P TIM M
SMathematical Programming Society Newsletter
Combinatorial Online Optimization in Practice
Abstract This paper gives a short introduction to combinatorial online
optimization. It explains a few evaluation concepts for online algorithms,
such as competitiveness, and discusses limitations in their application to
real-world problems. The main focus, however, is a survey of combinatorial
online problems in practice, in particular in large scale material flow and
flexible manufacturing systems.
Keywords: Online optimization, combinatorial optimization, real-world problems
By Norbert Ascheuer, Martin Groetschel,
Jorg Rambau, Konrad-Zuse-Zentrum fir
Informationstechnik Berlin (ZIB), Takustr.
S.... 7, D 14195 Berlin Dahlem; and
Nicola Kamin, ZIB, Herlitz PBS AG,
Berliner Str. 27, D 13507 Berlin)
In classical optimization, called offline optimization here, it is assumed that all input
data of an instance are available before solution algorithms are applied. In many appli
cations this is not realistic. Decisions have to be made before all data are known. Such
situations are often termed online. They arise in particular in processes that are con
tinuously running for a longer period of time. For instance, in material flow systems of
companies, transportation tasks arise throughout the day and decisions have to be
made before all jobs have been generated.
Online optimization is the task of finding "good," or "cheap," or "economic" deci
sions in online situations. An online algorithm provides such decisions. In practical ap
plications online algorithms are typically subject to additional constraints. For example,
they have to answer in real time, or must process a job (or request) within a given time
frame, sometimes they have access to limited computing resources only.
Competitiveness. A common concept to evaluate online algorithms is competitive-
ness. Here, an online algorithm has to act as follows: it always has to serve a request of a
sequence before the next request (or the next k different requests in a model with look
ahead) becomes visible. The idea behind the variants of this notion is the following:
compare the solution of the online algorithm under consideration with the solution
that some adversary would produce on the same set of data.
l "..The easiest case is the offline adversary, i.e., to him the complete sequence of re
quests is known in advance. For an algorithm Xlet CGs denote the cost Xproduces on
input sequence s. We assume for notational convenience that Cs is positive for all s
PAGE TWO -
conference notes 9
reviews 12 gallimaufry 15
MARCH 1998 PAGE
and that we want to minimize. A deterministic
online algorithm A is c-competitive if for any se
quence s of requests and any offline algorithm S
CAS< C- CS+ a
holds for real numbers c,a, both not depending
The goal is to find online algorithms that are
competitive in an optimal way, i.e., no other
online algorithm can have a better performance
ratio with the adversary. This concept is appli
cable both to deterministic and randomized al
gorithms, and it allows for provable statements
about the performance of an online algorithm.
A further advantage of this concept is that no
information about the distribution of the input
data is needed in order to make exact state
ments. Competitive analysis has been the sub
ject of many investigations concerning (mainly
elementary) online problems. (See, among oth
ers, Albers 1996; Albers 1997; Goemans 1994;
Irani and Karlin 1997; Motwani, Raghavan
1995; and Ottmann et al. 1994 for more infor
The competitiveness ratio is usually a pessi
mistic measure since the adversary is supposed
to be a bad person trying to fool the algorithm
by designing a particularly difficult sequence of
requests. Moreover, competitive analysis is based
on some hard restrictions to the model: one as
sumes that the next request does not become
available before attending to the current request.
In practice, however, there is often a dynami
cally growing and shrinking pool of requests of
unknown size visible to the algorithm.
Sometimes competitive analysis provides ab
solutely no insight into the quality of an online
algorithm. For instance, for a version of the
greeting card commissioning problem to be dis
cussed later, one can prove that all (reasonable)
online algorithms have the same competitiveness
factor K (the common capacity of the vehicles of
the system) (see Kamin 1998).
Furthermore, the competitive analysis is not
applicable if the decisions of the online algo
rithm have direct impact on the sequence of
Stochastic Optimization. Stochastic optimi
zation uses a model that seems to be close to
what we might call "reasonable acting under in
complete information." The decisions in the
online algorithm are made based on the optimal
solution of a stochastic program that has to be
solved beforehand. Usually, this is a linear pro
gram where the objective function is the expec
station of the cost function in the decision and
Automatic Guided Vehicles
Automatic Storage System
Figure 1. Sketch of the Factory Layout
This can only be done under the assumption
that the request data has a certain distribution.
Making decisions beforehand relying on statisti
cal data can be viewed as an offline model of an
online problem since the distribution allows for
computing the expectations before the actual re
quests occur. There are also models that perform
a certain number of alternating observation and
optimization steps in order to adjust the infor
mation about the distribution. These models
are, however, hard to evaluate in practice. Sto
chastic analysis is-as the name might suggest
anyway-focusing on the average behavior of the
algorithm. (See Pr6kopa 1995, and Kall and
Wallace 1994 for more background and applica
tions in this area.)
Since stochastic analysis requires an idea
about what the distribution of the incoming re
quests may look like, it is not applicable if one
cannot find any structure in the input data.
Moreover, if the probability is not concentrated
at the expectation, then the decisions that corre
spond to solutions from optimizing the expect
tion of the cost function may bear too large a
risk of failure. Usually, there is no guarantee
that the algorithm works well for any sequence
of requests that might occur.
Simulation. This is the approach used in prac
tice. A coarse mathematical model of the online
problem is developed and implemented in the
form of a simulation model. Several online algo
rithms are coded and experiments with real
world data are run on a computer to gain in
sight into the practical performance. In particu
lar, experiments are made to analyze high load
and failure situations. Usually those online algo
rithms that exhibit good average performance
and can somehow cope with "catastrophes" are
selected for use in practice.
A-posteriori Analysis. This approach uses
history. The actual sequences that came up in
the past are recorded and competitive analysis is
made based on the gathered data only. This
evaluation is often used to tune the online algo
rithms so that they perform better on these
known input sequences. One hopes that the old
input sequences are good representations of the
typical data and that hence the modified algo
rithms will show improved performance also on
future input sequences.
2. Application to Real-World
Besides the theoretical attractiveness of online optimi
zation problems, there is a broad variety of real-world
problems that can be modeled as an online problem.
In the sequel we outline problems we encountered in
joint projects with industry that were aimed at opti
mizing the internal material flow within a flexible
manufacturing system (FMS) and a distribution cen
ter. Wejust like to mention that, among others, there
exist further applications in computer science, vehicle
routing, scheduling, and telecommunication that will
not be discussed in this paper.
10 P T I M A 5 7
MARCH 1998 PAGE
2.1 Online Optimization of a
Flexible Manufacturing System
Siemens Nixdorf Informationssysteme AG
(SNI) maintains a production plant where all
their personal computers (PCs) and related
products are assembled. (See Figure 1 for a
sketch of the material flow within this FMS.)
Parts that are used to produce PCs (PCB, floppy
disk, cables, etc.) enter the FMS at the receiving
area in normed containers. They are brought by
automatic guided vehicles (AGV) into one of six
automatic storage systems (AUSS). The AUSS
serve as material buffer between the receiving
area and the assembly lines located at each side
of the AUSS. After assembly the PCs enter a test
area where for up to 24 hours test programs are
run in order to check the full functionality of
the PCs. After a manual test the PCs are packed
Optimization Problems. A profound analy
sis of the system showed that it offers a variety
of optimization problems, some of them of an
online character. These mainly are: scheduling
of transportation tasks within the AUSS; assign
ment of containers to storage locations; routing
and scheduling of the AGV; assignment of loca
tions in the test area; retrievals of PCs from the
test area. Here, the first question will be dis
cussed in more detail. Discussions of the other
topics can be found in Abdelaziz 1994, Ascheuer
1995, and Krippner Matejka 1993.
2.1.1 Stacker Crane Routing in the
Automatic Storage Systems
The AUSS are single-aisled with storage loca
tions on both sides of the aisle. In the lower part
there are buffer places where containers are pro
vided to the assembly line. A single stacker crane
has to fulfill all transportation tasks (jobs). So
far, a certain priority was assigned to each task
(storage, retrieval, buffer-refill, etc.). This prior
ity was only dependent on the type of the task.
Within one priority class, jobs were sequenced
due to a FIFO-rule. Although easy to imple
ment, this strategy resulted in a high percentage
of unloaded travel time.
Since every algorithm has to process all the jobs,
we can only control the unloaded moves of the
stacker crane. We suggested sequencing the tasks
in such a way that the total time needed for the
unloaded moves between thejobs is minimized.
This can be modeled as an asymmetric traveling
salesman problem (ATSP) where each job that is
not performed, together with the job that is
currently processed by the stacker crane, is
represented by a node in a complete digraph
D-( VA). W.l.o.g. we assume that the current job
corresponds to node 1. Each arc (ij) A,j,, 1,
represents the unloaded move between jobs iand
j. This arc is given a weight corresponding to the
time needed for the unloaded move from the
endpoint of job ito the starting point ofjobj.
To all arcs (, 1), i V\ 1, we associate weight 0.
Now, an optimal tour through the nodes of
DL( VA) corresponds to a sequence of the
transportation tasks with minimal total unloaded
This is an online problem since not all trans
portation tasks (resp. nodes for the ATSP) are
known in advance. They are generated during
the production period and neither generation
time nor start and end-coordinates are known
in advance, i.e., we have to solve an online-
A TSP. A detailed discussion of this topic can be
found in Ascheuer 1995. (See, e.g., Ausiello et
al. 1994a, and Ausiello et al. 1995a for competi
tiveness results on the online ATSP.)
Solution Approach. We decided to simply
ignore tasks that might be generated in the fu
ture and to solve a "static ATSP" as soon as a
new job is generated. In order to avoid the
stacker crane having to wait until we have fin
ished our calculations, we have implemented a
3-phase process. Whenever a new job is gener
ated we run the following optimization process:
Phase 1. Simple insertion heuristic. Try to
insert the new node as cheaply as possible
into the current sequence;
Phase 2. Run a more sophisticated heuris
tic. We have chosen a random insertion
Phase 3: Solve the ATSP to optimality.
This is done using a branch & bound
implementation of Fischetti and Toth
(Fischetti and Toth 1992).
Phase 1 runs in 0(n) time and is always com
pleted. For the typical problem sizes that occur
in our application (n 60), the computations
are done in fractions of a second. Even phase 3
was always completed within a few seconds.
After the completion of each phase, a se
quence is available that can be improved by one
of the subsequent phases. If, during the execu
tion of phase 2 or 3, a new job is generated,
then the whole process is stopped and restarted.
If the stacker crane has finished a task and asks
for a new one, the process is interrupted as well
and the best sequence so far is passed to the con
trol system of the stacker crane.
We tested several heuristics to be used in
phase 2 (See Abdelhamid, Ascheuer and
Gr6etschel 1998). It is easy to construct ex
amples where it does not always lead to the best
solution if each ATSP is solved to optimality.
The use of this strategy might construct se
quences that are "not good" with respect to the
nodes generated in the future. Nevertheless,
phase 3 empirically gives the best results on the
Computational Results. SNI provided data
for one week of production. During this period,
each generated task and each move of the
stacker crane was recorded at one AUSS. This
data was used to validate the simulation model.
Based on the SNI data we compared several
strategies for sequencing the jobs within the
Extensive computational tests showed that it
was possible to reduce the times needed for un
loaded moves by approximately 30% in heavy
load periods. As a result, this optimization pack
age was put in use at five AUSS and the results
were confirmed in everyday production. It
showed that even in the production environ
ment the optimization process could always fin
ish with phase 3.
Quality of the Online Solutions. A scien
tific question that arises is: how good are the so
lutions in comparison to an optimal offline so
lution? To evaluate the quality we performed an
a-posteriori analysis, i.e., we determined how we
would have sequenced the tasks had we known
which tasks were generated.
To this end we "collected" all jobs over a cer
tain time period and sequenced them optimally.
First note that the jobs cannot be sequenced ear
lier than they are generated. Moreover, the
completion of the jobs cannot wait too long as,
e.g., the production might be delayed. Thus, to
each job a time window is associated and we
only allow to visit a node within its time win
dow (ATSP-TW) (See, among others,
Desrochers et al. 1988 and Desrosiers et al.
1995 ). The ATSP-TW is a difficult combinato
rial optimization problem where it is even
strongly NP complete to find a feasible solution
(Garey and Johnson 1977, Savelsbergh 1985).
We have developed a branch & cut-approach
for the ATSP-TW (Ascheuer, Fischetti and
Graetschel 1997; Ascheuer, Fischetti and
Gr6etschel 1998) and a relaxation, namely the
ATSP with precedence constraints (Ascheuer,
Juenger and Reinelt 1997). The optimal solu
tions to these problems yield a lower bound to
an optimal online strategy if the same sequence
of nodes is generated. Computational tests based
on the production data from SNI showed that
there is still an online optimality gap of between
3-70%, with approximately 30% on average.
[ m0 -- P T. "A 5
MARCH 1998 PAGE /
We like to point out one important restric
tion of this a-posteriori analysis. This analysis is
based on the fact that the same sequence of
nodes is generated, independent of the way the
jobs are performed. This is not necessarily the
case for this application as the completion of a
certain job may have an influence on other gen
erated tasks. For example, consider the case that
a container delivered to the assembly line (task
A) contains parts that are not usable (e.g., they
are broken). As a result, the workers generate a
retrieval task B and order new parts (task C).
The sooner task A is performed, the earlier tasks
B and C are generated, and the earlier the time
window for B and C will become active. Thus,
there is no well defined optimal offline solution
to which we can compare the online solution.
As a consequence, competitive analysis cannot
2.2 Online-Optimization of a
The Herlitz PBS AG (WWW Herlitz) is the
main manufacturing firm for office supplies in
Germany. They maintain their Europe-wide dis
tribution center in Falkensee, close to Berlin. A
joint project is aimed at efficiently managing
their complete internal material flow. In a first
phase we have optimized one commissioning
area. We are currently working on the optimiza
tion of the whole pallet transportation system
consisting of a system of roller conveyors, ten el
evator systems and 18 AUSS.
Online optimization questions arise in the
following areas: routing of pallets; efficient con
trol of elevator systems; routing within the
AUSS; routing of commissioning vehicles.
2.2.1 Commissioning of Greeting
In this section we discuss one question in fur
their detail, namely the efficient commissioning
of greeting cards.
Description of the System. The cards are
stored in four parallel shelving systems (see Fig
ure 2). In accordance with the customers' or
ders, the different greeting cards have to be col
elected in boxes to be shipped to the customers.
Order pickers on eight AGV collect the orders
from the storage systems while following a cir
cular course. The vehicles are unable to pass
each other. Moreover, due to security reasons,
only two vehicles are allowed to be in the
middle aisles at the same time, whereas three are
allowed in the first and last aisle.
Gearnt Fahural Komninuioniemmbil
K .i4V a.IMa rInImhs N F a III -p
I 'I nrPI L I i v I i I6 I rl f r I I o I-J
-l . A. ..- .m -L. ,.\ ... Imua
Figure 2: Commissioning Area for Greeting Cards (screenshot from the simulation program)
At the loading zone each vehicle is "loaded"
with up to 19 orders. Afterwards, a dispatcher
decides when to send the vehicle onto the
course. After leaving this area the vehicles auto
matically stop at a position where cards have to
be picked from the shelf. Signal lights indicate
the position from where and to which box the
cards are picked.
The management was unhappy with the sys
tem since frequently vehicles ran into conges
tions and orders were completed late. For ex
ample, suppose that there is a vehicle that re
quires a lot of stops and the subsequent one only
has a few stops. In case the dispatcher sends
them onto the course, the fast vehicle will catch
up with the slow one immediately, resulting in a
congestion. As a consequence, the order pickers
of fast vehicles often left the AGV to smoke a
cigarette, etc., which resulted in further conges
Modeling. We suggested assigning the orders
to the vehicles in such a way that whenever a ve
hicle stops the order picker can collect as many
cards as possible; alternatively, for a given set of
orders, minimize the total number of stops to
fulfill these orders. In this way it is possible to
avoid some time-consuming deceleration, fine
adjustment, and acceleration phases for the ve
hicles. Besides minimizing the total number of
stops, we aim at reducing the time vehicles
spend in congestion. This can be modeled as a
mixed integer program. First computational test
showed that for some data sets provided by
Herlitz it took several hours of CPU-time just to
solve the linear relaxations of the MIP. Thus, an
exact solution approach was unsuitable for a de
ployment in the distribution center.
It could be shown that already the problem of
minimizing the total number of stops is NP
hard (Kamin 1998). Therefore, we implemented
several heuristics that reduce the total number of
10ta P T_ I" A 5
MARCH 1998 PAGE
stops required for the vehicles and evenly dis
tribute these stops among them. We used vari
ants of greedy and best-fit-algorithms with an
additional 2-exchange improvement heuristic.
In addition, we used a coarse simulation to de
termine the best starting time for each vehicle.
By this optimization-simulation approach, pre
dictable congestion are shifted to the loading
zone, where the order pickers can either have a
break or can be assigned other tasks.
Results. We implemented a very detailed simula
tion model for the whole commissioning area in
which we compared our approach to the one
used so far. Herlitz provided production data
from a period of about six weeks, which were
the basis for the comparison. The main results
are the following: a significant improvement
with respect to the completion times of the or
ders can be achieved; the number of vehicles can
be reduced from eight to six without any nega
tive impact on the system performance; conges
tions can more or less be avoided completely.
Vehicles run into congestion only for a few sec
More details can be found in Ascheuer,
Gr6etschel, and Kamin 1998; and Kamin 1998.
A prototype of the simulation approach is cur
rently tested by the support team of Herlitz for
its use as a decision support tool for the dis
Conveyor Modules in Large Scale Trans-
portation Systems. In connection with an
other cooperation with Herlitz AG, Berlin, we
are analyzing the following problem: the auto
mated pallet transportation system in a large dis
patch building of Herlitz in Falkensee has to
take care of a congestion-free flow of pallets
from/to ware-input, commissioning depart
ments, shelf system, and ware-output. Among
the building blocks for pallet transportation, the
following seem to be the most complex ones:
the automated shelf systems; the automated el
evator systems. Modules of these types are found
in many automated transportation systems.
One would like to describe how different
modules of such a system must be controlled in
order to work well together. The common prac
tice is to run very simple heuristics with empha
sis on avoiding congestion.
Prior to the investigation of the interplay be
tween the modules it is necessary to understand
the modules themselves. While for the auto
mated shelf systems we can use our experience
from the above mentioned project with SNI, the
S- AMSEL Verson IT: anumel_elevator
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a 10 Ie: 34
Inms -o. 12
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Figure 3: A Snapshot of the Animated Simulation of an Elevator System
elevator control problem is not well understood
so far. This is even the case in very elementary
settings, let alone real-world layouts with addi
tional restrictions to the flow from/to the eleva
A generic simulation environment for elevator
systems based on the event based simulation li
brary (AMSEL 1997) was designed in order to
test heuristic approaches to the problem (see
Conceptual Problems. Competitive analysis
is a mathematical performance measure of
online algorithms. It has the advantage of not
being dependent on the knowledge of the prob
ability distribution of the requests. However,
the online model that it is based on is too re
strictive for many real-world problems. For an
elevator system, e.g., there are usually many re
quests available at the same time, and the eleva
tor has the opportunity to make an offline
schedule based on the known information.
Moreover, not yet processed requests may be re
scheduled by the algorithm. This dynamic look
aheadshould be integrated into a generalization
of competitive analysis.
A very hard problem occurs if there is no cor
responding offline problem at hand. In these
cases even the definition of what should be an
optimal solution to the online control problem
is problematic. In control theory one computes
an optimal control at each point in time. Usu
ally one cannot ensure that these local optima
combine to a globally "optimal" solution if the
problem is discrete because the objectives are
not continuously dependent on the decisions.
0 P T_ I" A-5 7
3 Conclusion and Outlook
Online problems show up almost everywhere in
industrial production, logistics, etc. We have il
lustrated this by means of a few relevant ex
amples from practice. Online optimization
problems have, however, not received too much
attention from the mathematical programming
The field lacks "good" mathematical concepts
for decision support. From a practical point of
view, competitive analysis as well as similar ap
preaches rarely yield results that can guide deci
sion makers in the selection of which online
algorithm to use. Simulation experiments are
still the state of the art.
Nevertheless, by using the tools that have
been developed in combinatorial optimization
over the years, such as combining and modify
ing various heuristic and exact approaches for
associated offline problems, it is still possible to
improve considerably on what is currently done
in practice, as our examples show.
The research was supported by the Deutsche
Forschungsgemeinschaft (DFG) within the re
search cluster Echtzeit-Optimierung groler
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10 P T I M A 5 7
MARCH 1998 PAGE
W HEN I was asked to write this article, I thought immediately of the
amazing advances of the last dozen years relating to interior-point methods,
from Karmarkar's projective-scaling algorithm to the discovery and explore
tion of the concept of self-concordant barrier function by Nesterov and
Nemirovskii. (I don't think anyone will argue with the significance of these
developments; however, what follows definitely exhibits my own prejudices.
I hope the reader will find it interesting, even while deploring my taste!) As
exciting as these new approaches have been, I believe it's enlightening just
to consider how fascinating plain vanilla linear programming is, in all its
multi-faceted glory (pun intended).
I have long regarded the term "linear programming" as a misnomer, not
just because of the connotations of the word "programming," but also since
the subject is so much more complicated (and beautiful) than other linear
computational problems such as linear equations and linear least squares;
of course, this is due to the presence of inequalities. Inequalities lend a
piecewise-linear and nonsmooth element to a problem defined by linear
functions. Should we exploit the piecewise-linear nature of the problem
and use combinatorial methods like the simplex method, or approximate
the nonsmooth feasible region using a smooth but nonlinear function,
the barrier function, and use ideas from unconstrained (or linear
equality-constrained) optimization? Geometrically, what do typical high
dimensional polyhedra look like: are they like the mirrored balls hanging in
discos, for which edge-following methods like the simplex algorithm seem
very inefficient, while an interior-point approach appears very suitable; or
more like quartz crystals, with long edges running from one side of the
polyhedron to the other, where the reverse conclusion seems plausible?
As we celebrated the 50th birthday of the simplex method in Lausanne
last year, I was reminded again of how counterintuitive it is that such an al
gorithm is so efficient on large-scale problems. Of course, as George
Dantzig has remarked, our intuition about high-dimensional polyhedra can
be singularly misleading. Different viewpoints can give very contrasting in
dications: every student of mathematical programming should know the
simplex interpretation of the simplex method (Dantzig 1963), which led
Dantzig to believe that it could be efficient. But a vast amount of successful
computational experience over many years has perhaps left us blase about
the remarkable efficiency of edge-following algorithms.
One approach to explaining this efficiency is of course the probabilistic
analyses of the simplex method carried out by Borgwardt, Smale,
Haimovich, Adler, Karp, Megiddo, Shamir, and me in the early '80s
(Borgwardt 1987). But I want to highlight here a much older result on
polytopes, due originally to Caratheodory and rediscovered several times
since (see Griinbaum's classic book, Griinbaum 1967, for the history, or
the recent book by Ziegler, Ziegler 1995, which discusses the result in its
Oth chapter). This theorem challenges our intuition on polyhedra, giving
credence to the "many long edges" view that suggests the efficiency of the
simplex method. It also intriguingly points out behavior that starts in di
mension four, just beyond our power to visualize. Finally, the proof is very
Theorem. For every d f 4 and n> d, there is a d dimensional polytope
with n vertices which is 2-neighborly-every pair of vertices is connected by
I'll give the proof for d= 4 for simplicity; it clearly generalizes. Consider
the moment curve x() := (C t; j2; t') 4: t '. Note that the intersec
tions of this curve with the hyperplane x '4: a T = a0 correspond to the
roots of the polynomial -a + a t + 2 t + a3 f + a4 t, and conversely any
quartic (polynomial of degree at most four) corresponds to a hyperplane.
Choose any ti < 2 < < t, n t 5, and let the polytope Pbe the convex
hull of the corresponding n points on the moment curve. First note that
this is indeed a 4-dimensional set, as any hyperplane (corresponding to a
0 P T I M A -5 71
quartic) can contain at most four of the n points. Next, each of the n points
is indeed a vertex of P, since the hyperplane corresponding to (t t)2 con
tains the ih point x(t), with all others strictly on one side. Thus this hyper
plane supports the polytope precisely at the ih point. Finally, for any two
vertices, say x(t) and x(t), consider the hyperplane corresponding to the
quartic (t- t)2 (t- t2. This contains the two specified vertices, with all oth
ers strictly on one side, so it supports Palong the edge joining these two
vertices. The proof is complete.
Before leaving this example, we note two things. First, for d t 4, a similar
proof shows the existence of d/2 &neighborly polytopes with arbitrarily
many vertices; here the convex hull of anyset of d/2 Ivertices forms a sim
plicial face of the polytope. Second, the presence of edges linking every pair
of vertices merely hints at the simplex method's efficiency. While there is an
edge from any initial vertex to the optimal vertex, both are highly degener
ate, and many pivots may be necessary to reach an optimal basis. Moreover,
all vertices when projected onto the x,,x,) plane lie on the parabola x = x12,
and hence they are all "shadow vertices" under this projection; thus an un
luckily chosen parametric objective simplex method might go through every
vertex on its way to the optimal vertex.
I chose this result because it is simple, counterintuitive, insightful, and I
believe not widely known. Here is a list of some of the "runners up." I have
been very impressed with two results using convex programming to attack
discrete optimization problems: Lovasz's bound for the Shannon capacity
problem (Lovasz 1979) and Goemans and Williamson's .878-approxima
tion algorithm for the max-cut problem (Goemans and Williamson 1995),
both using semidefinite programming. I am very fond of the result, proved
independently by Adler, Karp, and Shamir, by Adler and Megiddo, and by
me, that a certain lexicographic parametric variant of the simplex method
requires an expected number of pivots growing only quadratically with the
smaller dimension of the problem, to solve an LP instance generated by a
particular class of probability distributions. (It may seem strange that Adler
is a member of two of these groups; however, at the time it was thought
that these two papers addressed different algorithms-only later was it real
ized that they were in fact the same.) For a discussion, see again Borgwardt's
book; unfortunately, the class of distributions is rather unsatisfactory. I
stand in awe at the intellectual achievement of Nesterov and Nemirovskii in
describing the class of convex programming problems for which efficient in
terior-point methods can be derived, although it is hard to point to a spe
cific encapsulating theorem (Nesterov and Nemirovskii 1994). Finally, I
value very highly the recent results of Kalai on subexponential bounds on
the expected number of pivots for a certain randomized pivot rule (Kalai
Let me turn now to open problems. Here again the field of interior-point
methods presents many possible choices: what is the "best" infeasible-inte
rior-point algorithm (a method that starts with an infeasible solution and
works toward both feasibility and optimality, or possibly toward detecting
primal or dual infeasibility)? What are the "right" search directions to use in
a primal-dual method for semidefinite programming (there seem to be at
least four or five candidates, see Todd 1997), or for general nonlinear pro
gramming? Is there an easily computed self-concordant barrier function for
a d dimensional polyhedron with n facets, n >> d, with parameter close to t.
(Existence is known due to results of Nesterov and Nemirovskii, but the
only known explicitly computable barriers have parameters n or ( no).)
And of course, explaining the good behavior of the simplex method still re
quires a definitive resolution: is there a polynomial simplex method, is the
bounded Hirsch conjecture true?
My favorite open problem is in fact the analogous question for interior
point methods for LP. This may seem paradoxical, since the methods are
provably polynomial-time. But the bounds on the worst-case number of it
erations to attain a given precision are O(n) or O( n), where n is the num
ber of inequalities, whereas the performance in practice is much better. In
deed, early computational experience led some to believe that there might
even be a constant bound, whereas results of Lustig et al. (Lustig, Marsten
and Shanno 1990) suggest that the growth in practice may be of order in n
(they solve "slices" of a fixed problem, with n varying up to about two mil
lion). In terms of In n, this is an exponential gap, as occurs for most variants
of the simplex method! And if the methods really took W (n) or W ( n) itera
tions, they would be hopelessly inefficient on very large problems. So we
still need to explain why primal-dual methods work as well as they do.
Some probabilistic analysis has been performed, but it does not explain
the drop from polynomial to logarithmic in n. On the negative side, we
now have results suggesting that there really is a large gap, i.e., that the gap
is notjust due to our inadequate analysis, but that pathological problems
exist. Results of mine, extended in joint work with Ye (Todd and Ye 1996),
show that for many primal-dual interior-point methods quite close to those
used in practice, there exist instances with n inequalities for which the
methods require W (n/3) iterations to achieve a very modest reduction in the
duality gap. So, at least for these methods, it is not possible to improve the
worst-case bound by much. Thus we are left with the challenging question:
can we explain the very slow growth rate in practice? The answer is left as
an exercise for the reader!
M. J. TODD
K. H. Borgwardt, The Simplex
Method, a Probabilistic
Analysis, Springer Verlag,
G. B. Dantzig, Linear Programming
and Extensions, Princeton
University Press, Princeton,
M. X. Goemans and D.P.
for maximum cut and
satisfiability problem using
Journal of the ACM42
(1995) 1115 1145.
B. Grfinbaum, Convex Polytopes,
Wiley Interscience, London,
G. Kalai, Linear programming, the
simplex algorithm and
simple polytopes, Math-
ematical Programming 79
(1997) 217 233.
L. Lovasz, On the Shannon capacity
of a graph, IEEE Transac-
tions on Information Theory
25 (1979) 1 7.
I. J. Lustig, R. E. Marsten, and D. F.
Shanno, The primal dual
interior point method on the
Cray supercomputer, in: T.
F. Coleman and Y. Li,
SIAM, Philadelphia, 1990.
Yu. E. Nesterov and A. S.
Nemirovskii, Interior Point
Polynomial Methods in
Theory and Algorithms,
SIAM, Philadelphia, 1994.
M. J. Todd, On search directions in
interior-point methods for
Technical Report No. 1205,
School of Operations
Research and Industrial
University (October 1997).
M. J. Todd and Y. Ye, A lower
bound on the number of
iterations of long-step and
algorithms, Annals of
Operations Research 62
(1996) 233 252.
G. M. Ziegler, Lectures on Polytopes,
Springer-Verlag, New York,
10 P T I M A 5 7
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S3 M A H Y J J A D I T H 3 a
) Internation Conference on Interval Methods and Their Application
in Global Optimization (INTERVAL '98)
April 20-23, 1998, Nanjing China.
) INFORMS National Meeting
April 26-29, 1998, Montr6al, Quebec, Canada.
) IPCO '98 Sixth Conference on Integer Programming and
June 22-24, 1998, Houston, TX.
) INFORMS International Meeting
June 28-July 1, 1998, Tel Aviv, Israel.
)4th International Conference on Optimization
July 1-3, 1998, Perth, Australia.
) APPROX 98 1 st International Workshop on Approximation
Algorithms for Combinatorial Optimization Problems
July 18-19, Aalborg, Denmark.
) Optimization 98
July 20-22, 1998, Coimbra, Portugal.
S2nd WORKSHOP ON ALGORITHM ENGINEERING, WA E'98
August 19-21, 1998, Saarbruecken, Germany.
) INFORMS National Meeting
October 25-28, 1998, Seattle WA.
) Sixth SIAM Conference on Optimization
May 10-12, 1999, Atlanta, GA.
MARCH998 PGE 1
Workshop on Non-Standard Methods
for Integer Programming
Utrecht, The Netherlands
April 20, 1998
The Worskshop is sponsored by ALCOM-IT, ES
PRIT Long Term Research Project No. 20244,
and by the graduate school IPA -Institute for Pro
gramming Research and Algorithmics.
Several problems that occur in areas such as tele
communication network design, routing, plan
ning and scheduling can be modeled as integer
programming problems. During the past decade
impressive algorithmic results have been obtained
for problems where the variables can take values
zero or one. In general integer programming,
much less is known about practical optimization
algorithms. During this workshop some recent ap
preaches (e.g., test sets, basis reduction, and group
relaxations) to solving general integer program
ming problems will be discussed.
The meeting will be held at the "Uithof Campus"
of Utrecht University, Centrumgebouw Zuid,
Heidelberglaan 1, lecture hall F125. Directions to
the workshop site are included online (http://
You can register by filling out the form on the
Workshop homepage (http://www.cs.ruu.nl/docs/
ipworkshop/index.html), or by sending an e-mail
to Karen Aardal (email@example.com). If you register
via e-mail, please include your address, phone
number and fax number.
The registration fee is NLG 30,00, and can be
paid to postal giro account number 1443360 in
the name of Karen Aardal. Please mention "IP
Workshop" on the payment form. Each registered
participant will be offered lunch, and coffee and
tea during the break. On-site registration is pos
sible, but we can only guarantee lunch for partici
pants that register before April 13, 1998.
The program includes lectures by Karen Aardal,
Imre Barany, Milind Dawande, Matteo Fischetti,
Robert Weismantel and Laurence Wolsey.
The full program can be found on the conference
web page. For further information, please contact
Karen Aardal (firstname.lastname@example.org).
Karen Aardal and Laurence Wolsey
Ettore Majorana Centre for Scientific Culture
International School of Mathematics
"G. Stampacchia" Workshop
Nonlinear Optimization and Applications
June 23 -July 2, 1998
The workshop aims to review and discuss re
cent advances and promising research trends
concerning theory, algorithms and innovative
applications in the field of nonlinear optimiza
tion. Both the finite and the infinite dimen
sional case will be of interest.
Topics include, but are not limited to:
Constrained and unconstrained optimization;
Convex analysis; Global optimization; Interior
point techniques for linear and nonlinear pro
gramming; Large scale optimization; Linear
and nonlinear complementarity problems;
Nonsmooth optimization; Neural networks
and optimization; Applications of nonlinear
As usual, the course will be structured to in
clude invited lectures and contributed lectures.
Proceedings including the invited lectures and
a selection of contributed lectures will be pub
lished. The following is the list of invited lec
turers: V.F. Demyanov, St. Petersburg State
University, St. Petersburg, Russia; M.
Fukushima, Kyoto University, Kyoto, Japan;
N.I.M. Gould, Rutherford Appleton Labora
tory, England; A. Ioffe, Technion University,
Haifa, Israel; O.L. Mangasarian, University of
Wisconsin, Madison, WI, USA; P. Marcotte,
University de Montreal, Montreal, Canada;
J.J. More, Argonne National Laboratory,
Argonne, IL, USA; J. Nocedal, Northwestern
University, Evanston, IL, USA; J.-S. Pang,
Johns Hopkins University, Baltimore, MD,
USA; P.M. Pardalos, University of Florida,
Gainesville, FL, USA; E. Polak, University of
California, Berkeley, CA, USA; L. Qi, Univer
sity of New South Wales, Kensington, NSW,
Australia; T. Rapcsak, Hungarian Academy of
Sciences, Budapest, Hungary; S.M. Robinson,
University of Wisconsin, Madison, WI, USA;
R.T. Rockafellar, University of Washington,
WA, USA; Ph. L. Toint, FUNDP, Namur,
Belgium; P. Tseng, University of Washington,
WA, USA; M.H. Wright, AT&T Bell Labora
stories, NJ, USA; S. Wright, Argonne National
Laboratory, Argonne, IL, USA; J. Zowe, Uni
versity of Erlangen-Nuernberg, Germany.
How to Participate
Persons wishing to attend the workshop
should write to: Prof. Gianni Di Pillo,
Dipartimento di Informatica e Sistemistica,
University di Roma "La Sapienza" via
Buonarroti 12, 00185 Roma, Italy (E-mail:
They should include date and place of birth,
together with present nationality, affiliation,
address and e-mail address. If they want to
contribute a lecture, they should also include
the title and abstract of the proposed lecture.
Young persons with only limited experience
should enclose a scientific curriculum vitae
and a letter of recommendation from the head
of their research group or from an experienced
person in the field. The total fee, which in
cludes full board and lodging (arranged by the
School), is US $800.
Closing date for application is April 30,
1998. Application by e-mail is strongly
Availability is limited. If necessary, admission
to the workshop will be decided in consult
tion with the Advisory Committee of the
School comprised of Professors F. Giannessi,
G. Di Pillo and A. Zichichi and will be com-
municated shortly after the closing date for ap
Participants must arrive in Erice on June 23
no later than 3 p.m. and will leave on July 2.
How to Reach Erice
Erice is situated in the northwest corner of
Sicily in the south of Italy. The easiest way to
reach it is to take a plane to Palermo or
Trapani. The Majorana Centre will then pro
vide transportation to Erice. More details will
be given to successful applicants.
More information about the workshop and a
sample data sheet for participants can be
found at the web site
Franco Giannessi, School Director
Gianni Di Pillo, Course Director
10 P T I M A 5 7
MARCH1998 PAGE 1
19th IFIP TC7 Conference on Sys-
tem Modeling and Optimization
July 12-16, 1999
Further information about this
meeting is now available on the
na/tc7con/). This web site includes
a list of conference topics, the
names of the plenary speakers, and
a call for submitted papers. Special
attention will be given to general
algorithms for optimization calcu
lations by reserving a parallel ses
sion throughout the meeting for
talks on this subject.
The following people have ac
cepted invitations to present the
plenary lectures: Martin P.
Bendsoe, Technical University of
Denmark; Robert E. Bixby, Rice
University, USA; Asen L.
Dontchev, American Mathematical
Society, USA; Nicholas I.M.
Gould, Rutherford Appleton Labo
ratory, GB; Julia L. Higle, Univer
sity of Arizona, USA; Rolf H.
Mahring, Technical University of
Berlin, Germany; Fadil Santosa,
University of Minnesota, USA; Jan
C. Willems, University of
Groningen, The Netherlands;
Henry H. Wolkowicz, University
of Waterloo, Canada; Stephen J.
Wright, Argonne National Labora
The deadline for submitted papers
is January 31st, 1999 (next year). If
you wish to receive future an
nouncements and reminders about
deadlines, please send an e-mail
International Congress of
August 18 27, 1998
The Second Announcement of
ICM'98 has been printed and ship
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tel reservations, and the handling of
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ganization company. When you
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The ICM'98 web site (http://
elib.zib.de/ICM98) has been rede
signed completely. The new version
basically follows the structure of
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ing months to reflect the progress
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If you click on "info" or on the
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rectly, you will find that informa
tion is now grouped into three
main categories (plus two addi
tional sections): General Informa
tion, Detailed Information, and In
formation Regarding the Organiza
tion of ICM'98.
In "General Information" you will
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registration form; the Second An
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The new category on the web site,
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However, the subsections will soon
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spending items in the Second An
Finally, the section "Information
Regarding the Organization of
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formation" about the various com-
mittees and institutions involved in
the preparation of the Congress, as
well as some items on the historical
development of ICM'98.
We hope that the new structure
will enable you to quickly find the
information for which you are
looking from now through the end
of ICM'98, August 18-27, in Ber
Martin Gr6etschel, President of the
ICM'98 Organizing Committee
0 P T I A -5 7
Developments in Global
edited by Immanuel E. Bomze,
Tibor Csendes, Reiner Horst and
Panos M. Pardalos
Kluwer Academic Publishers,
The objective of global optimization (GO) is to find
the "absolutely best" solution of (potentially)
multiextremal optimization problems. In recent years,
GO has found numerous applications in the sciences,
engineering and economics. Nonlinear approximation,
information retrieval, engineering design, extremal
energy models (in physics, chemistry, biology), and
I 1. 1 ...ul............I. I .... .. . .II.i I. .
tive examples from a dynamically expanding list of areas
in which GO has imminent relevance. (Several chap
ters of the book reviewed discuss further examples.)
The book is Volume 18 in the fast growing Kluwer
series on Nonconvex Optimization and Its Applica
tions. It is based on refereed contributions submitted
by participants of the Third Workshop on Global
Optimization (Szeged, Hungary, December 1995).
Thevolumeconsistsof 1' i. I.... .. i....
* Neumaier discusses NOP, a compact input format
to formulate nonlinear optimization problems
* Dallwig, Neumaier and Schichl describe GLOPT, a
program system developed for solving constrained
* Vrahatis, Sotiropoulos and Triantafyllou suggest a
new approach to solve "noisy" (i.e., imprecisely given)
* Ratz provides new results related to extended interval
Newton Gauss Seidel steps applicable in rigorous GO
* De Angelis, Pardalos and Toraldo describe global
optimality conditions and computational approaches
to the (indefinite) quadratic programming problem
(QP) under box constraints
* Bomze, Pelillo and Giacomini present a new algorithm
to solve the maximum clique problem (in a form lead
ing to general QP over a simplex)
* Stephens discusses finding lower and upper bounds on
the Hessian of a function over (box) search
(sub)domains, again in an interval GO context
* Strekalovsky and Vasiliev consider non-convex opti
mal control problems, related to the maximization of
a convex function of the terminal state
* Price presents a multistart clustering I ..I -, I algo
rithm which exploits gradient information
* H .. ,.. I i- .. ......... I .... -,, .... ..
speed of an integral approach for computing the es
sential supremum of an objective function
* Zabinsky and Kristinsdottir provide a Markov chain
analysis for combining pure adaptive search (an "ideal
algorithm) with passive random search
* Pinter describes a model development and solver sys
tem for continuous and Lipschitz GO
* Sergeyev presents an algorithm for solving one-dimen
sional GO problems in which the objective has a
* Dill, Phillips and Rosen propose a convex global
underestimation procedure, subsequently applied to
molecular structure prediction
* Pfening and Telek analyze renewal policies for slowly
degrading ("aging") mass service systems
* Bollweg, Maurer and Kroll study the numerical pre
diction of crystal structures by applying a simulated
* Garcia, Ortigosa, Casado, Herman and Matej describe
a parallel stochastic GO method applied to image re
* Holmquist, Migdalas and Pardalos study a greedy
randomized adaptive search heuristic to solve location
problems with economies of scale
* Imreh, Friedler and Fan provide an improved bound
ing procedure to solve complex process network syn
thesis problems by branch-and-bound
As can be seen even from this very sketchy descrip
tion, the book discusses a broad spectrum of GO top
ics, encompassing theoretical results, algorithm devel
opment, decision support systems, as well as a variety
of challenging practical applications.
This volume can be recommended to researchers and
graduate students working on the areas of mathemati
cal programming, operations research, computer sci
ence, applied sciences, economics and engineering.
JANOS D. PINTER
Linear Programming 1:
George B. Dantzig and
Mukund N. Thapa
This introductory book on linear programming is de
signed to be used in an undergraduate course. It surveys
linear programming and network flow problems, espe
cially the simplex algorithm and variants of it. The text
is the first of three volumes; the second will cover theory
and implementation and the third will cover structured
linear programs and planning under uncertainty. It
comes with a CD-ROM, which contains implement
tions of many of the algorithms described in the text.
One of the authors needs no introduction! The other
author was a Ph.D. student under Professor Dantzig in
the 1970s, and now runs his own company and also
teaches at Stanford University.
There is a twelve page preface by George Dantzig
discussing the early history of linear programming. This
is entertaining and interesting, and should provide good
motivation for the student. It also includes a list of recent
applications of linear programming and its extensions,
and shows its growing importance, and how it has
. ... 1. 1. .|1 1. ....|1. ... I. d. .. I. .|.II .. I I .. I. .... . I .
Chapter 1 is an introductory chapter, with several
examples of formulating linear programming problems.
Row and column approaches to developing formula
tions are discussed, and the student is taken through
them step by step.
Chapter 2 discusses the solution of simple linear
programming problems. It considers graphical solution
of problems with two variables or two constraints. The
approach for the two-constraint case gives motivation
for a definition of the dual problem. The dual problem
is not solved graphically, but is used as a check of
optimality. Fourier-Motzkin elimination is also dis
cussed, perhaps surprisingly for an undergraduate text,
since it is really a theoretical procedure, although it is
easy to understand conceptually.
The simplex method is the subject of Chapter 3. This
III.....hII..I ,,,,I,, h , hi,,,_, treatm ent
of upper and lower bounds and the revised simplex
method. The case of a linear program with an un
bounded optimal value is described by a theorem; it
would have been useful to also include an example. The
case where some artificial variables remain basic at the
end of Phase I is discussed thoroughly. To emphasize
the integration of the two phases, t I.. i, il ,. .. I. I
example in the chapter requires both Phase I and Phase
Interior point methods are discussed in Chapter 4.
This consists of a short treatment of the primal affine,
or Dikin's, method. There is only one rudimentary
picture. The treatment emphasizes the algebra of a single
iteration. It would have been nice to have seen, for
example, a discussion of the fact that Dikin's step is
equivalent to minimizing the objective function over an
inscribed ellipsoid. The choice of step length for which
the algorithm is guaranteed to converge should have been
discussed. There is no discussion of obtaining a dual
solution when using this method. It would be good to
have more treatment of interior point methods, espe
cially the polynomial time and practical potential reduc
tion and path following methods, since these methods
require only slightly more motivation than Dikin's
Chapter 5 covers duality. Finding the dual of any
system is described. The remaining material in the
chapter constitutes only four pages plus a bibliography
and exercises. This material should have been expanded,
with more examples and more illustration of its impor
tance and mathematical structure. For example, the
section on obtaining a dual solution from the final tab
leau is very brief and only considers the case where all
the artificial variables are in the final tableau. Theorems
of the alternative are not considered. This chapter would
have been improved by the inclusion of Section 7.1,
which considers shadow prices and provides more
motivation for duality, including discussion by means
of two long examples.
Chapter 6 discusses problemreformulations, include
i,,_ h ,,,,IIh ,,_ h , ,,, ,l.1.h _, .1 ...... ........... i,,, .l ,,,
and problems with piecewise linear objective functions.
The authors do not mention that splitting variables is
not a good technique with an interior point method
because the set of optimal solutions becomes un
bounded. Section 6.7 contains one notable error: it is
stated that a function is strictly convex if and only if its
Hessian is positive I. i.... . ....
The remaining sections of Chapter 7 are concerned
withsensitivityanalysis. U .... ...1i i i i.. 1 ...1 .. I ...
in the entries in the constraint matrix is discussed, with
analysis using the Sherman-Morrison-Woodbury for
mula. One of the weaknesses of the text is that the dual
simplex method is not discussed-it is instead delayed
to the second volume. The sensitivity analysis of the case
when a constraint is added is therefore somewhat cum-
bersome, requiring the use of a Phase I method.
The last two chapters cover network problems and
together constitute 110 pages, approximately a third of
the main body of the text. The use of figures is extensive
and helpful, and the text explains the various algorithms
well. Chapter 8 discusses the transportation problem.
A Phase I procedure for the capacitated transportation
problem is also considered. Chapter 9 considers general
networks. It covers a broad range of topics, including
augmenting path algorithms, the max flow/min-cut
theorem, the shortest path problem, the spanning tree
problem, and the network simplex algorithm. The dis
cussion of network simplex, in particular, is extensive.
The definition of "strongly connected" mistakenlyuses
chains rather than directed paths, a potentially confus
ing typographical error.
There are two appendices covering background
material in linear algebra and solving systems of equa
tions. Material on numerical implementation of the
solution methods is delayed to the second volume.
There are lots of exercises at the end of each chapter,
and also sprinkled throughout each chapter. Many of
these make extensive use of the CD-ROM, and there
are plenty of more traditional exercises. Some of the
exercises are quite challenging, having been drawn from
various Stanford Ph.D. comprehensive exams.
The CD-ROM contains implementations for a PC
of the primal simplex method, Dikin's algorithm, Fou
rier-Motzkin elimination, network simplex, Dijkstra's
algorithm, and other network algorithms. It is easy to
use and works well for small instances. The only prob
lem I noticed with it is that the implementation of
Dikin's algorithm may give incorrect answers for badly
There are selected bibliographies at the end of each
chapter. These generally contain lots of references to
seminal work in the '40s, '50s, and '60s, with more
limited coverage of more recent work. The References
section is somewhat lacking in selectivity. Not every
paper in this section is mentioned in the text; for ex
ample, nine papers by M. J. Todd are listed, but his name
does not appear in the index. The References section
0 P T I M A 5 71
contains sixteen pages of paperswhere Professor Dantzig
is the primary author.
The tone of the text is somewhat mathematical, and
the students must be comfortable with matrix notation
and introductory linear algebra. However, most proofs
are delayed until the second volume, as are details of
modifications required to handle large scale problems.
There are a number of examples, and these are oftenused
to illustrate theorems. Nonetheless, the textwould have
benefited from more examples in Chapters 5 through 7.
This text covers the main topics of linear program
ming, with supplementary material on networks. More
material should have been included on interior point
methods, since these arebecoming ever more important.
The text is clearly i 1 i.. .. 1. .. . 1 I. .. .... I
graduates with some mathematical sophistication. The
CD-ROM provides a useful accompaniment to the text.
Some instructors will find this text very suitable for an
introductory course in linear programming.
JOHN E. MITCHELL
Theory and Algorithms for
Linear Optimization: An
Interior Point Approach
by C. Roos, T. Terlaky and
Wiley, Chichester, 1997
Interior point methods applied to Linear Programs have
reached a high level of sophistication, with new scien
tific publications slowing down. This is a clear indica
tion that the :. I. I .. .. I / to be digested in book form.
It is therefore no surprise that several books on the topic
have recently been published; for instance, thebooksby
Saigal, Vanderbei, and Wright. The present book is
nevertheless an interesting addition to those books. It
consists of 20 chapters, grouped into four parts.
The first part covers the basics of Linear Optimiza-
tion (LO). Duality and polynomial solvability are intro
duced using the skew-symmetric model for Linear Pro
grams. Theoretical results, like the existence of strictly
complementary solutions, are derived through proper
ties of the central path. The complexity is obtained in
an elementary way through the Dikin direction.
Part 2 of the book takes a closer look at various ways
to solve LO through barrier methods. Chapters 6 and
7 are similar in structure and provide a thorough analy
sis of the Dhal Rarripr A//thodand the
Primal Duall 1 .,. I.. The
Newton-direction is introduced, and sufficient condi
tions for local quadratic convergence with full Newton
steps are investigated. The predictor-corrector approach
is analyzed in detail.
The second half of the book is devoted to more
advanced topics. In Part 3, the general T F .
Approach is presented in depth, with an analysis of the
Primal, the Dual and the Primal-Dual Newton Meth
ods. The last part, Miscellaneous Topics, consists of seven
chapters. Here one finds Karmakar's Projective Method,
adiscussionof Iii .. II .,11 I. i ..i .. ... fthecentral
path, higher order search directions, and parametric and
sensitivity analysis. The book closes with a chapter on
The book is carefully written, with an emphasis on
mathematical detail. It can serve as a basis for a modern
treatment of Linear Optimization atboth graduate and
undergraduate level. The most distinguishing feature of
the book lies in the convergence proofs of all the inte
rior pointvariants addressed. These are elementary, and
at the same time elegant. The most prominent topic not
addressed is infeasible interior point methods. In sum
mary, the book is recommended for anyone doing re
search or teaching in the field of interior point methods.
Optimization on Low Rank
Noncon vex Structures
Hiroshi Kono, Phan Thien Thach
and Hoang Tuy
Kluwer Academic Publishers,
The last decade has seen immense progress of global
optimization. Several general concepts and algorithmic
paradigms have been developed which proved useful in
solving small to medium sized global optimization
problems. A real breakthrough, however, has occurred
in solving global optimization problems with special
structures like lowrank nonconvex programming prob
lems. Such problems are characterized by the fact that
theybecome convexwhen avector of the formBxis fixed
where Bis a low rank matrix. Such problems occur in
many engineering or economic applications like loca
tion problems, design centering and others.
This monograph describes the theoretical found
tions, methods and algorithms as well as selected appli
The first six chapters are devoted to the foundations of
the field. Though self contained, some knowledge of
convex analysis is helpful. After a general introduction,
the notion of quasi-convexity is discussed, including
quasi conjugacy and quasi-subdifferentials. Starting
from typical examples, Chapter 3 develops the theory
of differences of convex functions and sets
(d.c.functions, d.c.sets). The authors continue with a
duality framework for important classes of nonconvex
problems and set the foundations for low-rank
nonconvex structures. The sixth and last chapter of Part
1 deals with global search methods and basic algorithms
The second part of this monograph develops numeri
cal methods and algorithms for solving typical special
classes of global optimization problems by exploiting
their low rank structure. In particular it deals with
parametric approaches for solving low rank nonconvex
quadratic programs and concave minimization prob
lems, with multiplicative programming problems,
and with monotonic problems. In addition, a price
directive decomposition approach is developed and
dynamic programming algorithms in global optimiza
tion are discussed.
The third part deals with selected applications. It
starts with a chapter on low rank nonconvex quadratic
programming problems; then continuous locationprob
lems are discussed; and finally, design centering and
related geometric problems are described. It closes with
a chapter on multiobjective and bilevel programming.
Throughout the text numerous new results are given.
" I...... . ,I . ... . . ..... .. . .-....... , ,14 0 0
titles will prove to be very useful. All in all, this mono
graph is a highlywelcome enrichment of the mathemati
cal programming literature which gives a comprehen
sive insight in a very active field.
TRAINER E. BURKARD
10 P T I M A 5 7
Tom Magnanti, who recently became Institute Professor at MIT, has also been awarded the degree
of doctor honors causa by IAG, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium...
Ravindra Ahuja (IIT, Kanpur), Stanislav Uryasev (Brookhaven) and Joe Geunes (Penn State) will
join the ISE Department and Center for Applied Optimization at the University of Florida in August,
1998... Peter Hammer (RUTCOR) received an Honorary Doctorate (Laurea Honoris Causa) from
La Sapienza University in Rome on March 23, 1998... Panos Pardalos has been awarded a three
year appointment to a University Research Foundation Professorship at the University of Florida.
Deadline forth nextO P T I M A May 15, 1998
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