Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00047
 Material Information
Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: October 1995
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Bibliographic ID: UF00090046
Volume ID: VID00047
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.


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Mathematical Programming

and the Web: To surfor not to surf

Advancements in computer technol-
ogy have influenced every aspect of
our lives: the way we work, the way
we communicate, the way we think,
and the way we spend our leisure
time. Those ofus who tried to keep
our involvement with computers to
the bare minimum could not stop the
Internet invasion four offices and
homes. This computer revolution
went the extra step when the World
Wide Web arrived on the net: One
can access most of the Internet's
wealth of information by simply
clicking on a highlighted text. But,
where does mathematical program-
ming fit into all of this? We, as
tioners, can benefit from the Internet
and its resources. Rather than
watching the boat sail away, we can
jump on it and enjoy a nice ride.
Oops, let me re-phrase what I said:
Rather than watching the board surf
away, let's jump on it and start
surfing the net.

Well, I am a mathematician, can you
start with the definitions?
The Internet is the largest computer network
in the world. Regional and local computer
networks are connected with each other to
form this giant computer network. The U.S.
Department of Defense started the effort, as
usual, by building ARPAnet about 25 years
ago. At the same time, many companies, such
as IBM, started building their own local net-
works. The National Science Foundation
then built its own network that linked the
universities in the U.S. with each other. In
1987, Merit Network Inc., which ran the
University of Michigan's network, took over
the role of upgrading the NSFnet. I guess
you know the rest of the story: Better con-
nections were built, many other networks
joined, and we ended up with the Internet.
The World Wide Web is the latest and most
sophisticated service on the Internet. Given
that the Internet is a big computer network,
the amount of information that can be stored
on its computers is massive. Many of the
Internet users make their ideas, thoughts, re-
search, and computer codes available for oth-
ers by storing them electronically in desig-
nated locations on the net. This information
can be retrieved using special computer pro-
grams which are called browsers in Web ter-
minology. The two most common Web
browsers are Mosaic and Netscape. Both are
available at no cost (at this time) on all com-
puter platforms. The World Wide Web and
its browsers seem to be the easiest and the
most flexible and fun tools on the Internet at
present. They are expected to become the
predominant method for surfing the net in
the next few years. PAGE TWO

report from the chair 3

article: stuctures
conference notes


book reviews


gallimaufry 16






PAGL 2 N0 47 OCTOBER 1995

the Web

So, what am I going to gain from this
Web thing?

This is a tough question to answer precisely.
When electronic mail started, many people
thought of it as a wasteful tool. Nowadays, most
of us use electronic mail to correspond with col-
leagues and friends, and no one can ignore its use-
fulness and importance. The same story holds
true for the Web. We are delaying our exposure
to it by making I'll ..I -i excuses, such as "I do
not have time for this" or "I am a theoretician
and do not have to deal with it." However, the
real reason for avoiding this tool as I as many
other computer tools is our lack of knowledge
and understanding of computers and our failure
to put a few hours into learning the new
As someone involved in mathematical program-
ming, this is the least that you can expect to get
from the Internet and its Web:
* RESEARCH. The Web permits the creation
of bulletins that are accessible by any person con-
nected to the Internet. These .n!. i e can be up-
dated easily and frequently. The moment an up-
date has taken place, the readers of a bulletin are
able to access the new version immediately.
Hmm, maybe I am not making myself quite clear
here; an example may help to explain my point.
Let us say that you are interested in interior point
methods for solving linear programs. There is a
bulletin, or a page, on the Web that contains all
recent papers and reports regarding the subject
InteriorPoint/). You can access this page by giving
its electronic address to your Web browser. Re-
member, these are articles that are dropped there
by their authors without going through the
lengthy refereeing process. You can save any of
these articles on your computer disk or print it
out. In return, you can leave an electronic copy of
your documents on the Web. Interested people
can read and print these documents and give their
feedback. There isn't any need to send hard cop-
ies by mail any more, to wait for somebody to re-
spond, and to beg for help from secretaries.
* TEACHING. This is an area from which
I can talk from personal experience (http://
I recently taught a simulation class and thought
about placing all the class material on the Web.
It seemed a little strange in the beginning: class
notes on the Web, homework assignments and
their solutions on the Web, illustrative figures on
the Web, and even grades on the Web. After a
couple of weeks, the students realized the conve-
nience of having the Web. They were able to ac-
cess everything related to the class from any com-
puter connected to the network. Nobody wanted
an extra copy of a handout or did not do the

homework because he/she was not in the class.
The material was out there in an organized con-
venient fashion.
* PUBLICATIONS. Many professional socie-
ties are moving towards electronic publishing
An author submits .I i article in an electronic
form through the Web. The article is sent elec-
tronically to the referees, the corrections and
comments are sent back to the author, and finally
the article appears on line. It seems like a dream,
but Hey, welcome to Cyberspace!
* OTHER ISSUES. Believe me, it is a com-
pletely different world out there. The amount of
information available is unbelievable. When I get
into my office in the morning, I read the newspa-
per, check the stock prices, and look at the
weather forecast on the Web. One can even order
a pizza on the Web in some places. Millions of
bytes are devoted to any subject you can think of;
they are waiting for you to click on their links and
activate them.
When can I start surfing the Net? (without
getting wet)

As a computer user, you can see that computer in-
terfaces are friendlier than ever. The Web brows-
ers are no exception: They are very simple to use.
* Start your browser (Mosaic or Netscape) by
double clicking on it. If you cannot find it on
your machine, ask your system administrator to
guide you to its location on your system. In a few
seconds, a new window appears on your screen.
* Go to the file menu and select Open Loca-
tion... or Open URL.... In the new box, type in
the: .11..
http://www.cs.rice.edu/ packy/mps/
then hit return. After a few seconds, the contents
of the window changes. Guess where you are!
* The underlined text is a link to another docu-
ment that contains more information about this
text. Click on the text and see what you get.
* There are many buttons and menu items,
such as Back and Forward, that can make your
ride smoother. If you get lost on the Web, do not
worry, just click on the Home button. When you
find something you like, you can add it to your
list of interests by selecting Add a Bookmark or
Add to Hotlist. Later, it can easily be retrieved
using View Bookmarks or Hotlist....
Here we go, you are an expert surfer after this
small lesson. Remember, surfing the Internet is
Markovian: It does not matter how long you have
been using the net. We are all experts.
Here are some pointers to Web pages that are re-
lated to Mathematical Programming:

* Open
to access the electronic system of the American
Mathematical Society
* Open
to access the Institute of Operations Research and
the Management Sciences home page
* Open
to access the page of the Society for Industrial and
Applied Mathematics
* Open
to access the Linkaping Mathematical Program-
ming Library
* Open
to access the NEOS Guide Optimization Tree at
Argonne National Laboratory
* Open
to access Michael Trick's Operations Research
* Open
to access the Opt-Net Home Page
Wow, this is getting interesting.
How can I spin my own Web?

Building your own Web page is not hard at all.
The Web is based on a technology c ll.. 1
hypertext. Usually, you need to create a file,
called index.html, that contains the information
which you want to place on the Internet. Believe
it or not, all that you need to know is out there
on the Web itself. A ,in 11 hint, whenever you see
something that you like, select the menu item
View Source... from your browser. A new win-
dow shows the commands used in creating the
original page. I found this feature to be the best
Hypertext tutor around. If you want to learn
more, access the home page of the World Wide
Web Initiative
It contains extensive documentation regarding
Hypertext and other Web related protocols.
Finally, you are warned that the Internet and the
Web can be addictive. Enjoy at your own risk!
-Samer Takriti
Digital Equipment Corporation


N? 47


PA~~n r~ltl 3 i 7 COER19



he society was formed in the
early 1970's, primarily to sup-
port a series of international
symposia that started in 1949.
The symposium that was held
in Ann Arbor in August 1994
under the chairmanship of
: John Birge and Katta Murty
.. was a memorable event. The
997 symposium will take
place in Lausanne; Tom
" Liebling and Dominique de
Werra are the organizers. For
the 2000 symposium, an advi-
sory committee chaired by Bob
Fourer welcomes proposals.
In addition to the general symposia, the society
also sponsors meetings on more specific topics.
The most prominent of these are the triennial
conferences on stochastic programming and the
IPCO conferences, on integer programming
and combinatorial optimization, held in every
non-symposium year. A stochastic program-
ming meeting was held last June in Nahariya,
Israel; the next one is scheduled for 1998, pos-
sibly in Vancouver. The fourth IPCO meeting
took place in Copenhagen at the end of May;
the fifth one will be held in Vancouver in June
Our journal, Mathematical Programming, is
the main publication outlet in the area of
optimization. In August 1994, the editors of
Series A and Series B, Bob Bixby and Bill
I',11H., I-.I ., I.. resigned after many years of dis-
tinguished service. They broadened the scope
of the journal, which now contains, in addition
to theory, more material on computation and
implementation. Bob also managed to decrease
the backlog of Series A quite substantially. His
successor is Don Goldfarb. Bill Pulleyblank,
who started Series B and made it a successful
series of special issues, was succeeded by John
Birge. In the coming years, the journal 11
have four volumes of three issues per year,
including at most three issues of Series B.

Then, as you will have noticed, OPTIMA be-
gan its second youth. The editorial staff, still
led by the founding editor, Don Hearn, was
expanded. The newsletter itself expanded from
three annual issues of twelve pages each to four
issues of sixteen pages.
Our prize program requires few comments.
The Fulkerson Prize, the Dantzig Prize, the
Beale-Orchard Hays Prize, and the Tucker
Prize are widely recognized awards. To facilitate
the work of future prize committees, the coun-
cil is considering clarifying the prize rules on
a few points.
A membership committee, chaired by George
Nemhauser, has been asked to advise the coun-
cil on issues regarding the recruitment of new
members, the relation between the society and
its regional and technical sections, and special
membership arrangements. The society has an
arrangement with its Hungarian members,
which, after 24 years, may require reconsidera-
tion. There is one regional section, the Nordic
Section, which achieves an admirable activity
level, and there could be more. There also is
one technical section, the Committee on Sto-
chastic ..._.; ,.1. ;1,, which, among other
things, organizes the main conferences in the
area. The society's first technical section, the
Committee on Algorithms, declared its mission
achieved and voted to disband in August 1994.
The membership committee will submit its re-
port shortly, and you will be able to read more
about it in one of the future issues of
The most recent activity is the establishment of
an MPS page on the WorldWide Web. As
most of you probably know better than I do,
this provides virtually unlimited possibilities.
Paying annual dues, signing up for symposia,
and consulting the membership list are some of
the first, easier options. At a later stage, the
table of contents of the journal, the text of
OPTIMA, an updated version of Phil Wolfe's
history of the society, and our constitution and
bylaws can be made accessible. Finally, the
journal is likely to be available on-line, which
will lead us into the era of electronic publish-
ing. But again, this refers to the future rather
than the past.

The society has fewer than 1,000 members,
who are evidently more interested in math-
ematical programming (or optimization, which
is a much better term) than in administrative
work. The organizational overhead is light. For
such a small and quiet group, it is remarkable
that it has achieved a truly international charac-
ter, with an active and high-level program of
meetings, publications, prizes, and regional and
technical sections.
While your primary business should stay in op-
timization, I want to encourage you to take
part in these activities. The society is in the
position to support initiatives and to provide
leverage. It can help to start up meetings and
sections of a regional or technical nature. And
OPTIMA needs your contributions. Our news-
letter is a volunteer effort, which can exist only
on the basis of feature articles, news items and
book reviews written by individual members.
Some words about our discipline. We often
hear that much of the research in optimization
that is being done in academia is baroque and
of no relevance to the outside world. Our work
does have its frivolous aspects. They give a cer-
tain charm to it. And research that is driven by
direct practical needs only and not by academic
curiosity is less likely to be innovative. But op-
timization as a whole is not baroque at all. It is
just reaching its maturity. It is ideally posi-
tioned in between mathematics, computing,
and practice. Modeling insights and computa-
tional methods developed during the last half
century have now come together with new
computer architectures and programming tech-
niques to enable us to solve truly large and truly
hard real-world problems. Optimization is very
much alive, and it is alive in its full breadth.
I want to thank all of my friends and colleagues
with whom I have had the privilege to work. In
particular, I express my gratitude to Michel
Balinski and George Nemhauser, past chairmen
of the society, to Les Trotter, our treasurer dur-
ing two terms, and to Rolf M6hring and Steve
Robinson, who chaired the Executive and Pub-
lications Committees. I hope that our succes-
sors will enjoy themselves as much as I did.
-Jan Karel Lenstra

- ----~--------~----~-`--

N- 47



Optimal Design of Engineering Structures

Aharon Ben-Tal
Arkadi Nemirovski
Technion- Israel Institute of Technology

structural design is an engineering discipline
aimed at creating constructions (bridges, cantile-
vers, inner skeleton of airplane wing, etc.) capable
of carrying external loads under different loading
scenarios. For example, a bridge should with-
stand forces corresponding to rush hour morning
traffic, rush hour evening i I t and perhaps an
earthquake. The criteria for "good" design are
either certain characteristics of rigidity, such as
stiffness and stability of the construction, or cost related measures,
such as total amount of material used, structure lifetime, or financial
cost of the construction. In this article we focus on discrete struc-
tures, so-called trusses. A truss consists of a finite number of thin
elastic bars, connected to each other at nodes (joints). Typical
examples are transmission towers and cantilever arms, but the most
widely known truss is doubtlessly the Eiffel Tower. When designing
a truss, an engineer bears in mind a set of active nodes where the
external forces are applied in the two-dimensional (2D) plane or in
three-dimensional (3D) space, a set of loading scenarios, where each
scenario reflects a particular d i, L tI .1 t 1 of the external forces, and a
set of fixed nodes (supports), such as the ground, a wall, or a ceiling
at which the construction can be supported. The final design is
given in terms of (1) the geometric location of the joints where bars
are linked, (2) the topology of the interconnections between the
nodes, and (3) the sizes (cross sections, areas, volumes) of the bars.
Layout optimization is concerned with the simultaneous selection of
all three design components: geometry, '. .I .4.1 i. and sizing, and it
constitutes one of the newest and most rapidly expanding fields of
structural design; see the excellent review article [10]. Although
mathematically and computationally the most challenging design
task, layout optimization is economically the most rewarding one,
as an efficient layout of a truss uses the given amount of material in
an optimal way to create the most rigid structure. Such a structure
can be significantly more stable than the layout obtained by ad hoc
To see how rigidity is determined, let us look at what happens when
a truss made of elastic material is put under a given load. External
forces cause a certain deformation of the truss, which means that the
free, i.e., the unsupported, nodes move slightly (called node /,'h- ,I...
ments) until the tension caused by the elongations of the bars
compensates for the external forces. As a result of the deformation,
the truss stores certain potential energy called compliance. The
compliance is thus a good global characteristic of the rigidity of the
truss with respect to a given load, i.e., the smaller compliance, the
more rigid is the structure, and serves as a reasonable objective
function. In the single-load case, the compliance is minimized with
respect to a unique loading scenario, and in the multi-load case, the
compliance corresponding to the worst possible scenario, out of a
given set of such nonsimultaneous loading scenarios, is minimized.

The approach to layout optimization as discussed in this article is
based on creating a fine mesh of potential node locations, allowing
all possible connections between all pairs of nodes. Thus, the
geometry problem is circumvented by solving a large-scale truss
'.." ... i,; design (TTD) problem which, fortunately, has surprisingly
good mathematical and computational characteristics; see the
review paper [21.
Mathematical model. The simplest formulation of a TTD problem is
derived as follows. Given a node set consisting of N elements in the
2D plane or 3D space, one can naturally associate an n-dimensional
space R" of virtual node displacements with it. Here, n is about 2N
for planar trusses and 3N for spatial ones ("about" since supported
nodes are not free to move in all directions). Displacements of the
nodes, and likewise the external loads, can be represented by
vectors in this space. It is convenient to identify the truss with an
m = 12N(N-1) dimensional vector t of bar volumes, where the entries
of this vector are indexed by the distinct pairs of nodes. The entry
corresponding to the unordered pair (j,k) of nodes is equal to the
volume of the bar linking nodes j and k. A zero entry means that the
corresponding pair of nodes is not linked. The tension/compression
caused by displacement x of the truss t is A(t)x for some n x n matrix
A(t). Consequently, the displacement caused by external load f R"
is determined by the equilibrium equation
and the compliance of the truss under the load is
In the linear elastic model of the material the matrix A(t) is linear in t,
A(t)=1 t'A,
where A, is a positive semidefinite symmetric matrix, in fact, a rank-
1 dyadic matrix
A =bb T.
The vector b. e R" contains the sines and cosines of the direction of
bar i and it also depends on some material characteristic, the so-
called Young modulus. Thus, a typical setting of the TTD problem is

(TTD) min max fx I A(t)x=f, t> 0, t,=V
fEF 1=1
where F C R" is a set of loading scenarios, and V is the given total
bar volume. The fact that the entries of t are allowed to take value
zero takes care of both the topology and sizing aspects of the design.
In the full layout optimization problem, the matrices A, depend on
the geometric positions of the nodes, and those are part of the
design variables.


N? 47



L'AGI 5 N~ 47 OCTOBER 1995



{dl ':

Figure ?.: In [I! ,_ t tii ,. -' 1 1 A MIAN
u n d ei h t ''t". i ' |_L,", ',, .h L 1 '. i 'l, '-
(a) gruulltd oL UiLLULt -VLtL I-vlllLall L"P'lubl
(only neighboring nodes are connected-192 bars
(b) optimal truss (96 bars), compliance = 16628.9

Figure 1: Geon, t i. ~. --, *
optimization fIr! .-i ', i,! *'.i "
shapes of opti ., I it, - l 1 11i,, I' .
possibility to cl....... IIl.. n J.. 1. I ,- 1 .1 1"-1!-
(a) ground structure; 120 potential bars
(b) fixed node set; optimal truss has 17 bars and compliance 8.944
(c) nodes can move in x-coordinate direction; optimal truss has 17
bars and compliance 6.563
(d) nodes can move in x and y coordinate directions; optimal truss
has 18 bars and compliance 6.326

Both the variables corresponding to the bar volumes and the
positions of the nodes are very important. In particular, even
relatively small variations of the node set may result in significantly
, ti.! I, shapes of the optimal truss; see Figure 1. Un F, .t iII. r i,. 1-..
even for fixed cardinality of the node set, the problem is nonconvex
in the positions of the nodes and may require techniques of
nonsmooth optimization. This approach is developed and used by
the group of Professor J. Zowe in Bayreuth (see [1,4]) and is illus-
trated in Figures 1 3 which are courtesy of Achtziger, Kocvara, and
Zowe. Here, in order to get a .:, .-p.ir.:.t,. r,, li;, tractable problem, we
are forced to fix the node set and treat t as the only design vector.
This approach is not as limited as it may sound. In fact, it allows us
to capture the geometry part of the design as well. By choosing
a fine 2D/3D grid as the node set, and allowing all possible links
between the nodes (this initial choice is called ground structure), we
can approximate the true layout optimization problem with
arbitrarily high accuracy. It can be proven that the number of
nonzero bar volumes in an optimal solution of TTD does not

Fil re : i : 1 ,1t. .. I i . .-1 W\
U ]l.l I l.n- l 'I I ,m .. ._I. \
(a o, -I i.dt. It, I, h .p .,,- 2 I.I

(b) optimal truss (24 bars), compliance = 3685.4 [22 percent of the
compliance for the optimal truss in Figure 2(b)]

exceed a certain value, which depends solely on the number of
active nodes and the cardinality of the set F of loading scenarios,
and not on the cardinality of the ground structure. Hence, an
optimal solution to TTD selects appropriate nodes and links
"automatically" from the structure, and solves, in principle, the
geometry, topology and sizing problems simultaneously.


N? 47


.-_ .jj_

n:~. :

PAGL 6 N~47 OCTOBER 1995

THE bad news is that in order to capture all components of the
design, we inevitably have to deal with large-scale TTD problems.
Indeed, even in the planar case, to approximate the actual, continu-
ous node universe within accuracy h, we should deal with a node
set of cardinality N=0(1/'. Moreover, in order not to impose
apriori restrictions on the '..p ...-_ we should allow basically all
possible links between the nodes so that the design dimension of
TTD will be m=0( --O(h-4). See Figures 2 and 3 for the effect of
using restricted versus rich topology. In the spatial case, the
situation is even worse: m=0(hl6). To get an impression of the sizes
of realistic instances of TTD problems, note that designing a simple
planar console with a quite moderate ground structure of 15 x 15 =
225 nodes results in TTD problem with m= 15,556 bars. This is by
order of magnitudes greater than the number (around 500) of bars
in the Eiffel Tower! Well, we should pay somehow for the fact that
we are not as ingenious as Eiffel...
Solving the truss topology design problem. The TTD problem is
the main subject of the research carried out during the last five
years in our Lab-the Optimization Laboratory of the Faculty of
Industrial Engineering and Management at Technion, Israel. What
attracted our attention was its challenging large-scale character
combined with its convexity properties and rich mathematical
structure. Moreover, it is indeed rewarding to deal with a large-scale
problem and yet to have the possibility to see, in the direct meaning
of the word, the solution! We started by processing the problem
in, ,tl.. i i, .. :11. which by itself was an exciting adventure. The goal
was to find an equivalent reformulation having smaller design
dimension, see [1]. What enabled us to achieve this goal was
extensive use of duality. It turned out that there are two
"computationally tractable" settings of TTD:
* The one where the set of loading scenarios is finite: (F=f ,...,f,}. In
this case, the dual of TTD is equivalent to the following minmax
quadratic-fractional problem:

k k xTAx k
(QF) minm 21ffx + V max X 'A > 0, x =1
R", keR j=1 i=l, ...,m J = ,=

In the single-load case (k=l) the X-variables disappear, and the
problem becomes simply a minmax problem with m convex
quadratic forms VxT Ax +frx of n variables x, see [3].

The one where F is a k-dimensional ellipsoid in R" given as the
image of the unit ball in Rk under the linear mapping u H- Qu,
where Q is an n x k matrix. In this case, using Fe 11.:i ,_l ii... i, it.. I! I
Ju1 Iili the problem is converted into the following semidefinite

(SD) min 2 Trace(QX) + V max Trace(XA-I'X A) IA > 0, Trace(A) = 1
A,X i= ,...,m i

where A is k x k symmetric matrix, X is n x k matrix, and the
constraint A 0 reads "A is positive semidefinite." Problem SD is
indeed a semidefinite program: it can be rewritten equivalently as

min{- .." + 'p xrbi)> 0, i= 1,...,m, Trace(A)=l 1

where b6 are the vectors involved in the dyadic representation A.=bbr
of the matrices A.

Note that both QF and SD are convex i ... ...... :.', problems, in contrast
to the original formulation TTD, which is not convex jointly in its
variables (x,t).
A second major advantage of the dual reformulations is a dramatic
reduction of the design dimension, kn + k-1 for QF and kn+ki -1 for
SD, instead of m=0(n2) for TTD. Note that k is usually small, and n
is in the order of hundreds. The huge original design dimension, of
course, does not disappear completely: now m becomes the number
of minmax components in the dual or, which is basically the same,
the number of smooth constraints in the inequality constrained
reformulation of the dual. Nevertheless, the "swapping of sizes"
that we get when passing from the original problem to the dual one
is very promising from the computational viewpoint, since a
majority of the available optimization methods are much more
sensitive to the design dimension of the problem than to the number
of constraints.
The minmax problem QF was something we could try to solve
S,,.1. i -I ,.. and we started with attempts to solve its single-load
version using available software. It turned out, however, that the
traditional methods like bundle, augmented Lagrangeans, and SQP,
are quite inefficient in that they were unable to handle "small" TTD
problems with tens of nodes and hundreds of bars. What actually
worked were interior point methods, and we strongly believe that
these methods constitute, if not the only, then definitely the most
appropriate tool for this kind of application. First of all, these
methods are theoretically efficient. With properly chosen log-type
penalties for the constraints, we get a polynomial time complexity
result (see [5,6]) as follows: to solve QF with accuracy e in the
objective, it suffices to perform O(Vm+k)ln(1/e) Newton-type
iterations having arithmetic cost C'.,,. -' each. When evaluating
this latter qu .111 i i.. one should take into account the nice structure
of the TTD data: due to its origin, A,=bbr with at most 4 (planar case)
or 6 (spatial case) nonzero entries in b.. The actual behavior of the
polynomial time interior point method was even better than could
be predicted by this complexity result; e.g., a single-load QF
problem coming from the ground structure with 225 nodes and
15,556 bars, was solved in 138 Newton steps [5].
Although promising, the polynomial time interior point method we
used was far from being the most efficient. The number of Newton
steps turned out to be sensitive to the number of loads and some-
times the computations lasted more than 500 hours. Therefore, we
definitely needed something more efficient. Intensive research
yielded two essential modifications of the standard interior point
scheme- the .:. *:...'. '. I'.'., multiplier (PBM) method [8] and the
truncated log-barrier (TLB) method [7]. As one can see from the table
below, both these methods solve single- and multi-load QF prob-
lems to high accuracy, i.e., 8-12 digits, in 30-50 Newton steps. The
number of Newton steps needed is basically independent of the
problem size.



Newton steps, Newton steps,
12 22
22 29
23 33
22 30
41 40
47 42



N? 47


E M E 17N"47OCOBR 99

i.. I-, PBM and TLB turned out to be very promising for general-
type convex optimization problems, not only for TTD.
Now it is time to present several nice pictures of trusses.

Figure 8: Optimal bridge with 21 x 17 nodes. The compliance is
403.981. i .. problem has 714 variables and 38896 constraints.

Figure 4: Optimal truss with 21 x 7 nodes. The compliance is
' 1 I .-,. The problem has 294 variables and 6574 constraints.

r*n* * *
n \ H

Figure 5: Optimal truss with 21 x 13 nodes. The compliance is
120.264. The problem has 546 variables and 22764 constraints.

. .1 . .

Figure 9: Optimal "high" bridge with 11 x 7 nodes. The compliance
is 283.755 The problem has 154 variables and 1828 constraints.
Figures 7 9 :i I,- i, t .- the same effect for bridges. The construction
is supported in the vertical direction at the very southwest and
southeast nodes, i.e.,"the river-., I. The single loading scenario
is formed by three equal vertical forces applied equidistantly on the
segment linking the supported nodes, i.e., "the road."

Figurte 10: Optimal bridge single-load formulation. The problem
has 96 variables and 730 constraints.

Figure 6: Optimal -.I truss with 21 x 13 nodes. The compliance is
106.145. The problem has 546 variables and 22764 constraints.
Figures 4 6: II. -n i.I the importance of a "rich" ground structure.
They deal with a single-load QF problem where the console to be
designed should transmit a single vertical force acting downward at
the middle node of the extreme right column to a vertical line of the
very left column, all of whose nodes are supported. We see that
enrichment of the ground structure makes the design close to the
solution of the corresponding continuous problem (a Mitchell truss).

Figure 7: Optimal bridge with 11 x 5 nodes. The compliance is
417.901. The problem has 110 variables and 934 constraints.

Figure 11: Optimal bridge minmax multi-load formulation. The
problem has 384 variables and 730 constraints.
Figures 10 11 demonstrate the difference between single- and
multi-load settings. Both designs relate to a 6 x 8=48-node ground
structure where links between all pairs of nodes are allowed. The
extreme southwest and southeast nodes are supported in vertical
direction. The design in Figure 10 corresponds to the case of a single
loading scenario where four equal forces are applied '..,., :i...,. '.-,'
and equidistantly on the segment linking the supported nodes. The
bridge in Figure 11 corresponds to a multi-load design with the

N 47




same four loads acting nonsimultaneously. In the latter case we
typically obtain a design with many more bars, one of which is far
more rigid.
Further research. It is now time to confess that our research up to
now has contributed more to large-scale convex optimization than
to practical truss topology design. From the practical viewpoint, the
indicated approach leads to designs which should serve as "refer-
ence I..' [. rather than to readily implementable constructions.
The reason is that QF only partly models the actual design con-
straints and that there are at least three important restrictions it does
not take care of:
* Bounds on node displacements. In the practical design, there are
restrictions on the movements that the nodes are allowed to take.
* Buckling. The linear elasticity model underlying TTD has
restricted applicability. For thin trusses, it is appropriate for a rather
wide range of external loads which extend the bar. In contrast to
this, the forces compressing the bar may cause arc-type or sine-type
deformations which a good design should avoid.
* Slabiilit, with respect to occasional loads. Problem QF takes
care only of the given loading scenarios. As a result, it may happen
that a small "( ... i. -i, -, I load may cause inappropriately large
deformations of the resulting construction.
Attempts to incorporate the "anti-buckling" restrictions and
restrictions on the node displacements straightforwardly lead to
essentially large-scale nonconvex optimization problems, which are
hardly tractable. One could prevent the indicated inappropriate
phenomena by imposing lower bounds on the bar volumes, which
basically preserves the nice convex structure of the problem. This
approach, however, has rather restricted value: it can be used only
in postoptimality analysis as it makes no sense to impose nontrivial
lower bounds on bar volumes before the topology of the construc-
tion is identified. In contrast to this, we can take certain care of the
stability of the resulting truss. The idea is as follows: let us pass
from the finite set of loading scenarios underlying the usual multi-
load TTD to an I/.I... ... of loads, thus thinking of stability of the
construction not only with respect to the I]. .d.1-. of interest," but also
with respect to all small enough "occasional loads." The most
natural way to construct such an ellipsoid is to take the "ellipsoidal
envelope" of the initial finite set, Fin, of loading scenarios and the
Euclidean ball, B, of all possible occasional loads of reasonable
magnitude, i.e., to take as F the ellipsoid of minimal volume
containing F,,,,u B. The immediate question is to which nodes the
occasional loads are applied. It could definitely not be the initial
node set of the ground structure, since it is natural to expect that the
majority of the initial nodes will not appear in the resulting truss.
There are, however, nodes which certainly will appear in this truss,
i.e., the active nodes to which the forces participating in Fll, are
applied, and we could choose B to be the ball in the subspace of
virtual displacements of the active nodes. With this approach, we
take from the very beginning certain, although incomplete, care of
the stability of the resulting construction. And, of course, we could,
and in our opinion also should, apply the outlined approach in the
postoptimality analysis, resolving the problem on the node set given
by preliminary design with "incomplete" stability constraints, i.e.,
with a "flat" ellipsoid of loads in the subspace of virtual displace-
ments of active nodes. When resolving the problem, we deal with
the "full-,. in-,t i- .i 1 ..I!'p-. of loads in the space of virtual
displacements of the new, reduced node set.

From the practical viewpoint there is nothing very specific with
ellipsoids. The only, and -11.. reason why we focus on ellipsoids
is the already indicated fact that the only .1 ..- ri i .. .!I, tractable
versions of TTD seem to be those related to the case when F is an
ellipsoid, or to the case where F is a polytope given by a list of its
vertices, which is the usual multi-load TTD resulting in QF
Mathematically, a TTD problem with an. Ili-.. ...I, I set of loads
results in a semidefinite program SD which seems to be more
difficult than QE Note, however, that in practical design the set of
given loading scenarios comprises a very small number (1-5) of
1 ... i. ..i loads, so that there are very few active nodes. As a
result, SD associated with the "pre-optimization stable formulation"
of TTD involves low-dimensional matrix inequality constraints and
is i. ,:1 as computationally tractable as the usual multi-load
TTD. In the postoptimality analysis we deal with "full-dimensional"
ellipsoid of loads, but this ellipsoid is associated with the reduced
node set given by the preliminary design, and we again may need to
deal with a semidefinite program of tractable size.
The outlined stable truss ', 1" 1.1 -,. design via semidefinite programming
seems to be quite promising. In particular, we hope that this setting
implicitly takes care of large node displacements and buckling
To illustrate the advantages of the "-r., i, truss topology design, let
us look at the following example. The left part of Figure 12 repre-
sents the results of the optimal single-load truss t. .] .1.... design on
a 9 x 9 square planar grid (81 nodes, 2040 tentative bars); the
extreme left nodes are completely supported, the remaining are free,
the external load is shown by the long arrow. The truss looks quite
attractive as its compliance with respect to the given load is 382.5. It
turns out, however, that the construction is highly unstable since,
when the initial load is replaced by a 10-times smaller "occasional"
one at the node shown by the short arrow on the picture, the
compliance becomes 18392.1-48 times larger. The "occasional" load
results in the displacement of the node where the load is applied,
which is almost 500 times larger than for the -.. .i ... load.

Figure 12: Single-load optimal design (left) and its postoptimal
"stable correction" (right).

'AGE 8

N 47


ISL 9l; N0 1 4 O B

The right part of Figure 12 is the truss given by postoptimal
"stabilization" of the solution. To be precise, we selected the bars
with relative volumes more than 1% from the aforementioned truss,
and chose, as the reduced node set, the nodes incident to the
selected bars. Then we resolved the problem, taking the 14 selected
nodes as the node set, allowing all 66 tentative links of the nodes,
and choosing as F the !lip- .1. I I.- .!"' l. of the initial load and
the ball consisting of all the 10-times smaller loads in the 20-
dimensional space of virtual displacements of our new node set.
The minmax compliance over our new 20-dimensional ellipsoid of
loads of the resulting construction is 395.6-which is 3.4% greater
than the minmax compliance of the first truss with respect to the
single scenario load. Moreover, the compliance of the -1 i-.l. truss
with respect to the original load is only 0.4% larger than the
compliance for the first truss.

[1] Achtziger, W., M.P. Bendsoe, A. Ben-Tal, J. Zowe (1992)
Si .1I ,1 I. ,i, displacement-based formulations for maximum
strength truss topology design," Impact of Computing in Science
and Fr ,' . .,'.- 4315-345.
[2] Bendsoe, M.P., A. Ben-Tal, J. Zowe (1994) "Optimization
methods for truss geometry and '. .1., .- design," Structural
Optimization, 7 141-159.
[31 Ben-Tal, A., M.P. Bendsoe (1993) "A new method for optimal
truss topology design," SIAM Journal of Optimization 3 322-358.
[4] Ben-Tal, A., M. Kocvara, J. Zowe (1993) "Two nonsmooth
approaches to simultaneous geometry and topology design of
trusses," in: Topology Design of Structures, Bendsoe, M.P. (Ed.),
Proceedings of NATO-ARW, Sesimbra, Portugal, 1992.
[5] Ben-Tal, A., A. Nemirovski (1992) "Interior point polynomial
time methods for truss topology design," Research Report 92/
3, Optimization Laboratory, Technion.
[6] Ben-Tal, A., A. Nemirovski (1994) "Potential reduction polyno-
mial time method for truss topology design," SIAM Journal of
Optimization 4 596-612.
[7] Ben-Tal, A., G. Roth (1994) "A Truncated log-barrier algorithm
for large scale convex programming and minmax problems,"
Research Report 1/94, Optimization Laboratory, Technion.
[8] Ben-Tal,A., M. ZtL..- -.1 (1993) i .:Iri /Barrier multiplier
methods: a new class of augmented Lagrangian algorithms for
large-scale convex programming problems," Research Report
6/93, Optimization Laboratory, Technion.
[91 Nesterov, Yu., A. Nemirovski (1994) Interior point polynomial
methods in convex iv.'., ',":'' .. SIAM Series in Applied Math-
ematics, Philadelphia.
[10] Rozvany, G., M.P. Bendsoe, U. Kirsch I '-i'-i "Layout optimiza-
tion of structures," Applied Mechanics Review 48 41-119.


Nordic Secon

The Nordic Section of the Mathematical Program-
ming Society was created at its first meeting in
Copenhagen, Denmark, in 1990. Stein Wallace,
who played an important role in the forming of the
section, was elected its first chairman. The stated
goal of the Section was to provide a framework for
inter-Nordic ....I ... .. r... within the field of
mathematical programming. We believe that the
Section has indeed contributed positively to the de-
velopment of the field and will hopefully continue
to do so for a long time. The present board consists
of Kaj Holmberg, Sweden; Kim Allan Andersen,
Denmark; and Dag Haugland, Norway.
At the first meeting it was decided to hold bian-
nual meetings of the Section, trying to bring to-
gether all researchers in the Nordic countries work-
ing on mathematical programming. The second
meeting was held in Trondheim, Norway, in 1992,
and the third was held in Link6ping, Sweden, in
1994 (in February so as not to conflict with The
International Symposium on Mathematical Pro-
gramming in Ann Arbor). The fourth meeting will
be held in 1996 in Aarhus, Denmark. These meet-
ings are . i 11 held over a weekend and are kept
simple in order to be self-financed with reasonable
registration fees (reduced, of course, for members
of MPS). There has been some funding for partici-
pating students obtained from NorFA I '"J..i. I
In addition to the biannual meetings, we have
started to produce a Newsletter of the Nordic Sec-
tion; Number 2 has recently been issued. Our goal
is to make the newsletter annual, but delays will
possibly occur. The newsletter mainly contains lists
of mathematical programming research reports and
theses produced in the Nordic countries, in order
that information about current research will be
available before the actual publication which, as we
all know, may take some time. These lists are par-
ticularly useful to people who cannot easily access
all the different journals in our field.
Another simple but very useful feature is the e-mail
list by which anyone can reach all the members of
the Nordic Section (address: mps@iok.unit.no).
We are now also beginning to collect information
about Nordic WWW-pages and ftp-sites contain-
ing mathematical programming material, 11 in or-
der to facilitate communication between Nordic
researchers and to make information easily avail-
-Kaj Holmberg

N? 47



mom= I

r(a $-i I.

; r E r E



0 ICCP-95-lnternational Con-
ference on Complementarity
Problems: Engineering &
Economic Applications, and
Computational Methods,
Baltimore, Maryland, U.S.A.
Nov. 1-4, 1995
Third Workshop on Global
Optimization, Szeged, Hun-
gary, Dec. 10-14, 1995
0 Conference on Network
Optimization, Center for
Applied Optimization,
Gainesville, Florida,
Feb. 12-14, 1996
Workshop on
Rutgers University
March 11-13, 1996
5th SIAM Conference on
Optimization, Victoria, British
Columbia, Canada,
May 20-22, 1996.
IPCO V, Vancouver, British
Columbia, Canada,
June 3-5, 1996
) IFORS 96 14th Triennial
Conference, Vancouver,
British Columbia, Canada,
July 8-12, 1996
International Conference
on Nonlinear Programming,
Beijing, China,
Sept. 2-5, 1996
XVI International
Symposium on Mathematical
Programming, Lausanne,
Switzerland, Aug. 1997


on Network


Gainesville, Florida
February 12-14, 1996
Advances in data structures and
0. 0 p.,.!.,. i. b ..l .1. ,- i 11r ,. devel-
opment of new algorithms have
made it possible to solve classes of
network optimization problems that
were formerly intractable. Among
these are problems in airline sched-
uling, transportation, satellite com-
munications, andVLSI chip design.
A conference on network optimiza-
tion problems, hosted by the Center
for Applied Optimization at the Uni-
versity of Florida, will bring together
researchers working on many differ-
ent aspects of network optimization
and on diverse applications in fields
such as engineering, computer sci-
ence, operations research, transporta-
tion, telecommunications, and manu-
facturing. It will provide a unique
opportunity for cross-disciplinary ex-
change of research.
The conference has receivedendorsements
from the Mathematical Programming
Society and the Institutefor Operations
Research andManagement Science, and
is beingheldin cooperation with SAM.
Additional information is available
from conference organizer Panos
Pardalos of the University of
Florida (pardalos@ufl.edu;
(904) 392-9011;
fax: (904) 392-3537).
Other organizers are Don Hearn
(hearn@ufl.edu) and Bill Hager


N- 47


8th Franco-Japanese and
Franco-Chinese Conference
Combinatorics and Computer
Brest, France
July 3-5, 1995

ticipants from 10 countries (Austria,
Canada, France, Germany, Italy, Japan,
Sweden, Switzerland, Taiwan, USA). The
scientific program consisted of 50 presen-
tations centering around: Graph theory
(graph-coloring, -decomposition, -genera-
tion, -recognition problems, homomor-
phisms, Slater's order, transversals, spectral
characterizations and problems on trees);
coding (p-adic, zigzag and block codes);
linear and integer programming (gravita-
tional and double description methods;
combinatorial algorithms for LP and LCP,
0;,.1. 1 d,. I ...... I, ,.. 1 h ,I
theory (Delauney and metric polytopes,
edge-coloring and probability, crossing of
hyperplanes on the torus); scheduling (job-
shop with task intervals, combinatorics of
scheduling optimization); approximation
I,_, h ,,, I ., ,,..,,,,,. I ,I, I,
lems); stochastic algorithms (GAs, simu-
lated annealing e.g. to calculate Ramsey
numbers); orders (contiguity orders, em-
bedding of bipartite orders); and matroids
(metric packing, pair-delta-matroids).
A larger part of the contributions were fo-
cussed on the efficient solution ofproblems
arising in important branches of computer
science such as parallel algorithms and ar-
chitectures (branch and bound, dynamic
programming, perfect matching in planar
graphs, recognition of consecutive ones);
distributed systems (task assignment using
network flows, threshold graphs and syn-
chronization, rerouting in DCS mesh net-
works); interconnection networks (embed-
dinggrids into d.. r ,i .... .i L I| I..
in meshs, design of bus and lightwave net-
works); pattern matching (on the
hypercube); and data compression.
Selected topics did include optimal strate-

1-vehicle routing, the guard problem in
spiral polygons, DNA sequencing and
Motley gems.
A refereed post-conference proceedings

series LNCS.
h,. ,. ,, i- ,. ,, , ,,, , ,111 .
held in Japan in 1996 and in Taiwan in

N 47


Volume 69, No. 1

B-issue: Nondifferentiable and
Large-Scale Optimization
Guest Editors: J.-P. Vial and
J.-L. Goffin
D.S. Atkinson and P.M. Vaidya,
"A itliu,, plane .iil l.'iiii for
convex programming that uses
i01.11,tl i centers."
O. Bahn, O. du Merle, J.-L. Goffin
and J.-P. Vial, "A cutting plane
method from analytic centers for
stochastic p "da*i, in, ""
D. den Hertog, F. Jarre, C. Roos
and T. Terlaky, "A sufficient
condition for self-concordance,
with application to some classes
of structured convex programming
K.C. Kiwiel, "Proximal level
litm,ll methods for convex
nlti I, hnal'hi' optimization,
sadle-point problems and
vari ional inequalities."
C. Len rchal, A. Nemirovskii
and Yu.\Nesterov, "New variants
of bundl& methods."
SYu. Nest rov, "C., ipli, t iir
estimate of some cutting plane
methods ased on the analytic
SYu.E. N sterov and A.S.
Nemir vskii, "An interior-point
methgd for "i, i I ,, I linear-
fraconal programming."
S L/andenberghe and S. Boyd,
I primal-dual potential
/ reduction method forproblems
involving matrix inequalities."

Vol. 69, No. 2

B. Chen and P.T. Harker,
"A continuation method for
monotone variational
A. Ebiefung, "Nonlinear
mappings associated with the
generalized linear
complementarity problem."

D.S. Hochbaum and S.-P. Hong,
"About -I ,r,.; polynomial time
i l.:, lit,, for quadratic optimi-
zation over submodular con-
R.D.C. Monteiro and S.J. Wright,
"Superlinear primal-dual affine
scaling algorithms for LCP."
M.X. Goemans, "Worst-case
comparison of valid inequalities
for the TSP."

Vol. 69, No. 3

J. Miao, "A quadratically
convergent O((k+l)+nL)-iteration
algorithms for the P.(k)-matrix
linear complementarity problem."
S.R. Tayur, R.R. Thomas and N.R.
Natraj, "An algebraic geometry
algorithm for scheduling in
presence of setups and correlated
J.-M. Cao, \, i--mii and
- 11,Il i iL condition for local
minima of a class of nonconvex
quadratic programs."
S.L. van de Velde, "Dual decom-
position of a single-machine
scheduling problem."
M. Bellare and P. Rogaway, "The
complexity of approximating a
nonlinear program."
Z.B. Zabinsky, G.R. Wood, M.A.
Steel and W.P. Baritompa, "Pure
adaptive search for finite global
A.I. Barvinok, "New il,., ilii,.'
for linear k-matroid intersection
and matroid k-parity problems."
H. Hamers, P. Borm and S. Tijs,
"On games corresponding to
, ,.in, i, iil. situations with ready



PACGE 12 N0 47 OCTOBER 1995

"i -----


Numerical Approximation of
Partial Differential Equations

by Alfio Quarteroni and Alberto Valli
Springer Series in Computational Mathematics 23
Springer-Verlag, Berlin, 1994
ISBN 3-540-57111-6

This book is devoted to the numerical solution of three important classes of second order

of mixed type, like advection-diffusion, Stokes, and Navier-Stokes receive considerable
attention. In particular, the latter ones play a central role in Computational Fluid Dy-
namics. Main emphasis is on finite element approximations, but other techniques, such
as collocation methods, are discussed as well.
W hen I started to read this book, I was,. l.i. I .1 1 .... 1 ,. ., doesn't
seem too many ifone attempts to cover the ins and outs of numerical evaluation ofPDEs.
One has to consider many special cases in order to obtain as well as to analyze efficient
and reliable computational schemes. The relevant literature in this field is enormous.
H ow eve. .. ,, Ii I il .. I,, il ..l I l ...I ,,,,,,,.., ],,, .. thingthatoneneeds
to know I1,... , 1 ;,, ,i.. I . P D E sin a .. ..,,,,! ', .. ... ... "" ..... I,.
been considered and has been presented in a very clear and understandable way. The authors
state in their introduction: "A sound balancing of theoretical analysis, description of al-
.; I .... ._,, r-'i ,..1'r, i. .. '"" i ... 1 concern. M anykindsofproblems
are addressed: linear and non-linear, steady and time-dependent, having either smooth
or non-smooth solutions. Besides model equations, we consider a number of (initial-)
boundary value problems of interest in several fields of application."
I ,rl I. Ii the authors have set high goals for themselves, I must admit that they have
succeeded well in their task. The book gives an impressing mix of theory, applications,
and implementation aspects. This -' ,, :/ illustrated by well-chosen computational
examples, relevant for large scale realistic problems.

N-- 47

"The theory is consis-
tently presented;
theorems are moti-
vated by the preceding
discussions: what are we
going to see in the next
theorem?, and what is it
goodfor?This makes it
worthwhile also for a
novice in the field to
study the relevant


The discussed and analyzed techniques are relevant for modern three-dimensional
modeling of physical problems. For instance, much attention has been paid to the itera-
tive solution of the linear finite elementsystems, besides the more traditional direct methods
which were more or less the methods of choice in classical two-dimensional finite element
computations. As far as I can judge, the described techniques represent the state of the
art: not only have methods been treated that were published as recently as in 1992, they
have also been implemented and the discussions by the authors seem to be supported by
. 1 ~. .. l ", -. f I. T .; i h.. ,11. h ,,111 .. i .. ,, I, I .. .
the book a very valuable source of information. Possibly I am slightly biased in my judg-
ment because of the elaborate treatment of Bi-CGSTAB (published in 1992), but I can
only conclude that the presentation of all relevant methods is very much to the point.
Discussions are supported by actual computational examples that help the reader to get
some feeling for the selection of methods for a particular given problem. This is necessary
since on more than one occasion there is a variety of approaches that may lead to an
acceptable solution. The choice of a particular method or approach usually determines
I, II . II,. .. .. I I ,..1 I ,,I 1 1 ,,,, .. rofam ethoddepends
on parameters that are not explicitly available to the user when solving realistic problems.
The book is also.. .. ii ..; . rl, . 1 .1: ,,, ... Il. c exercises are m missing.
The positive point for students is that unsolved problems are ..I........ i .i which
prevents the ..i. r... i,........ i l 1. 1. i. i ; i that virtually everything
in this field is well understood. The theory is consistently presented; theorems are mo-
tivated by the preceding discussions: .. '
is itgoodfor?This makes it worthwhile also for a novice in the field to study the relevant
theory. Both style and presentation are very helpful for attacking practical problems as
S. II 1,,,~I ... ,, .I-1 Ofcourse, thereareplacesinthebookwheretheexpertmight
occasionally frown, but the limits of the acceptable are never overstepped.
What I liked in particular is the attempts made by the authors to point out analogies between
approaches in widely different applications. For instance, Uzawa's scheme for the Stokes
problem is at the discretized lev .. .. .; ... .i ..... I ;', ..I P l . i i r.,
linear systems. This kind .1I 1 . .. is very appealing to me; not only does it contribute
to more insight, it also helps to clear up the apparent chaos of, at first sight, .I, It1. ..
approaches and techniques. It is also very helpful for memorizing some of the relevant
The authors state that "the book is addressed to graduate students as well as to researchers
in the field of numerical simulation of;. ,1;, ,I .11,11. I equations." I strongly believe,
as should be clear from my review, that Quarteroni and Valli have succeeded in their
mission, and I recommend the book to anyone who has interest in numerical mathemat-
ics, a central field in large scale scientific computing.




N? 47

"...an excellent text for
an advanced or seminar
course on optimization,
primarily addressed to
graduate students in math-
ematics, pure or applied,
computer science and
engineering schools."

Interior Point Approach to Linear,

Quadratic and Convex Programming

by D. den Hertog
Mathematics and its Applications
Kluwer Academic Publishers
Dordrecht, The Netherlands, 1994
ISBN 0-7923-2734-9.

This excellent book deals with recent developments in interior point algorithms for linear,
quadratic and convex prog ... ......L .. i,. I ,,i.i,. ,,..-.... i ,,,,. I ,, 's algorithm for
linear programming [1], this exciting area has been intensively and extensively studied
by many researchers. Most of the work is focused. C. r. ,.. :. 1_..;r!. . ,
and linear optim ization application, i ....I II .. -. I I i . ..... -
tion for solving nonlinear convex optimization problems. This foundation theory is mainly
due to Nesterov and Nemirovskii [2], but den Hertog simplifies and finalizes some of
the results. Thus, it seems much better to read this book before reading [2].
The main concept is the self-concordant barrier function on an open convex set, intro-
duced in Chapter 2 and analyzed in Chapter 3. This is a Lipschitian smoothness con-
dition of the Hessian with respect to a local Euclidean metric, plus a barrier for the
underlying convex set. The authors prove that Newton's method is effective on self-
concordant barrier functions. Thus, the problem of developing an efficient path-follow-
ingorpotential-reductionalgorithm"reduces" to .... ....... It concordantbarrier
for the constraint set (see Appendix A).
The authors devote the next chapter, Chapter 4, to reducing the complexity for linear
programming. I believe that his technique on adding and deleting constraints has an
important impact in practice as well. Chapter 5 is special; it unifies several popular interior-
po'.. i . .ri. .... I. ..'I It also presents a clear framework on how these algo-
rithms are related and brings mathematical insights to understand these l_. .1 ....
The .. I .1 .-. : clearly written. It is comprehensive and well-balanced on various
topics. It can make an excellent text for an advanced or seminar course on optimization,
primarily addressed to graduate students in mathematics, pure or applied, computer
science and engineering schools. On the other hand, researchers will also find it a valu-
able reference because the theorems contained in many of its sections represent the current
state of the art. In fact, the extensive bibliographic section .... l.., I ..;. ...;,- Fthe
book, quite complete and up to date. I believe this .,' 11 ......... a basic reference
for whomever is interested in convex optimization for years to come.

1. Karmarkar, N.K. "A New Polynomial-Time Algorithm for Linear
Programming," Combinatorica 4 (1984) 373-395.
2. Yu, E. Nesterov and A. Nemirovskii, "Interior Polynomial Algorithms
in Convex Programming," SIAM, Philadelphia, 1994.

a. __________________________________ I _________________


\C 5NI4 COBR19


Gatholic ( 't/l', J l1! 1'f I 'oL'tvh
Dlcp,'time'nt ofa. 1 tiiueiir l
[ 'ir. i-,-I ,-N ,.t1, ,.. ;l. i_ iniu

The Department of Mathematical
Engineering invites applications for an
academic appointment in mathematical
engineering, with preference for one of
the following topics:
* numerical analysis and scientific computing
" stochastic modelling
* discrete mathematics, combinatorics, graphs and
* mathematical system theory
* optimisation and variational calculus.

i he .aplli .ilnr i: I. a teaching load in ap-
plied mathematics. Applicants should demonstrate
both breadth of interest and promise in research
and teaching. Tenured positions will be considered
but appointment rank will depend on the candi-
dates and their records of accomplishment. Appli-
cants will be expected to teach in French, possibly
after a certain transition period.
The 550-year-old Catholic University of Louvain
is located on a campus created in 1972 in Louvain-
la-Neuve. The new campus now has more than
20,000 students and about 5,000 staff members.

F 0 R


Mail to:
The Mathematical Programming Society, Inc.
c/o International Statistical Institute
428 Prinses Beatrixlaan
2270 AZ Voorburg
The Netherlands

Cheques or money orders should be made
payable to The Mathematical Programming
Society, Inc., in one of the currencies listed
below. Dues for 1995, including subscription
to the journal Mathematical Programming,
are HFL100.00 (or USD55.00 or DEM85.00
or GBP32.50 or FRF300.00 or CHF80.00).
Student applications: Dues are /2 the above rates.
Have a faculty member verify your student status
and send application with dues to above address.

The Mathematical Engineering Department hosts
research programs in the different topics listed
above. The department also has close ties with the
,. i t, I i ... .-ions Research and Econometrics
I ... ..er for Systems Engineering and
I. I ,..... (CESAME) and the Institute
lied Mathematics (MAPA).
I .i. e a doctoral degree in en-
,,, ,, 1. .. postdoctoral experience,
i ..I ;cientitic publications record.
i I. i...... II start O ctober 1996.
S ., I .... I would furnish a curriculum vitae, a
.,! .1.. . s, an abstract of five selected pub-
I ..... 1 ... ....e and address of four referees in-
i. ..ii.. i i I, 'an referees who are not members
LiUL, Lbtlu 15 December 1995 to the Rector
of the university:
Recteur de l'Universitr Catholique de Louvain
1, Place de l'Universiti
B-1348 Louvain la Neuve
(The reference number of the position N 96/16
should be mentioned.)
Further information can be obtained from:
Professor Georges Bastin
Head of the Department of Mathematical Engi-
Catholic University of Louvain,
Batiment Euler
4, avenue Georges Lemaitre
B-1348 Louvain-la-Neuve
E.I i I' I
Fax: + 32 10 47 21 80


I wish to enroll as a member ofthe Society. My subscription is for
my personal use and notfor the benefit ofany library or institution.
I enclosepayment as follows:
Dues for 1995






-I -~--~----


N? 47

*AGE 15


OPTIMA is expanding to four issues
per year with publication dates keyed
to the academic semesters. The new
schedule will have issues in October,
December, March and June, with due
dates of Sept. 15, Nov. 15, Feb. 15
and May 15, respectively.As of August,
John Dennis (dennis@caam.rice.edu)
is the Chair of MPS, Clyde Monma
(clyde@bellcore.com) is Treasurer and
Steve Wright (wright@mcs.anl.gov) is
Chair of the Executive Committee.
Deadline for the next OPTIMA is
November 15, 1995.

Donald W. Hearn, EDITOR
e-mail: hearn@ise.ufl.edu
Utrecht University
Department of Computer Science
P.O. Box 80089
3508 TB Utrecht
The Netherlands
e-mail: aardal@cs.ruu.nl
Georgia Tech
Industrial and Systems Engineering
Atlanta, GA 30332-0205
e-mail: faiz@isye.gatech.edu
Department of Econometrics
i -,I h University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
e-mail: talman@kub.nI
Elsa Drake, DESIGNER


Journal contents are subject to change
by the publisher.


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




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