inequalities.
In this context, using local branching
constraints within a special-purpose branch-
and-cut code seems to be very suitable,
and interesting results have been obtained
by Hernandez-Prez and Salazar [10].

4 On-line Resources
A beta version of the LocBra code is
available for research purposes together
with a collection of hard MIP instances,
and some papers and slides illustrating the
method at the web page: www.or.deis.unibo.
Finally, a Local Branching heuristic
is now part of the arsenal of ILOG-
Cplex since version 9.1.

0 2000 4000 6000 8000 10000 12000

Figure 3: LocBra as a heuristic for instance B1C1S1 (solution value vs. CPU sec.).

1 UP I MdA7




[1] E. Balas, S. Ceria, M.
Dawande, F. Margot and
G. Pataki. OCTANE: A
New Heuristic For Pure 0-1
Programs. Operations Research
49(2), 207-225, 2001.
[2] Cplex. ILOG Cplex 7.0 User's
Manual and Reference Manual.
ILOG, S.A., 2001 (http://
[3] E. Danna, E. Rothberg, C. Le
Pape. Exploring relaxation
induced neighborhoods to
improve MIP solutions.
Mathematical Programming
102, 71-90, 2005.
[4] M. Fischetti, F. Glover, A.
Lodi. The Feasibility Pump.
Mathematical Programming
DOI 10.1007/s10107-004-
0570-3, 2005.

[5] M. Fischetti, A. Lodi. Local
Branching. Mathematical
f -. 23-47, 2003.
[6] M. Fischetti, A. Lodi,
"Repairing MIP Infeasibility
through Local Branching",
Technical Report RC23532,
IBM T.J. Watson Research
Center, 2005.
[7] M. Fischetti, C. Polo, M.
Scantamburlo. A Local
Branching Heuristic for
Mixed-Integer Programs with
2-Level Variables. Networks
44, 2, 61-72, 2004.
[8] F. Glover and M. Laguna.
General Purpose Heuristics
For Integer Programming:
Part I. Journal ofHeuristics 2,
343-358, 1997.
[9] F. Glover and M. Laguna.
General Purpose Heuristics
For Integer Programming:
Part II. Journal of Heuristics 3,
161-179, 1997.

[10] H. Hernandez-PNrez and J.J.
Salazar-Gonzalez. Heuristics
for the one commodity
Traveling Salesman Problem.
Transportation Science, 2003
(to appear).
[11] A. Lokketangen. Heuristics
for 0-IMixed-Integer
Programming. In P.M.
Pardalos and M.G.C.
Resende (ed.s) Handbook of
Applied Optimization, Oxford
University Press, 474-477,
[12] A. Lokketangen and F.
Glover. Solving Zero/One
Mixed Integer Programming
Problems Using Tabu
Search. European Journal of
Operational Research 106, 624-
658, 1998.

[13] N. Mladenovic and
P. Hansen. Variable
Neighborhood Search.
Computers and Operations
Research 24, 1097-1100, 1997.
[14] M. Nediak and J. Eckstein.
Pivot, Cut, and Dive: A
Heuristic for 0-1 Mixed Integer
Programming. Research Report
RRR 53-2001, RUTCOR,
Rutgers University, October
[15] M. Van Vyve and Y. Pochet.
A General Heuristic for
Production Planning
Problems. COREDiscussion
Paper 56, 2001.

Call for nominations 2006

Nominations are solicited for the George B. Dantzig Prize, administered
jointly by the Mathematical Programming Society (MPS) and the Society
for Industrial and Applied Mathematics (SIAM). This prize is awarded
to one or more individuals for original research which by its originality,
breadth and depth is having a major impact on the field of mathematical
programming. The contributions) for which the award is made must
be publicly available and may belong to any aspect of mathematical
programming in its broadest sense. Preference will be given to candidates
who have not reached their 50th birthday in the year of the award.

The prize will be presented at the 2006 SIAM Annual Meeting to be held
July 10-14, 2006, in Boston, Mass. Past prize recipients are listed on the
MPS Web site. The members of the prize committee are Robert Bixby
(Chair), Arkadi Nemirovski, Jong-Shi Pang, and Alexander Schrijver.

Nominations should consist of a letter describing the nominee's
qualifications for the prize and a current curriculum vitae of the
nominee, including a list of publications. They should be sent to the
chair of the committee and received by Nov. 15 2005. Submission of
nomination materials in electronic form is strongly encouraged.

Robert E. Bixby
ILOG, Inc. and Rice University
8 Briarwood Ct.
Houston, Texas, 77019

E-mail: bixby@ilog.com or bixby@rice.edu



45th session

Combinatorial Optimization:

Methods and Applications

JUNE 19-30 2006

University de Montreal. Requests for
participation or financial assistance must be
received before Feb. 28 2006.


0 P T I MA 2



The Lagrange Prize

in Continuous


Call for Nominations
Nominations are invited for the Lagrange
Prize in Continuous Optimization, awarded
jointly by the Mathematical Programming
Society (MPS) and the Society for
Industrial and Applied Mathematics
(SIAM). The prize will be presented at the
SIAM Annual Meeting in July 2006.

To be eligible, works should form the
final publication of the main results)
and should be published either (a) as
an article in a recognized journal or in
a comparable, well-referenced volume
intended to publish final publications only;
or (b) as a monograph consisting chiefly
of original results rather than previously
published material. Extended abstracts and
prepublications, and articles published in
journals, journal sections, or proceedings
that are intended to publish non-final
papers, are not eligible. The work must have
been published during the six calendar years
preceding the year of the award meeting.

Judging of works will be based primarily
on their mathematical quality, significance,
and originality. Clarity and excellence
of the exposition and the value of the
work in practical applications may be
considered as secondary attributes.

The Prize Committee for 2006 consists of
John Dennis, Nick Gould, Adrian Lewis,
and Mike Todd. Full details and prize rules
are given at www.mathprog.org/prz/Lagrange.
htm To nominate a publication for the prize,
please send a copy of the paper and a letter
of nomination by January 23, 2006 to :

Michael J. Todd
School of Operations Research
Rhodes Hall
Cornell University
Ithaca, NY 14853-3801

E-mail: miketodd@cs.cornell.edu

Electronic submissions are preferred.

Call for nominations 2006
The next A. W. Tucker Prize will be awarded at the XIXth International
Symposium on Mathematical Programming in Rio de Janeiro, 30 July-
4 August 2006, for an outstanding paper authored by a student.

The paper can deal with any area of mathematical programming. All students, graduate
or undergraduate, are eligible. Nominations of undergraduate students are welcome.
The Awards Committee will screen the nominations and select at most three finalists.
The finalists will be invited, but not required, to give oral presentations at a special
session of the symposium. The awards committee will select the winner before the
symposium and present the award prior to the conclusion of the symposium.

The paper may be original research, a particularly notable exposition or survey, a report
on innovative computer routines and computing experiments, or a presentation of a
new and ingenious application. The paper must be solely authored and completed
since 2003. A Ph.D. thesis qualifies. The paper and the work on which it is based
should have been undertaken and completed in conjunction with a degree program.

Nominations must be made in writing to the chairman of the awards committee
by a faculty member at the institution where the nominee was a student when the
paper was completed. Moreover, nominators should send one copy each of: 1) the
student's paper and a separate summary of the paper's contributions, written by
the nominee, no more than two pages in length and 2) a brief biographical sketch
of the nominee to each of the four members of the Tucker Prize committee:

Prof S. Thomas McCormick (Chair)
Sauder School of Business
University of British Columbia
Vancouver, B.C., Canada V6T 1Z2

Prof Monique Laurent
CWI, Centrum voor Wiskunde
en Informatica
Kruislaan 413
NL-1098 SJ Amsterdam, The Netherlands

Prof Jong-Shi Pang
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, New York 12180-3590, USA

Prof Riidiger Schultz
Department of Mathematics
University of Duisburg-Essen Campus
Lotharstr. 65
D-47057 Duisburg, Germany

The awards committee may request additional information.

The deadline for nominations is February 1, 2006.

Nominations and the accompanying documentation must be written in a language
acceptable to the awards committee. The winner will receive an award of $750 (U.S.) and
a certificate. The other finalists will also receive certificates. The Society will also pay
partial travel expenses for each finalist to attend the symposium. These reimbursements
will be limited in accordance with the amount of endowment income available. A
limit in the range from $500 to $750 (U.S.) is likely. The institutions from which the
nominations originate will be encouraged to assist any nominee selected as a finalist
with additional travel expense reimbursement. Previous winners and further information
about the Tucker Prize can be found at www.mathprog.org/prz/tucker.htm#winners.

10S T I MA 2



Call for nominations
The Fulkerson Prize Committee invites nominations for the Delbert Ray
Fulkerson Prize, sponsored jointly by the Mathematical Programming
Society (MPS) and the American Mathematical Society. Up to three
awards of US $1500 each are presented at each (triennial) International
Symposium of the MPS. The Fulkerson Prize is for outstanding papers
in the area of discrete mathematics. The prize will be awarded at the
19th International Symposium on Mathematical Programming to
be held in Rio de Janeiro, Brazil from July 30 to Aug. 4, 2006.

Eligible papers should represent the final publication of the main results)
and should have been published in a recognized journal, or in a comparable,
well-refereed volume intended to publish final publications only, during
the six calendar years preceding the year of the Symposium (thus, from
Jan. 2000 through Dec. 2005). The prizes will be given for single papers,
not series of papers or books, and in the event of joint authorship the
prize will be divided. The term "discrete mathematics" is interpreted
broadly and is intended to include graph theory, networks, mathematical
programming, applied combinatorics, applications of discrete mathematics
to computer science, and related subjects. While research work in these
areas is usually not far removed from practical applications, the judging of
papers will be based only on their mathematical quality and significance.

Further information about the Fulkerson Prize, including a list of previous
winners, can be found at www.mathprog.org/prz/fulkerson.htm
and at www.ams.org/prizes/fulkerson-prize.htmt.

The Fulkerson Prize committee consists of Noga Alon (Tel-Aviv University),
Bill Cunningham (U. Waterloo) and Michel Goemans (MIT), chair.

Please send your nominations (including reference to the nominated
article and an evaluation of the work) by Jan. 31, 2006 to the committee
chair. Electronic submissions to goemans@math.mit.edu are preferred.

Michel Goemans
MIT, Room 2-351
Department of Mathematics
77 Massachusetts Avenue
Cambridge, MA 02139

E-mail: goemans@math.mit.edu

Beale-Orchard-Hays Prize
Call for nominations 2006
The Mathematical Programming Society invites
nominations for the Beale-Orchard-Hayes Prize
for Excellence in Computational Mathematical
Programming. For details of rules and eligibility,
please see www.mathprog.org/prz/boh.htm

Nominations can be submitted either electronically
or in writing, but not a combination of the two. The
nomination must include a cover letter with the title,
authors, and publication details of the nominated
paper or book. If submitted electronically, the final
published version of the nominated publication should
be attached to the message. If in writing, please send
four copies of the paper or book. Supporting justification
and any supplementary materials are welcome but
not mandatory. The screening committee reserves
the right to request further supporting materials
from the nominees. The deadline for nominations is
March 17, 2006. Nominations should be mailed to:

Stephen Wright
Computer Sciences Department
University of Wisconsin
1210 W. Dayton Street
Madison, WI 53706 USA
E-mail: swright@cs.wisc.edu

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1ing m a u

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journal Mathematical Programming,
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10S PTI 72





Center for Applied Optimization
401 Weil
PO Box 116595
Gainesville, FL 32611-6595 USA


Jens Clausen
Informatics and Mathematical Modelling,
Technical University of Denmark
Building 305 room 218
DTU, 2800 Lyngby
Tlf: +45 45 25 33 87 (direct)
Fax: +45 45 88 26 73
e-mail: jc@imm.dtu.dk

Alberto Caprara
DEIS Universita di Bologna,
Viale Risorgimento 2,
I 40136 Bologna, Italy
e-mail: acaprara@deis.unibo.it

Donald W. Hearn

Christina Loosli
University of Florida

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