Title: Optima
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Title: Optima
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Publication Date: March 2008
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Mathematical Programming Society Newsletter

Structure Prediction and
Global Optimization
Marco Locatelli
Dip. Informatica, Univ. di Torino 'I 'i
Fabio Schoen
Dip. Sistemi e Informatica, Univ. di Firenze I- ',
February 26, 2008
"Every attempt to employ mathematical methods in the study of chemical
questions must be considered profoundly irrational and contrary to the spirit
in chemistry. Ifmathematical analysis should ever hold a prominent place
in chemistry an aberration which is happily almost impossible it
would occasion a rapid and widespread degeneration of that science."
Auguste Comte
Cours de philosophies positive
"It is not yet clear whether optimization is indeed useful for biology
surely biology has been very usefulfor optimization"
Alberto Caprara
private communication

1 Introduction
Many new problem domains arose from the study of biological and
chemical systems and several mathematical programming models
as well as algorithms have been developed. We are slightly more
optimistic than Comte and Caprara and still believe that the use
of mathematical programming tools can be valuable in biology,
chemical-physics as well as in the study of innovative materials. Of
course we perfectly agree on the fact that a lot of research stimuli
for the optimization community originated from those fields.
In this paper we concentrate our attention on the problem of structure
prediction: given some information on the composition of a complex
molecule we would like to predict the structure that the molecule
will most likely assume. Such a problem is a very relevant one as the
properties of, e.g., biomolecules are intimately related to their three-

continue on page 2

A new column
for Optima
by Alberto Caprara
Andrea Lodi
Katya Scheinberg

This is the first issue of 2008 and it
is a very dense one with a Scientific
contribution on computational biology,
some announcements, reports of
conferences and events, the MPS Chairs
Column. Thus, the extra space is limited
but we like to use few additional lines
for introducing a new feature we are
particularly happy with. Starting with
issue 76 there will be a Discussion
Column whose content is tightly related
to the Scientific contribution, with
a purpose of making every issue of
Optima recognizable through a special
topic. The Discussion Column will take
the form of a comment on the Scientific
contribution from some experts in
the field other than the authors or
an interview/discussion of a couple
of experts in the area or some other
short contribution which may reflect
alternative points of view related to the
special topic.

We hope our readers will enjoy the new
column and we strongly encourage
feedbacks especially in terms of
suggestions for topics to be covered in
future issues.
continue on page 17

- -P a G Opio n 2 A- -C- am



MARCH 2008


0P I A7

MARCH 2008

Structure Prediction and Global Optimization

dimensional conformation; it is quite
well accepted nowadays that this most
stable conformation corresponds to a
global minimum of a suitable function
which represents the free energy of the
molecule. Given our only limited capability
of capturing the essential phenomena
in a manageable energetic model, and
given the fact that various factors (e.g.,
thermodynamic and kinetic factors) concur
to determine the actual structure, it is
widely believed that being able to detect
the global minimum as well as other low-
lying local minima is an important issue.
This observation immediately leads to the
application of global optimization: in order
to predict the structure, first a mathematical
model of the total potential energy, the
energy function E, is defined, and then
the function Eis (globally) minimized.
Defining a reasonable model E for the
energy is in general an extremely complex
task; many classical models include sums of
terms which account for various interactions
inside the molecule. In general, some terms
are related to bonded interactions (forcing
pairs of bonded atoms to stabilize around a
fixed distance, the bond length, or favoring
triplets of bonded atoms to form specific
angles or quadruplets to find an equilibrium
around some known dihedral angles).
Other terms account for weaker interactions
between non-bonded pairs; usually these
interactions have the following form:

: = I)

q q., (21

where term (1) represents the van der Waals
interaction and depends on the distance R
between any pair of non-bonded atoms; as
A, > 0 and B, > 0, this term is composed
of a repulsive term and an attractive one.
Term (2) represents the Coulomb, or
electrostatic, interaction and, again, depends
on the pair distance R and on the electric
charges q, and q, of the two atoms: as it is
well known, atoms whose electric charges
are opposite in sign contribute a negative
(attractive) term to the energy, while the

opposite is true for pairs of atoms with
charges of the same sign. In most energy
models, covalent bonds are considered to
be too strong to be broken or modified
at physiological temperatures, so the only
degrees of freedom of complex molecules
can be considered as those associated with
non bonded pairs. This is the reason why in
many applications terms (1)-(2) are the only
ones taken into account in optimization, as
all the other terms are assumed to contribute
a constant term to the total energy.
Globally minimizing E turns out to be
a very challenging task. Several discretized
and simplified versions of the problem
have been proven to be NP-hard (see,
e.g., [8]). The main source of difficulty is
not necessarily the dimensionality of the
problem (in some cases the number of
control variables is very small) but the huge
number of local (and not global) minima of
the energy function E, which rules out any
trivial Multistart approach. Moreover, in
some cases a single function and/or gradient
evaluation of E may be extremely expensive.
A very deep survey on the models which
are currently used to describe the energy of
complex molecules can be found in [49];
an in-depth analysis on the characteristics
ofnanoclusters can be also found in [2].
We cannot close this introduction without
citing some recent developments in the
field of energy modeling, which we cannot
survey here but which are of great interest
for mathematical programming. We refer to
approaches which, differently from classical
ones which start "ab initio" and try to form
models according to first principles, are
based on the desire of finding a model for
which the structures which are observed
in nature are indeed global minima of the
model, while structures obtained through
perturbation of the observed ones are
not. Models can be built through linear
combination of suitable base functions;
given the enormous amount of knowledge
already available in protein databases,
the parameters of these models can be
obtained through the solution of huge linear
programs. We do not comment any more on
this subject, but refer the interested reader
to [47].

In the following we will present in some
detail the most important models for atomic
and molecular clusters and will give a
short introduction to the more challenging
problems of protein docking and protein
folding. Then, in Section 3 we will
introduce some basic ideas underlying many
of the global optimization approaches used
to solve these problems.

2 Structure prediction problems
In this section we review some well
known structure prediction problems.

2.1 Cluster optimization
In cluster optimization we are given N
particles (atoms or molecules) and an energy
function E, which depends on the relative
positions of the particles in the 3D-space; we
aim at detecting the global minimum of the
energy function. Different energy functions
have been proposed in the literature. Within
the field of atomic clusters usually only non-
bonded interactions are accounted for and in
the simplest models, particles are considered
to be charge-free. Such potentials only
depend on the distance R, between pairs i,j
of atoms, thus, for a cluster of N atoms, the
function to be minimized is the following:

E= Y E paiR (R,,)
i=1 j>i

Different models only differ from each
other for the definition of the pair potential
function Epa,,. In the field of atomic
clusters the most popular energy potential
is the Lennard-Jones (LJ) one. The LJ
pair potential can be defined as follows:

I 2
E ,(R, ) = Vi J(R,) =- h" h.
*J E

This potential produces very accurate
representations of real clusters like, e.g.,
some noble gases or some metals like
gold and nickel. But the interest of the
LJ potential also lies in the fact that


10 PTr i A_

MARCH 2008

it has been widely employed as a test
system to develop and gain insight into
new algorithmic techniques to be later
extended to other molecular conformation
problems. Another popular potential for
atomic clusters is the Morse one. The
Morse pair potential is defined as follows:

E,,,,(R,) = VM(R,; p) =
(exp{p(l R,)} 1)2 1.

The shape of this potential is quite similar
to the LJ one, but it allows for a greater
flexibility: a small p value models those
situations where the repulsive force as
the distance between two atoms is driven
to 0 is a mild one, while large p values
models situations where such a force is very
strong. Also, large p values correspond to
short range forces, which quickly vanish
outside a restricted neighborhood of the
minimum. In Figure 1 we plot both the
LJ and a few Morse pair potentials.
Besides producing an accurate
representation for some real clusters such
as those of C60 molecules (p = 13.6) and of
alkali metals (p = 3.1), the Morse potential
also offers a more varied test system with
respect to the LJ one. While the above
models always assume that all atoms in the
cluster are equal, other models have also
been proposed for clusters where atoms of
different types are present, in particular
for binary clusters, i.e., clusters with two
distinct atom types. As an example, we
mention binary Lennard-Jones clusters ([18,
19, 37]), i.e., clusters formed by a mixture
of two different atom types, which can be
modeled through


( (Ri Rij9 hi Y)

where E, and oy are suitable constants
which depend only on the types of
atom i and atomj. Note that from the
optimization point of view, binary clusters
turn out to be particularly interesting
because they mix continuous aspects
(atom positions) and combinatorial ones
(atom types). Besides binary Lennard-

o0 ; 1

-0.2 -



Figure 1: Illustration of Lenn

Jones, other potential energy model have
been proposed in the literature like, e.g.,
the Gupta model analyzed in [43, 44].
All the above mentioned potentials
are extremely challenging for global
optimization methods. The number of
local minima is conjectured to grow at
least exponentially with the number N of
atoms and, for Morse potential, the problem
becomes more and more difficult as the
parameter p increases because the energy
landscape becomes more and more rugged.
In the binary LJ clusters things are also
complicated by the combinatorial aspects
introduced by distinct atom types. In the
Cambridge Cluster Database (CCDB) ([7]),
the main database in the field of cluster
optimization, putative global minima are
reported for up to N < 150 and for 300 < N
< 1000 atoms for LJ clusters; for the more
challenging Morse clusters only results up
to N= 80 atoms are reported; finally, for
binary LJ clusters results for N up to 100
and different oij values are reported.
Many other cluster optimization problems
together with lists of their putative global
minima are reported in the CCDB. Among
them we recall Dzugutov clusters (see [17,
21]) and C6, fullerenee) clusters: the latter
are indeed molecular clusters, but C6,
molecules are extremely close to spheres and
thus many optimization methods consider
them just as single particles in space.
In the field of molecular clusters we
mention water clusters, in which some water
molecules interact; models that capture

1.2 1.4 1.6
lard-Jones and Morse pair potentials

the fundamental interactions among water
molecules are based on the assumption
that a single molecule has a prescribed
shape which cannot be altered; the energy
contribution is thus dependent on the
relative positions of H and O atoms of
different molecules. Among the best known
models for water clusters we cite the TIP4P
and TIP5P potentials. These molecular
clusters are much more difficult to optimize
with respect to atomic clusters because
of the additional orientational degrees of
freedom; indeed, the energy contribution of
a pair of water molecules does not depend
only on the distance between the geometric
centers of the molecules, but also on the
relative rotation of one with respect to the
other. In other words, while the potential
is still given as a sum of contributions (van
der Waals and electrostatic) due to the
relative distances between pairs of atoms,
as each molecule is considered to be rigid,
the "natural" degrees of freedom of each
molecule are the position of its center and
the rotation with respect to a prefixed
orientation. Thus, all pairwise distances
can be seen as functions of these degrees of
freedom. The CCDB reports putative global
minima for water clusters with no more
than N = 21 molecules and, according to
the current literature on the subject, there
are still some doubts that the published
structure at N = 21 (represented in Figure 2)
is indeed the global minimum.


0P I A76

MARCH 2008

Figure 2: Putative optimal custer for TIP5P21

2.2 Molecular distance geometry
The three-dimensional structure of real-life
proteins is usually determined by means
of Nuclear Magnetic Resonance (NMR)
and X-ray diffraction experiments. From
these experiments we obtain a subset
of all possible pairwise distances, from
which we aim at finding a conformation
of all the atoms in such a way that
the constraints imposed by the subset
of known distances is satisfied. More
complex versions of this problem also
include information about angles.
If complete information were available
(all distances were known), the problem
could be solved in O(I3) time (where N
is the number of atoms) by eigenvalue
decomposition of the distance matrix;
however, when incomplete information on
distances is available the problem becomes
strongly NP-hard (see [45]).
To further complicate matters, distances
are not usually known exactly but only
lower and upper bounds can be retrieved
from the experiments.
Since the introduction of the EMBED
algorithm in [9], many different techniques
have been proposed to solve this problem,
including graph reduction [30], geometric
build-up [13], Semidefinite Programming
[4]. Global optimization techniques
have also been proposed ( see, e.g., [36,
51]). Indeed, the problem can be easily
reformulated as that of globally minimizing
the cumulative relative error
[" ix{(,fi- I xi | |

(i- ..r ,,_ L *i J

[' (. II X X. || -, } (3

or, alternatively, the cumulative
absolute error:

Z [nmax{(0, II X, X, i||}]2

IO X i X 12 2 112
+ [max{0. X, x- Xj2 ,2 }]"

Here X1, . ,X are the positions of the
atoms, l, and u, are respectively the known
lower and upper bound for the distance
between atoms i andj, and D is the subset of
distances for which lower and upper bounds
are available.

2.3 Protein-protein docking
and protein folding
Protein-protein docking is the process by
which a large and complex biomolecule
interacts with another one by forming a
single complex; such complexes form the
bases of most activities of all living bodies.
Being able to predict the correct docking
of two specific proteins is considered to be
one of the most important challenges in
computational biology for the next years. In
rigid docking, the two interacting molecules
are considered as rigid bodies; of course,
this cannot be true in practice, but often
the relative position of two proteins docked
in this way can be used as a starting point
for a flexible docking phase, in which both
proteins are allowed to change their shape.
In rigid docking we can formulate the
problem as one of minimizing the energetic
contribution of pairs composed of atoms
belonging to the two different proteins
- in fact internal contributions account for
a constant term in the energy if the shape
of each molecule is kept fixed. It can be
easily understood that rigid protein docking
is a low-dimensional global optimization
problem: we can formulate the problem
assuming that one of the two molecules
is kept fixed, and thus there are only six
degrees of freedom: three translation and
three rotation parameters which enable
to identify the relative position of the two
molecules. Despite the low dimension, the

complexity of rigid protein-protein docking
is formidable, both because of the very
rugged energy landscape, characterized by
an enormous number of local optima, and
as a consequence of the large number of
interactions which have to be computed for
each energy evaluation: proteins are usually
composed of thousands of atoms and thus,
given the relative position of two proteins,
millions of pairwise interactions (van der
Waals and Coulomb) have to be computed.
As it can be easily understood, when
relaxing the assumption of rigidity, the
number of degrees of freedom enormously
increases and the (flexible) docking problem
becomes a large scale global optimization
one. In [5] an approach is presented in
which rigid docking is used as a tool for
generating good starting conformations for
flexible docking. A survey of some
global optimization approaches for
docking can be found in [12].
Protein folding is concerned with the
determination of the three-dimensional
structure of a protein given the so-called
primary structure, i.e., the aminoacid
sequence. A protein is, in fact, composed
of a linear set of aminoacids; it is widely
accepted and confirmed by several
experiments and observations that the 3D
structure of a protein is largely determined
by the sequence of its aminoacids. In other
words, proteins which are composed by the
same sequence of aminoacids are considered
equal and all assume roughly the same
three-dimensional shape. Being able to
predict the conformation of a protein built
from a specific sequence of aminoacids is
considered as a fundamental problem in
computational biology; if we were able to
solve this problem, we could simulate the
folding of many synthetic proteins until
we find one which folds in a prescribed
and stable shape. The literature on protein
folding methods is enormous; here we
just refer the reader to [11, 20, 23, 46, 48]
for some general references and to [22]
for an interesting example of how global
optimization and integer programming
can be effectively used to design a novel
biomolecule which might lead to the design
of a more effective treatment for specific


10 MTAl l200 PAGE6

3 Computational approaches
In this section we report some observations
and related techniques which allowed to
greatly increase the efficiency in solving
some of the problems presented above.
3.1 Funnel structure
In spite of their huge number, the local
minima of the energy functions are not
randomly displaced. Indeed, we can group
them into a few (not necessarily disjoint)
large sets called funnels. If we denote a
funnel by S, then every local minimum
XO E S is the starting point of (at least)
one sequence of local minima within S

X - X, = X X, SVi

such that V i it holds that:
E(X ) > E(X1,), i.e., the sequence
is monotonically decreasing;
X is "reachable" from X,, which
basically means that a small
neighborhood of X has a nonempty
intersection with the region of
attraction of the local minimum X,

All the sequences with the above properties
within a given funnel have a common
end point, denoted by X ,in (4),
called the funnel bottom. In Figure 3 we
report an example of a one dimensional
function with a funnel structure.

0.2 I







-2 -1.5 -1 -0.5

Note that the global minimum of an
energy function is always the funnel
bottom of one of its funnels. Therefore, the
problem of detecting the global minimum
is equivalent to the problem of detecting the
lowest funnel bottom. The key observation
is that the number of funnels is usually
very small compared to the number of local
minima. The simplest instances are those
with even a single funnel (a typical example
is L55, i.e., the LJ instance with N= 55
atoms). Hard instances are those with many
funnels (this is typical for Morse instances
with large p values) but also those with few
funnels when the funnel whose bottom
is the global minimum is very narrow
(typical examples are LJ38 and LJ5). Simple
algorithms, such as Basin Hopping and its
variants (see [32, 50]), are able to reach quite
efficiently a funnel bottom starting from
a given local minimum. In the simplest
instances with a single funnel (whose funnel
bottom must be the global minimum) every
run of these algorithms quickly leads to the
global minimum, no matter which is the
starting point, in spite of the huge number
of local minima. In the hardest cases it may
be necessary to run the algorithms many
times in a Multistart fashion from different
(usually randomly sampled) starting points
before reaching the global minimum.

3.2 Geometric properties and
landscape deformation
Local and global minima of the energy
functions have particular geometrical
structures in the 3D-space. Some approaches
are based on conjectures about the
geometrical structure of global minima.
Some care is needed when making explicit
use of such conjectures within an algorithm.
One of the first methods to solve LJ
instances ([38], later refined by [53]) was
based on the conjecture that global minima
for instances with a relatively small number
of atoms have an icosahedral structure. Based
on this conjecture, the search for global
minima was carried on an icosahedral lattice.
Though successful on many instances, the
limit of this approach, as well as of any
other approach making apriori assumptions
about the structure of global minima, is its
biasedness: it only explores a portion of the
search space, the one containing minima
with icosahedral structure, but is completely
unable to detect global minima with a
different structure. This was confirmed by
the later discovery of new putative global
minima with non-icosahedral structure:
LJ,3 (FCC structure, see [14, 24, 40]),
LJ/7,102 104 (decahedral structure, see [14,
15]), LJ98 (a new and quite unexpected
structure, the Leary tetrahedron, see [31]).
Another possible way to exploit
geometrical properties comes from
observing that global minima of LJ and
Morse instances (no matter if they have
icosahedral, decahedral, close-packed or
any other structure) are compact figures
with particular shapes related to the three
eigenvalues of their moment-of-inertia
ten-sor. These shapes are usually spherical
(all eigenvalues are equal), prolate (one
eigenvalue is larger with respect to the other
two), or oblate (one eigenvalue is smaller
with respect to the other two). Figure 4
reports an example of each of these three

0 0.5 1 1.5 2

Figure 3: Illustration of a funnel

MARCH 2008


0P I A7i

MARCH 2008

(a) (b)
Figure 4: Cluster shapes: (a) spheric
LJU (b) prolate for M0, (c) oblate f

Based on this observation, the fol
modification of the energy function
introduced (see [16, 33]):
i.e., Fis the sum of the original
energy function E and of a geometr
penalization term h. For LJ and Mo
clusters h is defined as follows

S(max {.(l ,_ )2 + ) l(y, y

+ ,, z,) D})2 (6)

where (x, y,, z) are the three coordi
atom i, D is a parameter underestin
the diameter of the cluster (largest
between atoms within the cluster),
w ,w are parameters belonging to t
interval [0, 1]. This penalization ter
the effect of compressing the cluster
its compactness, but the compressio
different along the three axes. The
function Fwas employed to define
phase local searches: during the firs
phase a local minimum of the modi
function Fis detected; then, in the
phase the local minimum ofF dete
in the first phase becomes the start
point of a local search with the orig
energy function E. Parameters w ,w
the key ones. These are strictly relat
the eigenvalues of the moment-of-in
tensor (see [16]) and can be used to
different shapes. For instance, if we
to favor spherical shapes we set w =
= 1 so that the first phase of two-ph
local searches will favor local mining
a spherical shape with respect to loc
minima with other shapes. Therefore
appropriately selecting parameters u
w we can bias the search towards s
geometrical shapes. However, while
previously mentioned approaches bi

is introduced by making a priori assumption
on the structure of global minima with
the risk of being unable to detect some
global minima whose structure is not one
of the a priori assumed, here we do not
restrict to particular shapes, but, by means
of properly chosen bias parameters w, and
it is possible to drive the search towards
(c) all possible shapes. In practice it has been
al for observed that two-phase local searches
or M61. (with few choices of the parameters w,
and ) employed within the monotonic
lowing variant of Basin Hopping (see [32]) reduces
was by orders of magnitudes the effort for
detecting the hardest (non-icosahedral) LJ
(5) instances (see [35]) and the very difficult
Morse instances with p = 14 (see [16]).
ic Here we have only discussed the geometry
Irse of LJ and Morse instances. While the above
discussion can be extended to other clusters,
some cases need a special attention. For
}2 instance, another potential, the Dzugutov
one (see [21]), has global minima which
do not have compact shapes but have
polytetrahedral structures. Of course, in this
nates of case the geometric penalization (6), which
ataing favors compact shapes, is not suitable, but it
distance is possible to think about other definitions
and for h which are suitable for this potential.
he Water clusters represent a particularly
m has interesting case from the geometrical point
favoring of view because of the strong competition
n is between different three-dimensional (prism,
modified cage) and also bi-dimensional (book, 6-
two- ring) structures. Also in this case a suitable
t definition for the geometric penalization
ified term h is a possible subject for future
second research.
acted Finally, we remark that the introduction
of the geometric penalization term h induces
final a deformation of the energy landscape,
are which somehow "simplifies" it (in particular,
ed to it reduces the number of local minimizers
ertia and, even more important, of funnels).
favor This is not the only possible way to induce
want deformations. Among others, we recall
S here the smoothing technique employed in
ase [36] to solve molecular distance geometry
a with problems. This is basically a deformation
al of function (3) depending on a parameter.
re, by For some initial value of the parameter
1 and the function is deformed into a convex
special one, while in the following iterations the
in the parameter is progressively reduced and
asedness the resulting function is locally optimized

starting from the local optimizer obtained
in the previous iteration. In the last iteration
the parameter is fixed to 0 and the function
corresponds to the original one.

3.3 Population-based approaches
and dissimilarity measures
Population-based approaches, where a
population of clusters (basically, local
minima of the energy function) is grown,
have been widely employed in cluster
optimization (see e.g. [3, 10, 28, 33, 41]
for LJ clusters, [26, 42] for Morse clusters,
[29] for water clusters, [25, 39, 43, 44] for
binary clusters). Besides the usual operations
(mutation, crossover), a key element in these
approaches is the dissimilarity measure,
which measures the dissimilarity between
two given clusters. One of the limits of all
those approaches, like Basin Hopping and
its variants, where at each iteration a single
cluster is kept in memory, is the fact that in
some cases different runs of these approaches
often converge to the same funnel bottom,
although this might not correspond to
the global minimum, but only to one
which is more easily reached. Population-
based approaches are able to avoid this
phenomenon by keeping the population
diversified through a dissimilarity measure.
A basic requirement for such a measure
is to be able to recognize the equivalence
(dissimilarity measure equal to 0) between a
cluster and every possible result of rotations
and/or translations of the cluster itself. In
population-based approaches each newly
generated cluster is compared only to similar
clusters within the current population
and replaces one of them if it has a lower
energy. The choice of the dissimilarity
measure is essential for the efficiency of the
approach. In [33] a measure for LJ instances
is proposed. In [26] different measures are
tested and compared over some hard LJ
and Morse instances and the results make
clear the high impact of the choice of the
measure on the efficiency of the approach.
We also refer to [27] for an experimental
analysis giving some insight into the results
of the population based approach presented
in [26]. Some measures have also been
introduced to discriminate between different
geometrical structures. For instance, the g
value in [28], based on a projection of the
cluster over a plane, is small for close-packed


10PTr I"M A-76

structures, larger for decahedral structures,
and even larger for icosahedral structures.

3.4 Direct mutation
Direct mutation is a special mutation
operator employed in population-based
approaches (see, e.g., [28, 41]). We discuss
it separately because of its relevance
especially when the number N of atoms
increases. In spite of its importance in the
field of cluster optimization, it has been
observed that the performance of the Basin
Hopping algorithm degrades as N increases.
The main reason for this behavior is the
mechanism to generate a new candidate
local minimum in the neighborhood of the
current local minimum. In Basin Hopping
this is obtained by randomly perturbing
all the coordinates of the current local
minimum. This way Basin Hopping often
quickly reaches a local minimum which
only slightly differs from the global one,
but then the final improvement towards the
global minimum is a very difficult, time-
consuming step, because of the perturbation
at each iteration of all the atoms in the
current solution, which disrupts the whole
structure of the solution. Therefore, the
key to improve the performance is to find
other more structured and less random
(or even deterministic) moves, which are
defined as direct mutations. Often a direct
mutation simply removes a single atom
from a position (typically a position where
the atom does not give a "good enough"
contribution to the total energy) and tries
to place it in a new and better position.
Direct mutation can thus be regarded as
a "fault correction" mechanism. In [28]
Hartke observes that if direct mutation is
employed "the resulting overall speedup
can be so large that it makes all the
difference between an efficient solution and
impractically long computation times".
Dynamic Lattice Searching (DLS)
[6], where only atoms with high energy
contribution are moved over a (dynamic)
lattice made up by their own positions plus
all possible vacant sites, can be viewed as a
direct mutation operator.
Finally, we include in the field of direct
mutation operators also the combinatorial
moves employed with binary clusters (see,
e.g., [19]). Such moves are the swap one (if
atoms i andj are of different types, their

types are exchanged), and the change one
(the type of a single atom i is changed).
We remark that these moves are extremely
important when dealing with binary (or,
more generally, multi-atomic) clusters,
because they allow to take into account the
combinatorial nature of these problems.

4 Conclusions and further remarks
In the previous section on computational
approaches we did not mention methods
for protein docking and protein folding
problems. There are several reasons for this
omission. First, the literature on methods
for protein conformation is so large that
we cannot include even a short survey in
this paper. Second, when dealing with
proteins not only methods are different
but, perhaps more important, models are
widely different it is not clear which model
is good enough for protein prediction and
docking; quite often structures generated
by the minimization of an energy model
have to be "manually" refined by expert
biologists who have sufficient experience
to visually analyze the shape of complex
proteins. The difficulties associated with
global optimization of different models,
each one depending on suitably calibrated
parameter sets, makes the comparison
between algorithms quite a difficult task.
This is the reason why in this survey we
chose to give substantial space only to
the treatment of cluster optimization,
where models are well defined and widely
accepted. We may also add that many of the
ideas we find in the optimization of clusters
and, in particular, the notion of "funnel
landscape" and the methods to explore
funnel bottoms, are commonly found in the
literature on protein conformation. Thus,
it seems that a good method for cluster
optimization coupled with a good model
for the evaluation of the free energy of a
protein will yield a promising approach
to solve protein conformation problems.
Although this paper was focused on
molecular conformation problems, we are
confident that some of the ideas presented
here might find an application in other
fields. As an example, let us consider
the classical disk packing problem:
although vaguely resembling a molecular
conformation problem, this one is indeed
quite different. First, it is based on 2D

shapes; second, the model is a constrained
one and thus local optimization, sampling
and perturbation have to take into account
the constraints. Finally, there is no "energy"
to be minimized we might think that
non-overlapping circles contribute 0 to
the energy and overlapping ones have +oo
penalty, but, in any case, the energetic
model is radically different from those seen
in this paper. Despite these differences, the
authors successfully applied some of the
techniques described in this paper within
the "Circle packing contest" (see [1, 54]);
in that contest participants were required
to guess the positions of N non overlapping
circles of prescribed different radii included
in a smallest circular container. A quite
effective algorithm was obtained by properly
mixing the idea of funnel exploration, the
use of populations to avoid too greedy
searches and combinatorial moves (direct
We are quite confident that many other
structure determination problems might be
tackled with success through clever use of
global optimization techniques.

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MARCH 2008


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in binary Lennard-Jones
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structure of binary Lennard-
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[21] C. A. Floudas, J.L. Klepeis,
J.D. Lambris and D.
Morikis, De novo protein
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Basel, 1994), Vol. III, 503-509
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Jones T.V., Roberts C., Darby
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Schoen F., An experimental
analysis of population
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38(3), 351-370 (2007)

[26] Hartke B., Global cluster
geometry optimization by a
phenotype algorithm with
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scaling with cluster size,
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Doye, Tethrahedral global
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Optimization and Applications
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and M. Nanias, The protein
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optimization of force fields,

1 iPTfJI A 7

MARCH 2008


By Tom Luo, University of Minnesota
Shuzhong Zhang, The Chinese University of Hong Kong

A first of its kind workshop on Optimization and Signal Processing
(WOSP2007) was recently held on the campus of the Chinese
University of Hong Kong (CUHK) during the period December 19
- 21, 2007. With financial support from the Department of Systems
Engineering and Engineering Management, The Shun Hing
Institute of Advanced Engineering at CUHK, and the Huawei
Technologies Ltd., the workshop has brought together some of
the world's leading experts from both signal processing and the
optimization communities, as well as technical representatives from
leading information technology industry. It provided a valuable
forum for the algorithm developers and engineering practitioners to
share research ideas and identify important topics of future research.
In the past thirty years, the work-horse algorithms in the field
of digital signal processing and communication have been the
gradient descent and the least squares algorithms. While these
algorithms have served their purpose well, they suffer from slow
convergence and sensitivity to the algorithm initialization and
stepsize selection, especially when applied to ill-conditioned or
nonconvex problem formulations. This is unfortunate since many
design and implementation problems in signal processing and
digital communication naturally lead to nonconvex optimization
formulations, the solution of which by the gradient descent
algorithm usually works poorly. Moreover, some applications
require real-time implementation in DSP chips or large scale
deployment across a distributed network. Simply put, the need
for efficient and robust optimization algorithms is greater than
ever in the field of signal processing and communications.
In recent years, the field of optimization has witnessed a
significant surge in the research of interior point methods and
convex conic optimization. A set of extremely powerful algorithms
and highly reliable software packages have been developed. This

on-going work has substantially enlarged the set of signal processing
problems that can be reliably solved in an efficient manner. For
the optimization community, signal processing provides a rich
source of application problems to which the advanced optimization
knowledge and algorithms can bring a strong and immediate
impact. Some of the signals processing problems have led to
significant theoretical advances in optimization. Through close
collaboration with researchers from signal processing, optimizers
can help recognizing and solving convex problem formulations;
utilizing the theory of convex optimization to characterize and gain
insight into the optimal solution structure and to derive bounds
on performance; deriving convex relaxations of hard, non-convex
problems; and developing powerful general purpose or application-
driven specific algorithms, including those that enable large scale
optimization by exploiting the problem structure.
The goal of WOSP2007 was to promote this burgeoning field of
interdisciplinary research. This small workshop has attracted 187
registered participants; most of them are graduate students from
various universities of Hong Kong. This plus a delegation of 20
technical representatives from Huawei Technologies Ltd located in
Shenzhen, as well as many non-registered participants, created a
large audience that easily exceeded the maximum seating capacity
(245) of the lecture theatre. Some participants had to stand or sat
on the floor to listen to the talks. The technical program consisted
of tutorial lectures as well as in-depth technical presentations
showcasing the success of applications of optimization in signal
processing. The lecture materials, including the slides of the
presentations, can be found in the website of the workshop:

Frontiers in Bioscience, 9,
3296-323, (2004).
[45] M. Wagner, J. Meller, R.
Elber, Large-scale linear
programming techniques for
the design of protein folding
potentials, Mathematical
ProgrammingB, 101, 301-318,
[46] David J. Wales and Harals
A. Scheraga, Global
Optimization of Clusters,
Crystals, and Biomolecules,

Science, 285, 1368-1372,
[47] D. J.Wales, Energy Landscapes
with Applications to Clusters,
Biomolecules and Glasses,
Cambridge University Press,
Cambridge, (2003).
[48] D. J. Wales andJ. P. K. Doye,
Global optimization by
basin-hopping and the lowest
energy structures of Lennard-
Jones clusters containing up
to 110 atoms, J Phys. Chem.

A, 101, 5111-5116, (1997).
[49] Williams G.A., Dugan J.M.,
Altman R.B., Constrained
global optimization for
estimating molecular
structure from atomic
distances, J.Comp.Biol., 8,
523-547 (2001)
[50] Xiang Y., Cheng L., Cai
W., Shao X., Structural
distribution of Lennard-Jones
clusters containing 562 to
1000 atoms, J Phys. Chem. A,

108, 9516-9520 (2004)
[51] G. L. Xue, Improvements
on the Northby Algorithm
for molecular conformation:
Better solutions, Journal of
Global Optimization, 4, 425-
440 (1994).
[52] Al Zimmermann's Circle
Packing Contest, www.
CircePacking, 2006.


MARCH 2008



50 Years of Integer


Jon Lee
IBM T.J. Watson Research Center
Yorktown Heights, New York, USA

e-mail: jonlee@us.ibm.com

The year 2008 marks the fiftieth
anniversary of the birth of integer
programming. Naturally, you are now
wondering what seminal event occurred
in 1958 that we now refer to as the birth
of the subject. During that year, Dr.
Ralph E. Gomory devised and published
a short paper [26] that really set the field
of integer programming in motion. In case
you follow mathematical programming
only very casually, in that paper Gomory
described his cutting-plane algorithm for
pure integer programs, and he announced
that the method could be refined to give a
finite algorithm for integer programming.
A published proof of his finiteness result is
contained in [32]. Gomory gave a cutting-
plane algorithm for the mixed integer
problem [29], and this approach was only
shown to be quite effective many years later
[2]. It is interesting to note that now, half
a century after they were first introduced,
even Gomory's finite cutting-plane method
for the pure integer case is being re-
examined and is showing new promise [5].
To commemorate this occasion, on 7
January 2008 a special workshop was held
at Centre Paul Langevin, Aussois, France.
This special workshop was part of the 12th
Combinatorial Optimization Workshop,
7-11 January 2008, which is held each
year at Aussois. It is fair to say that the
Aussois workshop is the yearly event for
presenting and keeping up with the latest
developments in integer programming and
combinatorial optimization coming from
the operations research community. In light
of this, it was natural for the celebration of
fifty years of integer programming to be
staged as part of an Aussois workshop.

Before getting into the specifics of the
workshop, some brief words about the
venue are in order. Centre Paul Langevin is
located at Aussois, a small Savoyardvillage
type ski resort in the French Alps close
to the Italian border. It is situated in the
Maurienne valley, at the gateway of the
Vanoise National Park, starting from
1500m of altitude (to 2750m high). Besides
providing a measure of isolation, which
helps make for a good workshop, on the
occasion that participants need a short
diversion from the mathematics, there are
opportunities for skiing and snowboarding
(a passion of the present author!).

The center has a conference room with 196
seats, enough lodging for most participants,
and dining facilities. The center is named
for Paul Langevin (born 23 January 1872),
who was a prominent French physicist.
Besides his scientific activities, Langevin
was a founder of the Comit6 de vigilance
des intellectuals antifascistes, and he was
also president of the Ligue des droits de
l'homme (Human Rights League) from
1944 to 1946. He died in Paris on 19
December 1946, two years after living
to see the Liberation of Paris. Langevin
was buried at the Pantheon (in Paris).

The organizers of the workshop were:
Michael Jiinger (Universitit zu Kiln),
Thomas Liebling (Ecole Polytechnique
Federale de Lausanne), Denis Naddef
(Ecole Nationale Suprieure d'Informatique
et de Mathdmatiques Appliqutes de
Grenoble), William Pulleyblank (IBM
Corporation), Gerhard Reinelt (Universit"at
Heidelberg), Giovanni Rinaldi (Istituto
di Analisi dei Sistemi ed Informatica,
Roma), and Laurence Wolsey (Universit6
catholique de Louvain). Already one can
see that the organizers themselves are
leaders in the subject, providing some
of the leading work over the course of a
few decades, spanning such key areas as
polyhedral combinatorics, branch and

0PTI A76




MARCH 2008

cut, matching, the TSP, lot sizing, and
computational integer programming.

The workshop began after lunch, with
a session, chaired by Tom Liebling, of
invited survey talks on some themes in
integer programming that have withstood
the test of time. In this session, there were
talks by Gerard Cornudjols ("Polyhedral
Approaches to Mixed Integer Linear
Programming"), Bill Cook ("50+ Years of
Combinatorial Integer Programming"),
and Laurence Wolsey ("Decomposition and
Reformulation in Integer Programming").

In the early evening, George Nemhauser
led us through the history of the first
twenty years of integer programming,
walking us through a list of milestone
papers in that time period. Nemhauser
kindly agreed to allow us to reprint that
list here (see references [23]-[61]). Of
course such a list is subjective and will
inevitably suffer from some omissions,
but all of us interested in the field would
be well served by studying these papers.
Further information regarding the early
days of integer programming and associated
topics can be found in [16] and [22].

Nemhauser took 1954 as his starting
point, highlighting both the seminal paper

of G.B. Dantzig, D.R. Fulkerson and S.
Johnson [23] on cutting planes for the TSP
and the fact that counting is hard (even
for integer programmers). It is noteworthy
that Gomory acknowledged in [26] the
influence of [23] and [17] on his work:

"The algorithm closely resembles the
procedures already used by Dantzig,
Fulkerson and Johnson, and Markowitz
and Manne to obtain solutions to
discrete variable programming problems.
Their procedure is essentially this. Given
the linear program, first maximize the
objective function using the simplex
method, then examine the solution. If
the solution is not in integers, ingenuity
is used to formulate a new constraint
that can be shown to be satisfied by
the still unknown integer solution but
not by the noninteger solution already
attained.... What has been needed
to transform this procedure into an
algorithm is a systematic method for
generating the new constraints."

Nemhauser's review nicely set up a panel
session, led by Bill Pulleyblank, with six of
the pioneers who have been so influential,
especially during that early period. The
panelists were: Egon Balas, Michel Balinski,
Jack Edmonds, Arthur M. Geoffrion,
Ralph E. Gomory and Richard M. Karp.

Left Right: Balinski, Gomory, Karp

Alan J. Hoffman, Harold Kuhn and Ailsa
H. Land were also invited to be panelists,
but unfortunately they were not able to
attend. Susan Powell was kind enough to
deliver some remarks on behalf of Land.

Egon Balas is University Professor of
Industrial Administration and Applied
Mathematics and the Thomas Lord
Professor of Operations Research at
Carnegie Mellon University. He was
awarded the John von Neumann
Theory Prize (INFORMS) in 1995.
Some of the fundamental work of Balas
includes implicit enumeration [35] and
disjunctive programming [1]. Much
more about him can be found in [20].

Michel Balinski is a Directeur de Recherche
(m&rite), CNRS, cole Polytechnique,
Paris. He was founding Editor-in-Chief of
the journal Mathematical Programming and
participated in founding the Mathematical
Programming Society. Balinski was awarded
a Lester R. Ford Award (Mathematical
Association of America) in 1976 for his
paper [3]. He is well known for his work
on routing, apportionment and voting,
and set partitioning approaches.

Jack Edmonds was a professor in the
Department of Combinatorics and
Optimization at the University of


0P I A76

MARCH 2008



Waterloo, Ontario, from 1969. He retired
from that position in 1999. Edmonds
was awarded the John von Neumann
Theory Prize (INFORMS) in 1985.
Edmonds began his fundamental work
at the National Bureau of Standards.
He is particularly well known for his
contributions to polyhedral combinatorics
[39], branchings [44], matroids [51],
matching [38], flows [4] and the notion
of polynomial time [38]. For more details
on Edmonds' illustrious career, see [14].

Arthur M. Geoffrion is the James A. Collins
Professor of Management Emeritus at the
UCLA Anderson School of Management,
Los Angeles. In 2000 he was awarded
the George E. Kimball Medal, for service
to INFORMS and to the profession of
operations research. Geoffrion is well
known for his contributions to implicit
enumeration [45] and to decomposition
schemes and their connections to
Lagrangian duality [42, 55]. In 1978
Geoffrion co-founded INSIGHT, Inc., a
management consulting firm specializing
in optimization-based applications in
supply-chain management and production
planning. In 1982 he founded what has
become the INFORMS Roundtable.

Ralph E. Gomory is President Emeritus of
the Sloan Foundation, which he led 1989-
2008. Before that, Gomory spent nineteeen
years at IBM, beginning in the summer of
1959. He was named IBM Fellow in 1964,
Director of the Mathematical Sciences

Department in 1965, and IBM Director
of Research in 1970, a position which he
held until 1986. Gomory became IBM Vice
President in 1973 and Senior Vice President
in 1985. In 1986 he was named IBM Senior
Vice President for Science and Technology,
retiring in 1989. Gomory was awarded the
Frederick W. Lanchester Prize (INFORMS)
in 1963, the John von Neumann Theory
Prize (INFORMS) in 1984, and the
National Medal of Science (by the President
of the United States) in 1988. In addition
to his work on cutting-plane methods and
also corner polyhedra [10, 11, 12], Gomory
also made fundamental contributions
to column generation [6, 7] and to the
concept of providing a structure that
solves many closely related instances of an
optimization problem [9]. Much more on
Gomory's career can be found in [13].

Alan J. Hoffman is a Fellow Emeritus
of IBM Research. He has been a
Member of the National Academy of
Sciences since 1982 and was awarded
the John von Neumann Theory Prize
(INFORMS) in 1992. Hoffman has made
fundamental contributions to the theory
of total unimodularity [24], polyhedral
combinatorics and our understanding of
greedy algorithm; [18] and [21] provide
wonderful opportunities to find out much
more about the man and his work.

Richard M. Karp is the Class of 1939
Chair and University Professor at the
University of California at Berkeley. In

addition, he is a Research Scientist at the
International Computer Science Institute at
Berkeley. Karp was awarded the Frederick
W. Lanchester Prize (INFORMS) in
1977, the Delbert Ray Fulkerson Prize
in (AMS and MPS) in 1979, the Turing
Award (ACM) in 1985, the John von
Neumann Theory Prize (INFORMS) in
1990, and the National Medal of Science
in 1996. Karp's work that relates to integer
programming includes Lagrangian duality,
subgradient optimization and the TSP
[49, 53], reducibility of combinatorial
problems [56], and efficient algorithms
for network-flow problems [4].

Harold W. Kuhn is a Professor Emeritus
of Mathematics at Princeton University.
He was awarded the John von Neumann
Theory Prize (INFORMS) in 1980.
He is particularly well known for his
contributions to game theory and to
nonlinear programming, and for the
Hungarian method for the assignment
problem [15]. More information
about Kuhn can be found in [19].

Ailsa H. Land and Alison G. Doig proposed
in 1957 and published in 1960 [28] what
is considered the origin of branch and
bound as a general technique. Land is
Professor Emeritus of Operational Research
at the London School of Economics.

Many of the panelists noted the strong
support, in the early days, of the Rand
Corporation and Princeton University.



As a result of the Cold War, Government
funding was quite high for science and
mathematics, so there were ample resources
to support optimization. Many of the
panelists told how the influence of George
Dantzig, Ray Fulkerson, Alan Goldman,
Alan Hoffman and Alan Tucker should
not be underestimated. For some, the
passion came from the beauty of what
they were investigating, for others the
applications and the associated drive to
make a difference were motivating factors.

After a long day of listening, we reached
our fill of food for thought, and the
evening concluded with a banquet.

From Tuesday through Friday, the workshop
was run on its traditional model. That is,
there were no parallel sessions, and the
program was made during the meeting.
As usual, there were two morning sessions
and one late afternoon session to leave
time in the afternoon for "collaboration"
(broadly defined to include joint efforts at:
proving theorems, enjoying the beautiful
landscape, and skiing and snowboarding).

On Tuesday, the program was completely
filled with talks concerning cutting planes.
Some highlights included: Ralph Gomory
("40 Years of Corner Polyhedra"), Matteo
Fischetti ("Looking inside Gomory") and
Jean-Philippe Richard ("Group Relaxations
for Integer Programming"). A bit tiring for
those of us with a short attention span, but
ample testimony to the fact that the subject
remains a hot area even after fifty years.

On Wednesday, topics included some
very exciting and less traditional topics
whose computational value is only just
now being well explored. For example:
Frangois Margot ("Symmetry in Integer
Programming"), Franz Rendl ("Semidefinite
Relaxations for Integer Programming")
and Kurt Anstreicher ("A Computable
Characterization of the Convex Hull of
Low-Dimensional Quadratic Forms").

On Thursday, some highlights included
two very different aspects of issues

concerning nonlinearity and discrete
choices: Robert Weismantel ("Nonlinear
Integer Programming") and Michel
Balinski ("The Majority Jd.],, nr ).

Friday's program included excellent
talks, rewarding those who could stay
for the full week. For example: Fritz
Eisenbrand ("Integer Programming and
Number Theory") Richard Karp ("How
Hard are the NP-hard Problems?"),
Andrea Lodi ("Computation in Integer
Programming") and Tom McCormick
("Strongly Polynomial Algorithms for
Bi-Submodular Minimization").

The workshop was generously sponsored
by ADONET -Algorithmic Discrete
Optimization Network (a Marie
Curie Research Training Network
of the European Union), ALMA,
ILOG, IBM and Springer Verlag.

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Cornu'ejols and N. Natraj
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[4] J. Edmonds and R.M.
Karp (1972). Theoretical
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ACM 19 (2), 248-264.
[5] E. Balas, M. Fischetti and A.
Zanette (2008). "Looking
inside Gomory," presented

at the 12th Combinatorial
Optimization Workshop,
7-11 January 2008, Aussois,
[6] P.C. Gilmore and R.E. Gomory
(1961). A linear programming
approach to the cutting stock
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[7] P.C. Gilmore and R.E. Gomory
(1963). A linear programming
approach to the cutting
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Operations Research 11,
[8] R.E. Gomory (1960). Solving
linear programming problems
in integers, Combinatorial
Analysis (R.E. Bellman and
M. Hall, Jr. eds, American
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[9] R.E. Gomory and T.C. Hu
(1961). Multi-terminal
network flows, J. Soc. Indust.

Appl. Math. 9(4), 551-570.
[10] R.E. Gomory (1965). On the
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linear programs. Proceedings
of the National Academy of
Sciences of the United States
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[11] R.E. Gomory (1967). Faces
of an integer polyhedron.
Proceedings of the National
Academy of Sciences of the
United States of America, 57,
[12] R.E. Gomory (1969).
Some polyhedra related to
combinatorial problems.
Linear Algebra and its
Applications 2, 451-558.
[13] E.L. Johnson (2005). IFORS'
Operational Research Hall
of Fame Ralph E. Gomory,
International Transactions
In Operational Research 12,

[14] M. Jiinger, G. Reinelt and
G. Rinaldi, eds. (2003).
Combinatorial Optimization
Eureka, You Shrink! Lecture
Notes In Computer Science
2570, Springer-Verlag.
[15] H.W. Kuhn (1955). The
Hungarian method for the
assignment problem, Naval
Research Logistics Quarterly
2, 83-87.
[16] J.K. Lenstra, A.H.G. Rinnooy
Kan and A. Schrijver,
eds. (1991). History of
Mathematical Programming:
A Collection of Personal
Reminiscences. North-
[17] H.M. Markowitz and
A.S. Manne (1957). On
the solution of discrete
programming problems,
Econometrica 25.
[18] C. Micchelli, ed. (2003).
Selected Papers of Alan

MARCH 2008


Hoffman, with commentary,
World Scientific Publishing
Company, Singapore.
[19] G. Owen (2004). IFORS'
Operational Research Hall
of Fame Harold W. Kuhn
International Transactions
in Operational Research 11,
[20] G.K. Rand (2006). IFORS
Operational Research
Hall of Fame Egon Balas,
International Transactions
In Operational Research 13,
[21] U.G. Rothblum (1989), Alan
J. Hoffman Biography,
Linear Algebra and Its
Applications 114/115, 1-16.
[22] K. Spielberg and M.
eds. (2007). History of
Integer Programming:
Distinguished Personal Notes
ad Reminiscences, Annals of
Operations Research 149.

1973) (Compiled by
George L. Nemhauser)
[23] G.B. Dantzig, D.R. Fulkerson
and S. Johnson (1954).
Solution of a large scale
traveling salesman problem,
Operations Research 2,
[24] A.J. Hoffman and J.B.
Kruskal (1956). Integral
boundary points of convex
polyhedra. linear inequalities
and related systems, H.W.
Kuhn and A.J. Tucker eds.
Princeton University Press,
[25] G.B. Dantzig (1957). Discrete
variable extremum problems,
Operations Research 5,
[26] R.E. Gomory (1958). Outline
of an algorithm for integer
solutions to linear programs,
Bull. of the American
Mathematical Society 64,
[27] G.B. Dantzig (1960). On the
significance of solving linear
programs with some integer
variables, Econometrica 28,
30- 34.

MARCH 2008

[28] A.B. Land and A.G. Doig
(1960). An automatic
method for solving discrete
programming problems,
Econometrica 28, 497-520.
[29] R.E. Gomory (1960). An
algorithm for the mixed
integer problem, RM-2597,
The Rand Corporation.
[30] J.F. Benders (1962).
Partitioning procedures
for solving mixed variables
programming problems,
Numerische Mathematik 4,
[31] H. Everett III (1963).
Generalized lagrange
multiplier method for
solving problems of optimal
allocation of resources,
Operations Research 11,
[32] R.E. Gomory (1963). An
algorithm for integer solutions
to linear programs. Recent
Advances in Mathematical
Programming, R. L. Graves,
P. Wolfe, eds. McGraw-Hill,
[33] J.D.C. Little, K.G. Murty,
D.W. Sweeney and C. Karel
(1963). An algorithm for the
traveling salesman problem.
Operations Research 11,
[34] M. Balinski and R. Quandt
(1964). On an integer
program for a delivery
problem, Operations Research
12, 300-304.
[35] E. Balas (1965). An additive
algorithm for solving linear
programs with zeroone
variables. Operations
Research 13, 517-546.
[36] M. Balinski (1965). Integer
programming: methods, uses,
computation, Management
Science 12, 253-313.
[37] R.J. Dakin (1965). A tree-
search algorithm for mixed
integer programming
problems, The Computer
Journal 8, 250-254.
[38] J. Edmonds (1965). Paths,
trees, and flowers, Canad. J.
Math. 17, 449-467.
[39] J. Edmonds (1965). Maximum
matching and a polyhedron
with 0,1-vertices, J. Res. Nat.
Bur. Standards 69B, 125-130.

[40] R.E. Gomory (1965). On the
relation between integer and
noninteger solutions to linear
programs, Proc. Nat. Acad.
Sci. 53, 260-263.
[41] R.D. Young (1965). A
primal (all integer) integer
programming algorithm, J.
of Res. Nat. Bur. Stds. 69B,
[42] R. Brooks and A.M. Geoffrion
(1966). Finding Everetts
lagrange multipliers by linear
programming, Operations
Research 14, 1149-1153.
[43] P.C. Gilmore and R.E.
Gomory (1966). The theory
and computation of knapsack
functions, Operations
Research 14, 1045-1074.
[44] J. Edmonds (1967). Optimum
branchings, Journal of
Research of the National
Bureau of Standards 71B,
[45] A.M. Geoffrion (1969).
An improved implicit
enumeration approach
for integer programming,
Operations Research 17,
[46] R.E. Gomory (1969).
Some polyhedra related to
combinatorial problems,
Linear Algebra Appl. 2, 451-
[47] E.M.L. Beale andJ. Tomlin
(1970). Special facilities for
nonconvex problems using
ordered sets of variables.
Proceedings of the 5th
International Conference
on Operational Research,
J. Lawrence ed. Tavistock
Publications, 447-454.
[48] D.R. Fulkerson (1970). The
perfect graph conjecture and
pluperfect graph theorem.
Proceedings of the Second
Chapel Hill Conference on
Combinatorial Mathematics
and its Applications,
R.C. Bose et al. eds. University
of North Carolina Press,
[49] M. Held and R.M. Karp
(1970). The traveling salesman
problem and minimum
spanning trees, Operations
Research 18, 1138-1162.

[50] E. Balas. Intersection Cuts
A new type of cutting plane
for integer programming,
Operations Research 19,
[51] J. Edmonds (1971). Matroids
and the greedy algorithm,
Mathematical Programming
1, 125-136.
[52] D.R. Fulkerson (1971).
Blocking and antiblocking
pairs of polyhedra,
Mathematical Programming
1, 168-194.
[53] M. Held and R.M. Karp
(1971). The traveling salesman
problem and minimum
spanning trees: Part II,
Mathematical Programming
1, 6-25.
[54] R.S. Garfinkel and G.L.
Nemhauser (1972). Integer
Programming, Wiley.
[55] A.M. Geoffrion (1972).
Generalized benders
decomposition, J. of
Optimization Theory and
Appl. 10, 237-260.
[56] R.M. Karp (1972).
Reducibility among
combinatorial problems.
Complexity of Computer
Computations, R.E. Miller
and J.W. Thatcher eds.
Plenum Press 85-103.
[57] L. Lovasz (1972). Normal
hypergraphs and the perfect
graph conjecture, Discrete
Mathematics 2, 253-267.
[58] V. Chvatal (1973). Edmonds
polytopes and a hierarchy
of combinatorial problems,
Discrete Mathematics 4,
[59] J. Edmonds and E.L. Johnson
(1973). Matching, Euler tours
and the chinese postman,
Mathematical Programming
5, 88- 124.
[60] S. Lin and B.W. Kernighan
(1973). An effective heuristic
algorithm for the traveling
salesman problem, Operations
Research 21, 498-516.
[61] M.W. Padberg (1973). On the
facial structure of set packing
polyhedra, Mathematical
Programming 5, 199-215.
IBM T.J. Watson Research Center

e-mail address: jonlee@us.ibm.com


10PTr IM A_

MARCH 2008

MPS Chair's Column
Steve Wright
17 February 2008

It's a pleasure to contribute my second
column to Optima and an even greater
pleasure to see our society's newsletter
back on a regular production schedule.
We are much obliged to Andrea Lodi and
the editorial and publishing teams for
making this happen. This issue of Optima
contains an article by Jon Lee on the
2008 Aussois workshop, which included a
commemoration of the 50th anniversary
of the publication of Ralph E. Gomory's
1958 paper that founded the area of integer
programming. Jon's article makes clear that
integer programming remains a thriving
field, with many of the founders remaining
actively engaged in research alongside
extremely talented younger generations.
I note with sadness the passing of two
individuals who meant a great deal to
our community. Alex Orden, who was
Chair of MPS in 1983-86 and a founding
council member of the society, passed
away in Chicago on 9 February 2008.
Alex was renowned for his work on linear
programming, in particular, as co-author
with Dantzig and Wolfe of the paper on
the generalized simplex method and as the

inventor of the product-form inverse of the
basis matrix. Gene H. Golub of Stanford
University died on 16 November 2007
after a short illness. Gene was a giant of
numerical analysis and scientific computing,
and his research laid important foundations
for numerical optimization. Even more
importantly, in both his leadership capacities
and personal life, he was a great supporter of
the optimization community and of many
individual optimization researchers.
I salute our former Chair Rolf Moehring,
who celebrated his 60th birthday on 16
February 2008. The occasion was marked
by a symposium organized by his colleagues
in Berlin. Rolf continues to work hard
on behalf of MPS by heading an ad hoc
membership committee whose charge is to
find new ways for the society to serve its
members and the optimization community.
With the vast changes to the academic
landscape brought on by electronic
publishing, and with the increasing size and
diversity of our own research community,
it is time for us to step back and think hard
about how MPS should adapt and evolve.
Our membership stands at 1150, a record

level, due in large part to new members who
joined through their participation in the
most recent ISMP and ICCOPT meetings.
Part of our challenge is to serve these new
members well enough that they will renew
in future years! If you have any suggestions
concerning services that you would like
to see MPS provide, or new roles that we
should be playing, please contact Rolf or
myself. And please, if you have not already
done so, renew your MPS membership for
Finally, I draw your attention to
provisions in the society's by-laws (posted
on our web site mathprog.org) about the
establishment of regional and technical
sections of MPS. If you and a group of like-
minded optimization colleagues see benefits
in organizing yourselves, possibly with a
view to holding regional meetings or topical
conferences, or to establishing a web-based
community around some interest area, feel
free to consult with us about the possibility
of MPS affiliation. The arrangements
outlined in our by-laws are quite flexible.

Application for Membership

I wish to enroll as a member of the Society.
My subscription is for my personal use and not for the benefit of any library or institution.
F I will pay my membership dues on receipt of your invoice.
F I wish to pay by credit card (Master/Euro or Visa).







Mail to:
Mathematical Programming Society
3600 University City Sciences Center
Philadelphia, PA 19104-2688 USA

Cheques or money orders should be
made payable to The Mathematical
Programming Society, Inc. Dues for
2008, including subscription to the
journal Mathematical Programming,
are US $85. Retired are $40.
Student applications: Dues are $20.
Have a faculty member verify your
student status and send application
with dues to above address.

Faculty verifying status



Call for not

George B. I


MARCH 2008 PAGE 16

nination of the 2009

Dantzig Prize in

:al Programming

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 2009 International Symposium on Mathematical
Programming, to be held August 23-28, 2009, in Chicago, Illinois, U.S.A. Past
prize recipients are listed on the MPS Web site. The members of the prize committee
are Jong-Shi Pang (Chair), Yuri Nesterov, Alexander Schrijver, and Eva Tardos.

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

Jong-Shi Pang
Department of Industrial and Enterprise Systems Engineering
University of Illinois at Urbaba-Champaign
117 Transportation Building MC-238
104 S. Mathews Ave.
Urbana Illinois 61801
e-mail: jspang@uiuc.edu
and received by 15 November 2008. Submission of nomination
materials in electronic form is strongly encouraged.

1i PTr- IM A7_

MARCH 2008

Discussion Column

Commentary on "Structure prediction and global optimization"

by *Roy L. Johnston

Global optimization is of undoubted and
increasing importance in most areas of
science and engineering. The problem of
global optimization the determination of
the absolute maximum or minimum of a
function (or indeed a process) depending
on a large number of variables is very
difficult, even for discrete integer-valued
problems, let alone for continuous, real-
valued problems. Global optimization can
be regarded as the process of searching
a multi-dimensional landscape for the
highest "mountain peaks" or lowest "valley
bottoms". While this does not present
too many problems in the everyday 3-
dimensional world in which we live, in the
higher dimensions (i.e. number of variables)
commonly encountered in important
scientific and engineering problems, the
situation is far harder. Thus, apart from
the cases of simple convex mathematical
functions, or discrete problems small
enough to be grid searched (or for which
a branch-and-bound search tree-pruning
algorithm can be applied), one can never
be certain that the lowest (or highest) value
that is found is really the global optimum.
This is to be contrasted with the certainty
with which we can locate and identify
local minima and maxima, utilising (either
analytical or numerical) gradients and
curvatures of the function in question.
Over the past 30 years or so, many
approaches have been developed to increase
the likelihood of finding global optima,
including techniques such as simulated
annealing, Monte Carlo methods (e.g. the
Basin Hopping approach) and the growing
class of Evolutionary Algorithms (e.g.
Genetic Algorithms). Of course, the "No
free lunch theorem" ensures that while
certain methods may be particularly good

*School of Chemistry, University of
Birmingham,Edgbaston, Birmingham B15
2TT,United Kingdom

for certain classes of problems, no one
approach is guaranteed to work in every
In the article "Structure prediction
and global optimization", Locatelli and
Shoen describe the application of global
optimization to several problems, taken
from chemistry and biology, involving
the prediction of structure: finding the
minimum energy configuration of a cluster
(or nanoparticle) composed of atoms or
molecules; determining the lowest energy
folded conformation of a model protein
molecule; and the docking of protein
molecules. In these cases, the inter-
atomic interactions are described by quite
simple potential energy functions, which,
nevertheless, reproduce essential aspects
of the underlying physics. The difficulty
in finding the global mimimum energy
arises due to the high dimensionality of
the problem (for example, the number
of variables for an N-atom cluster is 3N,
corresponding to the Cartesian coordinates
of all the component atoms) and the
consequent very high number of possible
configurations (local minima).
As well as providing an important survey
and bibliography of applications of global
optimization to the cluster optimization
problem (which they concentrate on),
Locatelli and Shoen describe a number
of approaches and algorithms which they
(among others) have employed to improve
the success rate and efficiency of global
optimization techniques. For example,
the recognition of funnel-like features on
energy landscapes is used to rationalise the
differences between certain clusters whose
(putative) global minima are easy to find
with those which are far more difficult to
find. Consideration of the energy landscape
also explains the relatively high success of
methods, such as Basin Hopping and hybrid
Genetic Algorithms which incorporate local

minimization. The danger of assuming a
growth sequence (i.e. extrapolating from
known global minima to predict those of
larger clusters) is pointed out, but it is also
shown that biasing can be used in a positive
sense by modifying the energy or "penalty"

function (deforming the landscape) to
include a geometrical term in order to direct
the optimization towards structures with
a certain desired packing arrangement or
overall shape.
The efficiency of population-based search
methods (such as Genetic Algorithms, and
parallel tempering) is due to the parallel
exploration of the configuration landscape.
Locatelli and Shoen show that these
methods are most successful when coupled
with the use of dissimilarity measures, so as
to ensure (or at least encourage) exploration
of diverse regions of the landscape. The
process of "direct mutation", which involves
more deterministic permutations, perhaps
utilising prior knowledge about the system,
can also be used within a population-based
search method to move (ideally) towards
the global minimum. The Dynamic Lattice
Searching method (wherein high energy
atoms in the clusters are preferentially
moved) is one such example of this
In conclusion, I believe that Locatelli
and Schoen have presented a concise yet
informative introduction to the field of
structure prediction as an exercise in global
optimization, along with recent techniques
for understanding and improving the
optimization procedure. Applications of
many of these methods to other problems
are already widespread and likely to grow
considerably in the future. In their final
section, by way of an example, the authors
show that their approach has been very
successfully applied to the 2-D disk packing


0P I A76

MARCH 2008

Ettore Majorana Centre for

Scientific Culture

International School of Mathematics
"G. Stampacchia"
Erice Sicily, Italy
48st Workshop: Nonsmooth Analysis,
Optimization and Applications
May 9 17, 2008
Lecture-Hall: San Rocco

Sponsored by the:
Italian Ministry of University
and Scientific Research
Sicilian Regional Government
University of Calabria
Centro "Enrico Fermi"
Italian National Research Council,
Institute of High Performance
Computing and Networking,
Rende (CS)

The need of providing satisfactory answers to several questions posed by diverse advanced
application fields, basically in Engineering, Mechanics and Economics, has been the
strongest motivation to tackle mathematical problems where differentiability of the involved
functions is no longer guaranteed. Nonsmooth Analysis has then grown considerably and
it is now a well established area of modern Mathematics. The main battlefield, where the
theoretical findings of Nonsmooth Analysis are tested, is the design of effective algorithms
for solving a wide range of optimisation problems. Starting from the pioneering works in
the Sixties on the minmax problems, several application fields have benefited from the
development of Nonsmooth Analysis. We list here the study of large scale programming
problems via decomposition techniques, the Lagrangian relaxation of integer extremum
problems, the numerical solution of variational inequalities, the extremum problems with
equilibrium constraints, several classification and approximation problems, structural
design, nonsmooth mechanics, etc. The aim of the Workshop is to bring together people
working on both sides of Nonsmooth Analysis and Optimization and Applications to
discuss the state-of-the-art and the possible future developments. Some tutorials will also
be given to encourage young scientists to approach such an exciting research area. The
Workshop is dedicated to Vladimir F. Demyanov, on the occasion of his 70th birthday.

The workshop will be held in Erice,
Sicily, Italy at the "E. Majorana" Centre
for Scientific Culture. The Centre is
located in the pre-mediaeval city of
Erice and the lecture halls are located
in two restored monasteries and the
ancient Palazzo Ventimiglia former
residence of Viceroys of Sicily.

Persons wishing to attend the
Workshop and possibly to contribute
a lecture should contact:

Professor Manlio Gaudioso
D.E.I.S. Universita della Calabria
Via Pietro Bucci, Cubo 41C
87036 Rende (CS), Italy.
e-mail: gaudioso@deis.unical.it

1. Data and place of birth, together
with present nationality;
2. Affiliation;
3. Address, e-mail address.

Young people with only limited experience
should enclose a scientific curriculum
vitae and a letter of recommendation
from the head of their research group or
from a senior person active in the field.
Application by e-mail is strongly
encouraged. Closing date for application:
March 15, 2008. Participants are
expected to arrive in Erice on
May 9, no later than 5 p.m.

M. Gaudioso and D. Pallaschke
Directors ofthe Workshop

F. Giannessi
Director ofthe School

A. Zichichi
Director ofthe Centre

Numerical methods for nonsmooth
Nonsmooth optimization and integer
Nonsmooth dynamics
Nonsmooth analysis
Learning methods


MARCH 2008

Ettore Majorana Centre for Scientific Culture

Minmax problems
Decomposition methods
Variational inequalities
Online and incremental methods
Nonsmooth mechanics

The workshop will consist of invited lectures
and contributed lectures. Invited lecturers
who have confirmed the participation are:

Adil Bagirov
University ofBallarat, A

J.-P. Crouzeix
University Blaise Pascal, Clermont-Ferrand, F

Asen L. Dontchev
University ofI.'L,/ga. USA

Rosalind Elster
Universitat Autbnoma de Barcelona, E

Francisco Facchinei
University di Roma "La Sapienza" I

Antonio Frangioni
University di Pisa, I

Masao Fukushima
University of Kyoto, J

Alexei Gaivoronski
Norwegian University of Science and
Technology, Trondheim, NO

Giorgio Giorgi
University di Pavia, I

Angelo Guerraggio
University dell'Insubria, I

Alexander loffe
Technion, Haifa, IL

RWTHAachen University, D

Alexander Kurzhanski
University of California, Berkeley, USA

Juan Enrique Martinez Legaz
Universitat Autbnoma de Barcelona, E

Boris Mordukhovich
Wayne State University, USA

Massimo Pappalardo
University di Pisa, I

Panos Pardalos
University of Florida, Gainesville, USA

J.P. Penot
University of Pau, F

R.T. Rockafellar
University of Washington, USA

Stefan Scholtes
University of Cambridge, UK

Michel Thera
University de Limoges, F

Xiaoqi Yang
Hong Kong Polytechnic University, PRC

10PT A76






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


Andrea Lodi
DEIS University of Bologna,
Viale Risorgimento 2,
I 40136 Bologna, Italy
e-mail: andrea.lodi@unibo.it

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

Katya Scheinberg
IBM T.J. Watson Research Center
PO Box 218
Yorktown Heights, NY 10598, USA

Donald W. Hearn

University of Florida

Journal contents are subject to
change by the publisher.

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