Title: Optima
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Title: Optima
Series Title: Optima
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Language: English
Creator: Mathematical Programming Society, University of Florida
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Place of Publication: Gainesville, Fla.
Publication Date: March 1999
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1 P T I M A
Mathematical Programming Society Newsletter

Planning Implants of Radionuclides for the

Treatment of Prostate Cancer: An Application

of Mixed Integer Programming

Eva K. Lee
Georgia Institute of Technology
E mail: evakylee@isye.gatech.edu
Richard J. Gallagher
Columbia University
Marco Zaider
Memorial Sloan Kettering Cancer Institute
March 10, 1999

0 P -- 1*1

Planning Implants of Radionuclides

for the Treatment of Prostate

Cancer: An Application

of Mixed Integer Programming

1 Introduction

In recent years, technical advances in medical
devices have led to the increasing use of
radioactive implants as an alternative or supple
ment to external beam radiation for treating a
variety of cancers. This treatment modality,
known as brachytherapy, involves the placement
of encapsulated radionuclides ("seeds") either
within or near a tumor [4]. In the case of
prostate cancer, seed implantation is typically
performed with the aid of a transrectal ultra
sound transducer attached to a template con
sisting of a plastic slab with a rectangular grid
of holes in it. The transducer is inserted into
the rectum and the template rests against the
patient's perineum. A series of transverse images
are taken through the prostate, and the ultra
sound unit displays -superimposed on the
anatomy of the prostate -the grid on the tem-
plate. Needles inserted in the template at appro
private grid positions enable seed placement in
the target at planned locations.
Despite the advances in devices that assist in
accurate placement of seeds, deciding where to
place the seeds remains a difficult problem. A
treatment plan must be designed so that it
achieves an appropriate radiation dose distribu
tion to the target volume, while keeping the
dose to surrounding normal tissues at a mini
mum. Moreover, a planning technique should
enable controlling the dose at any given point
in or near the implantation.
Traditionally, to design a treatment plan, sev
eral days (or weeks) prior to implantation the
patient undergoes a simulation ultrasound scan.
Based on the resulting images, an iterative
process is performed to find a pattern of needle
positions and seed coordinates along each nee
dle which will yield an acceptable dose distribu
tion. Adjustments are typically guided by
repeated visual inspection of isodose curves
overlaid on the target contours. This iterative
manual process is lengthy, sometimes taking up
to eight hours to complete. Moreover, the large
number of possible source arrangements means
that only a small fraction of possible configura
tions can actually be examined.
There have been a number of research efforts
directed at developing computational approach
es to aid in brachytherapy treatment planning.
Among them, Silvern [12] and Yu and Schell
[16] proposed genetic algorithm approaches,
and Sloboda [14] proposed an approach based
on simulated annealing. One shortcoming of

these heuristic search methods is that they do
not provide a mechanism for strictly enforcing
clinically desirable properties within the models
(e.g., strict lower and/or upper bounds on the
dose delivered to specified points near the
In this article, various integer programming
models for finding a good seed configuration in
brachytherapy treatment planning are proposed
and applied to the planning of permanent
prostate implants. The basic model, described
in Section 2, involves using 0/1 indicator
variables to capture the placement or non
placement of seeds in a prespecified three
dimensional grid of potential locations. The
dose delivered to each point in a discretized
representation of the diseased organ and neigh
boring healthy tissue is modeled as a linear
combination of these indicator variables. A
system of linear constraints is imposed in an
attempt to keep the dose level at each point to
within the specified target bounds. Since it is
physically impossible to satisfy all dose con
straints simultaneously, each constraint uses a
variable to either record when the target dose
level is achieved, or record the deviation from
the desired level. These additional variables are
embedded into an objective function to be opti
mized. A description of this MIP approach and
preliminary computational experiments with it
have appeared in medicaljournals [3, 8, 13].
Although not the focus of this article, it is also
noteworthy that for external beam radiation
treatment planning, linear programming
approaches have been proposed as far back as
1968 [1, 5, 10, 11].
Besides the likelihood of generating superior
treatment plans to those generated via tradition
al manual methods, one potential advantage of
using computational optimization approaches
to treatment planning is speed and the conse
quent possibility of generating treatment plans
immediately prior to implantation. It is often
the case that the position of the diseased organ
in the operating room differs from the position
in the pre-implant simulation images. In such a
case, there may be a need to change the plan in
the operating room. One goal of an automated
treatment planning system is to be able to assist
physicians and radiation physicists in obtaining
good treatment plans "on the fly." Hence, it is
imperative that the optimization component of
an automated system obtain good solutions
quickly. The numerical results presented in

MARCH 1999


S T A 6 1

Section 3 indicate that "good" solutions can be
obtained via the MIP approach within 5 to 15

2 Mixed Integer Programming

Our basic model involves using 0/1 variables to
record placement or non-placement of seeds in a
prespecified three-dimensional grid of potential
locations. In the case of prostate cancer, the
locations correspond to the projection of the
holes in the template onto the region represent
ing the prostate in each of the ultrasound
images. If a seed is placed in a specific location,
then it contributes a certain amount of radiation
dosage to each point in the images. (The dose
contribution to a point is proportional to the
inverse square of the distance from the source.)
Thus, once the grid of potential seed locations is
specified, the total dose level at each point can
be modeled. Let x be a 0/1 indicator variable for
recording placement or non-placement of a seed
in grid position. Then the total radiation dose
at point Pis given by

D ( P X, ,, (1)

where X is a vector corresponding to the coordi
nates of grid points, denotes the Euclidean
norm, and D(Q) denotes the dose contribution
of a seed to a point r units away. The target
lower and upper bounds, Lp and Up for the
radiation dose at point Pcan be incorporated
with (1) to form constraints for the MIP model:

D \P- X I|)x L

D P-X, )x Up.

Of course, not all points Pin the images are
considered. The images are discretized at a gran
ularity that is conducive both to modeling the
problem accurately and to enabling computa
tional approaches to be effective in obtaining
solutions in a timely manner. For discretizations
that provide accurate modeling, it is typically
not possible to satisfy desired dose constraints at
all points simultaneously. This is due in part to
the proximity of diseased tissue to healthy tissue.
Also, because of the inverse square factor, the
dose level contribution of a seed to a point less
than 0.3 units away, say, is typically larger than
the target upper bound for the point.

One approach of addressing this difficulty is
to identify a maximum feasible subsystem. This is
the essence of our first MIP model. By introduce
ing additional 0/1 variables one can directly
maximize the number of points satisfying the
specified bounds. In this case, constraints (2) are
replaced by

D (P-X, x Lp -N, vp)

D (P-XX x,-Up MI p),

where Vp and Wp are 0/1 variables, and Mp and
Np are suitably chosen positive constants. If a
solution is found such that p = 1, then the right
hand side of the first inequality in (3) is zero;
and hence, the lower bound for the dose level at
point Pis not violated. Similarly, if wp= 1, the
upper bound at point Pis not violated. In order
to find a solution that satisfies as many bound
constraints as possible, it suffices to maximize
the sum of these additional 0/1 variables; i.e.,
maximize p (vp+ Wp). In practice, achieving
the target dose levels for certain points may be
more critical than achieving the target dose lev
els for certain other points. In this case, one
could maximize a weighted sum: p (ayv +
bpW, where the more critical points receive a
relatively larger weight. Using a weighted sum
was important for the prostate cancer cases to be
discussed in Section 3. Since there were signifi
cantly fewer urethra and rectum points com-
pared to the number of points representing the
prostate, to increase the likelihood that the for
mer points achieved the target dose levels, a
large weight was placed on the associated 0/1
The role of the constants Np and Aif in (3) is
to ensure that there will be feasible solutions to
the mathematical model. In theory, these con
stants should be chosen suitably large so that if
Vpor Wp is zero, the associated constraint in (3)
will not be violated regardless of how the 0/1
variables x are assigned. In practice, the choice
is driven by computational considerations of the
optimization algorithm being used and/or by
decisions by the radiation oncologist. For a
branch-and-bound algorithm, it is advantageous
computationally to assign values that are as tight
as possible. The medical expert can guide the
selection of the constants by either assigning
absolute extremes on acceptable radiation dose
levels delivered to each point (note that Up+
Mp is the absolute maximum dose level that will

be delivered to point Punder the constraints in
(3), and Lp- Np is the absolute minimum), or
by estimating the number of seeds needed for a
given plan. In the latter case, if the number of
seeds needed is estimated to be between k, and
k2 (k, k2), say, then the constant N can be
taken to be Lpminus the sum of the smallest k,
of the values D (|IP- X|), and the constant AM
can be taken to be the sum of the largest k2 such
values minus U, Selection in this fashion will
ensure that no plan having between k, and k2
seeds will be eliminated from consideration.
An alternative model involves using continue
ous variables to capture the deviations of the
dose level at a given point from its target
bounds and minimizing a weighted sum of the
deviations. In this case, the constraints (2) are
replaced by constraints of the form

D IP -X ) +yp Lp
D P -X )x Zp Up,

where yp and zp are non-negative continuous
variables. The objective for this model takes the
form: minimize (apyp+ bp bz), where apand
bp are non-negative weights selected according
to the relative importance of satisfying the asso
ciated bounds. For example, weights associated
with an upper bound on the radiation dose for
points in a neighboring healthy organ may be
given a relatively larger magnitude than weights
associated with an upper bound on the dose
level for points in the diseased organ.
One enhancement that we have not yet
explored, but that could be incorporated into
either of the above models, is the allowance of
alternative seed types. There are a variety of
radioactive sources that are used for brachythera
py, including palladium-103, iodine-125,
cesium-137, iridium-192, and gold-198, each of
which has its own set of exposure rate constants.
(Pd-103 or I-125 are commonly used for treat
ing prostate cancer.) Typically however, a single
seed type is used in a given treatment plan. This
fact is, in part, due to the difficulty of designing
treatment plans with multiple seed types. The
allowance of multiple seed types can easily be
incorporated into the MIP framework -one
need only modify the total dose level expression
(1) as

D (IP- X )x,. (5)

MARCH 1999


10 P I -M A 11

Table 1. Lower and Upper Bound Specifications as Multiples
of Target Prescription Dose

Rectum Urethra Contour Uniformity

Lower Bound 0 0.9 1.0 1.0
Upper Bound 0.78 1.1 1.5 1.6

Table 2. Problem Statistics

Model 1 Model 2

Pt Rows Cols 0/1 Vars Rows Cols 0/1 Vars

1 4398 4568 4568 4398 4568 170
2 4546 4738 4738 4546 4738 192
3 3030 3128 3128 3030 3128 98
4 2774 2921 2921 2774 2921 147
5 5732 5957 5957 5732 5957 225
6 5728 5978 5978 5728 5978 250
7 2538 2658 2658 2538 2658 120
8 3506 3695 3695 3506 3695 189
9 2616 2777 2777 2616 2777 161
10 1680 1758 1758 1680 1758 78
11 5628 5848 5848 5628 5848 220
12 3484 3644 3644 3484 3644 160
13 3700 3833 3833 3700 3833 133
14 4220 4436 4436 4220 4436 216
15 2234 2330 2330 2234 2330 96
16 3823 3949 3949 3823 3949 126
17 4222 4362 4362 4222 4362 140
18 2612 2747 2747 2612 2747 135
19 2400 2484 2484 2400 2484 84
20 2298 2406 2406 2298 2406 108

Table 3a. Solution Statistics for Model 1 (Maximization)

Pt Initial First Heuristic Best Best
LP Obj. sees Ob. LP Obj. IP Obj.

1 1888013.3 245.2 1752286 1873433.93 1766609
2 1809964.8 672.7 1736946 1796642.43 1736946
3 687448.6 43.6 587712 633843.00 593228
4 803564.9 338.7 753672 802134.58 765115
5 2855667.4 1345.9 2638679 2835825.38 2649950
6 2925181.3 1349.2 2805284 2907792.52 2805284
7 651682.5 54.7 582314 639160.50 598630
8 1132430.4 669.8 1062561 1112930.1 1075670
9 677253.3 194.7 639527 669073.94 641643
10 286986.4 25.5 252368 274188.69 257492
11 2585974.0 366.8 2453886 2529795.54 2462279
12 983328.6 70.7 804213 945400.35 817875
13 862373.4 52.0 744450 827676.91 795149
14 1611020.9 476.2 1509329 1590484.06 1531009
15 438667.7 29.5 376087 428376.60 396064
16 1273297.8 163.6 1170743 1248805.82 1204870
17 892239.9 52.2 747929 817014.30 757446
18 683918.1 71.0 581684 666083.02 592861
19 425871.6 14.9 341328 403235.91 376179
20 360474.3 14.1 288973 343623.43 309499

Here, xj is the indicator variable for placement one can ensure that target dose bounds at specif specified. For the 20 cases considered, the aver
or non-placement of a seed of type i in grid ic points are satisfied by fixing the associated age numbers of points in each category were:
location, and D(r) denotes the dose level con "feasibility" variables (vp wp yp z) to appropri uniformity 1305, contour 461, urethra 8, and
tribution of a seed of type i to a point r units ate values. In the numerical work reported in [8] rectum 9. The lower and upper bounds for each
away. In this case, a constraint restricting the we used this approach to ensure that the dose point type were specified as multiples of the tar
number of seeds implanted at grid point is also delivered to all points representing the urethra get prescription dose. These are tabulated in
needed: x 1. It remains to be tested did not exceed a specified upper bound. Table 1.
whether the added flexibility of allowing multi Numerical tests were performed using two
pie seed types will have a substantial impact on 3 Computational Strategies and distinct models. Model 1 utilized constraints (3)
the number of points at which target dose levels Clinical Experiments and the associated objective max (a, vp+ bp
can be satisfied. Nevertheless, it is an intriguing w); and Model 2 utilized constraints (4) and
possibility. Computationally, the optimization We tested our MIP approach using data from the objective min (apyp+ bpz). Various com-
problem may prove to be more difficult due to twenty prostate cancer cases. In each case, binations of objective function weights for each
the increased number of 0/1 variables, iodine-125 was used as the radioactive source, of the two models were tested. Here, we present
Besides the basic dosimetric constraints, other and four separate categories of points, correspon results based on one set of weights for each
physical constraints can be incorporated into our ding to distinct anatomical structures, were spec model. Detailed analysis of the two models and
basic models. One could incorporate constraints ified. Contour points defined the boundary of the a study of the sensitivity of resulting plans to
to control the percentage of each tissue structure diseased organ in each of the slices; the regions selected weights can be found in [7].
satisfying specified target bounds. Alternatively, enclosed by each boundary were populated with For both models, it is advantageous to place
one could -if desired -constrain the total num- uniformly spaced points, termed uniformity relatively large weights on the objective function
ber of seeds and/or needles used. Note also that points; and points representing the positions of variables associated with urethra, rectum, and
the urethra and rectum in each slice were also

MARCH 1999


S1*PTI A 6

MARCH 1999

Table 3b. Solution Statistics for Model 2 (Minimization)

Pt Initial First Heuristic Optimal bb Elapsed
LP Obj. sees Obj. IP Obj. nodes Time

1 29973430.5 21.7 440437196.1 93550763.6 377 9706.0
2 19921521.4 34.7 179171112.9 49156651.9 9184 378857.0
3 11333869.7 5.2 97625273.7 50517325.3 4051 27724.0
4 2597572.3 18.7 189610043.6 21005621.8 1377 27485.0
5 73684327.8 112.4 467410325.8 93828192.8 1293 748292.3**
6 36902037.2 105.3 524058129.4 64216816.0 5293 1136221.7
7 45848681.6 6.5 302836935.1 118325071.3 712 4655.5
8 17614469.1 32.3 250057575.6 73399636.5 62373 1863362.0
9 14691002.3 17.3 344540093.9 57209440.5 1643 41212.1
10 28197622.0 2.1 90862556.4 55251869.2 883 2619.1
11 172211617.5 29.1 616562230.8 293530404.3 643 14741.8
12 292898229.2 11.5 785823995.0 518235776.6 1985 35718.5
13 163007095.9 4.3 21671699.9 77173221.5 481 2817.6
14 40303495.4 27.1 378940132.7 119586431.2 1408 58654.2
15 89432119.5 5.5 236921860.0 191780731.4 10838 55913.8
16 78434032.7 14.1 244541089.6 148828362.1 1282 25969.0
17 830974566.8 2.7 717574515.4 799657523.1 25 178.2
18 155505947.5 9.6 700452425.7 351076662.5 82118 554737.2
19 73628152.3 2.1 204208781.0 149604823.5 377 1207.8
20 45968824.5 1.8 57904156.7 15635930.3 415 1222.5

** Not optimal

contour points. For the results reported herein,
the objective function weights for the variables
associated with uniformity points were set equal
to 1; those associated with contour points were
set equal to the ratio of the number of uniform
ty points to the number of contour points; and
those associated with urethra and rectum points
were set equal to the number of uniformity
points. Selecting such large weights for the ure
thra and rectum points essentially ensures that
the dose contribution to these points will lie
within the specified bounds. The heavy weight
for the contour points assists in achieving pre
scription isodose curves that conform well with
the boundary of the diseased prostate.
The dosimetric constraint matrix in both
models is completely dense and has coefficients
ranging in magnitude from tens to tens of thou
sands. Table 2 shows the problem statistics for
Models 1 and 2. Here, Rows and Cols indicate
the number of rows and columns, respectively,
in the constraint matrix; and 0/1 vars indicates
the number of 0/1 variables.
Although the problem size is only moderate,
even solving the initial linear programming
relaxation is memory-taxing, often resulting in a
process having a total size of 300MB (including
text, data, and stack), and total resident memory
approaching 300MB. Computational experience
with these instances has demonstrated that they
are extremely difficult to solve to optimality,

requiring strenuous computational effort to
improve the objective value by a marginal
amount. Even obtaining good feasible solutions
which are clinically acceptable is difficult.
The numerical work reported herein is based
on a specialized branch-and-bound MIP solver
which is built on top of a general-purpose mixed
integer research code (MIPSOL) [6], using
CPLEX 6.0 as the intermediate LP solver. The
general-purpose code, which incorporates pre
processing, reduced-cost fixing, cut generation,
and fast heuristics, has been quite effective in
solving the instances reported in MIPLIB3 [2].
For the prostate cancer instances reported in this
paper, specialized heuristics and branching
schemes were implemented to quickly obtain
good feasible solutions [7]. We also experiment
ed with a reduction and approximation
approach in an attempt to devise efficient com-
putational strategies to improve the solution
process for these instances [9].
None of the instances for Model 1 were
solved to proven-optimality, whereas for Model
2, all except one were solved to optimality. In
Tables 3a and 3b the solution statistics for both
models are given. The instances were solved on
an UltraSparc II 168 Mhz workstation. We set
the running time limit to be 10,000 CPU sec
bonds for Model 1. In each table, the column
labeled Pt denotes the patient case; the column
labeled Initial LP Obj. lists the optimal objective

value of the initial LP relaxation; and the
columns First Heuristic (sees, Obj.) list the
elapsed time when the heuristic procedure is first
called and the objective value corresponding to
the feasible integer solution returned by the
heuristic. For Table 3a, the columns Best LP Obj.
and Best IP Obj. report, respectively, the LP
objective bound corresponding to the best node
in the remaining branch-and-bound tree and the
incumbent objective value corresponding to the
best integer feasible solution upon termination
of the solution process (10,000 CPU seconds).
In Table 3b, the columns Optimal IP Obj., bb
nodes, and Elapsed Time report, respectively, the
optimal IP objective value, the total number of
branch-and-bound tree nodes solved, and the
total elapsed time for the solution process.
Using the reduction approach alluded to earli
er, the running time for Model 2 decreased by 5
to 100 times for the 20 instances, with an aver
age decrease of 28.3 times. The readers are
referred to [7, 9] for more details regarding this
To contrast the performance of our solver on
these instances and to illustrate their level of dif
ficulty, in Table 4 we provide a solution profile
for case Pt 1 using the MIP solver of CPLEX 6.0
(with pseudo-cost branching, which appears to
be the best among the possible options). We
note that none of the instances were solved to
optimality using CPLEX 6.0. For Model 1


S10 P T- 1*1

Table 4. Solution Statistics for Pt 1 Running on CPLEX 6.0

CPU sees Best Best bbnodes
elapsed IP obj. LP obj. searched

Model 1
6784.82 914006 1888012.12 2100
14196.93 914006 1888011.90 8100
25141.08 914006 1888011.66 15100
35972.33 995417 1888011.59 21600
62448.75 995417 1888011.58 27400
Cuts added 13173

Model 2
106.60 1047338492.9 3.5015e+07 61
5008.38 440437196.1 7.3056e+07 2241
10037.33 108100907.2 8.0022e+07 6241
15185.81 93550763.5 8.3096e+07 15001
20357.77 93550763.5 8.4342e+07 25001
32736.21 93550763.5 8.5909e+07 50001
45911.94 93550763.5 8.6919e+07 77321
46884.43 93550763.5 8.6987e+07 79341

Table 5. Clinical Significance of the MIP Generated Plans

Pt Prostate Activity
Vol. (cc) (mCi)

conformity coverage No.

1 49.1 0.592 1.20 .973 40
2 53.6 0.450 1.16 .994 51
3 34.2 0.334 1.18 .945 51
4 31.0 0.400 1.17 .985 42
5 68.7 0.590 1.21 .985 50
6 68.1 0.450 1.20 .986 64
7 26.7 0.400 1.25 .970 39
8 40.8 0.450 1.21 .983 44
9 28.9 0.500 1.28 .988 32

10 16.6 0.468 1.29 .939 28

11 66.1 0.520 1.12 .964 85
12 38.3 0.544 1.23 .951 58
13 39.9 0.450 1.22 .986 70
14 48.2 0.450 1.17 .989 76
15 24.3 0.550 1.18 .980 42
16 45.3 0.592 1.15 .975 57
17 50.7 0.463 1.11 .874 72
18 26.4 0.500 1.29 .970 51
19 25.4 0.450 1.15 .964 48
20 25.6 0.400 1.16 .977 57

instances, great computational effort was exert
ed, only to yield marginal improvement in the
objective value. Instances for Model 2 were
slightly more manageable, although the objec
tive value improvement eventually stalled (e.g.,
after 80,000 nodes for Pt 1). The columns CPU
sees elapsed, Best IP obj., Best LP obj., and bb
nodes searched record, respectively, the time
elapsed within the solution process, the incum
bent objective value corresponding to the best
integer feasible solution, the corresponding best
LP value from the remaining branch-and-bound
nodes, and the number of nodes solved. For
Model 1, we report the solution process up to
62448.75 CPU seconds and for Model 2, the
solution process is observed up to 46884.43
CPU seconds.
It is noteworthy that using our reduction
approach we solved the Model 2 instance for Pt
1 in 1012.8 CPU seconds. In addition, this
same instance was solved in 1312.24 CPU sec
onds when CPLEX 6.5 was used, running on an
UltraSparc-II 296 Mhz machine.
Despite the computational .i.ili. .,,- high
quality clinically desirable treatment plans were
obtained using both models. In Table 5 we
report some clinically relevant statistics. For the

results reported in the table, the treatment plans
are those associated with the first feasible solu
tion obtained from our specialized solver applied
to Model 1. The cases are categorized according
to the target prescription dose (100Gy, 120Gy,
or 160Gy). Prostate Vol (cc) records the volume
of the prostate, Activity (mCi) is the activity rate
of the implanted seeds, and conformity and cov
erage are measures of the quality of the generate
ed plans. Conformity is defined as the ratio of
the volume of the prescription isodose surface
determined by the plan to the portion of the
target volume within this surface. Coverage
measures the ratio of the target volume within
the prescription isodose surface to the entire tar
get volume. For an ideal plan both the conform
ity and coverage indices should be 1. A con
formity index greater than 1 provides a measure
of the amount of healthy normal tissue receiving
the prescription dose or greater. In particular, a
smaller conformity index implies that nearby
healthy tissue is exposed to less radiation, thus
reducing the probability of complications.
Compared to traditional manual planning meth
ods, plans derived via the MIP approach use
fewer seeds (20-30 fewer) and needles, and pro
vide better coverage and conformity indices [7, 8].

4 Concluding Remarks

The computational work presented herein
demonstrates that a mixed integer programming
approach to brachytherapy treatment planning
can produce high-quality treatment plans for
prostate cancer cases. The MIP models provide
the flexibility to enforce clinically critical dosi
metric conditions, and to prioritize dose level
achievement for vital organs and tissues near the
diseased structure.
Although the mixed integer programming
problem instances are difficult to solve to opti
mality, with our specialized heuristic procedure,
good treatment plans were returned within 15
CPU minutes. This suggests that incorporation
of an MIP-based optimization module into a
comprehensive treatment planning system for
use in the operating room is feasible. With this
in mind, work is now in progress to interface
the optimization module with standard clinical
evaluation tools (e.g., tools for displaying iso
dose curves and cumulative dose-volume-his
tograms). A major advantage of such an online
system is that it would allow the generation of
an alternative plan in the event that unforeseen
circumstances arising during implantation pre
vent strictly following the pre-implant simula

MARCH 1999


S T A 6 1

MARCH 1999

tion plan. In this scenario, the system must be
able to accommodate the fact that some seeds
may have already been implanted based on the
pre-implant plan, and that other pre-selected seed
locations must now be disallowed. With respect
to the MIP models, this only requires fixing
selected binary variables to zero or one.
This work has the potential to have a direct
positive impact on treatment success, as well as in
eliminating the time-consuming task of generate
ing treatment plans via traditional manual
approaches. Interested readers can refer to the
publications [3, 8, 13, 15] for additional medical
and clinical insights regarding this research.
Studies detailing the integer programming
aspects, including computational strategies for
solving the associated MIP instances, are reported
in [7, 9].

5 Acknowledgments

The first author was supported in part by
National Science Foundation CAREER grant
9501584/9796312. The second author was sup
ported in part by National Library of Medicine
Training Grant LM07079. The computational
work for this research was performed on SUN
equipment made available through SUN
Microsystems Academic Equipment Grant SUN
AEG EDUD-US-970311. The authors thank
Professor Ellis Johnson and Professor George
Nemhauser for their comments on an earlier ver
sion of this article. In addition, we acknowledge
CPLEX, a division of ILOG Inc., for providing
CPLEX 6.0 for this research. We also thank
Professor Robert Bixby for profiling the Model 2
instance for Pt 1 using the recently released
CPLEX 6.5. Finally, the authors would like to
thank Dr. David Silvern for his assistance in pro
viding the data used in the numerical work pre
sented in Section 3.



[1] Bahr G.K., Kereiakes J.G., Horwitz H.,
Finney R., Galvin J., Goode K., The
method of linear programming applied to
radiation treatment planning, Radiology
1968; 91:686-693.
[2] Bixby R.E., Ceria S., McZeal C.M., and
Savelsbergh M.WP, An Updated Mixed
Integer Programming Library: MIPLIB 3,
[3] Gallagher R.J., Lee E.K., Mixed integer
programming optimization models for
brachytherapy treatment planning, Journal
of the American Medical Informatics
Association, Symposium Supplement (D.R.
Masys, ed.) 1997; 278-282.
[4] Interstitial Collaborative Working Group.
Interstitial Brachytherapy. Physical,
Biological, and Clinical Considerations.
Raven Press, New York, 1990.
[5] Langer M., Brown R., Urie M., Leong J.,
Stracher M., Shapiro J., Large scale opti
mization of beam weights under dose
volume restrictions, International Journal
ofRadiation Oncology and Biological
Physics, 1990; 18:887-893.
[6] Lee E.K., Computational experience of a
general purpose mixed 0/1 integer pro
gramming solver (MIPSOL), preliminary
report, Georgia Institute of Technology,
[7] Lee E.K., Gallagher R.J., Johnson E.L.,
Nemhauser G.L., Zaider M., Mixed inte
ger programming approaches to treatment
planning for brachytherapy -application
to permanent prostate implants, Technical
Report, Georgia Institute of Technology,
[8] Lee E.K., Gallagher R.J., Silvern D., Wuu
C.S., Zaider M., Treatment planning for
brachytherapy: an integer programming
model, two computational approaches and

experiments with permanent prostate
implant planning, Physics in Medicine and
Biology 1999; 44(1):145-165.
[9] Lee E.K., Johnson E.L., Nemhauser G.L.,
Computational issues for a mixed integer
programming approach to treatment plan
optimization for radiation therapy.
Working paper, 1999.
[10] Legras J., Legras B., Lambert J-P.,
Software for linear and nonlinear opti
mization in external radiotherapy, Comp.
Prog. Biomed. 1982; 15:233-242.
[11] Rosen I.I., Lane R.G., Morrill S., Belli
J.A., Treatment plan optimization using
linear programming. Phys Med Biol, 1991;
[12] Silvern D.A., Automated OR prostate
brachytherapy treatment planning using
genetic optimization. Ph.D. Thesis, 1998.
Department of Applied Physics, Columbia
University, New York.
[13] Silvern D.A., Lee E.K., Gallagher R.J.,
Stabile L.G., Ennis R.D., Moorthy C.R.,
Zaider M., Treatment planning for perma
nent prostate implants: genetic algorithm
versus integer programming, Medical &
Biological Engineering & Computing 1997;
35:(Supplement, Part 2):850.
[14] Sloboda R.S., Optimization of brachyther
apy dose distributions by simulated
annealing, Medical Physics 1992; 19:955
[15] Wuu C.S., Zaider M., Lee E.K., Ennis
R.D., Schiff P.B., Dosimetric criteria for
selecting a source activity and a source
type (125for 103Pd) in the presence of irreg
ular pellet placement in permanent
prostate implants. Working paper, 1998.
[16] Yu Y., Schell M.C., A genetic algorithm
for optimization of prostate implants,
Medical Physics 1996; 23:2085-1091.


a SA, Crei-Melo Universit


THIS is the first of a series of articles on the history of some of
the leading academic programs in optimization throughout
the United States. For the first article I chose the Operations
Research Group at the Graduate School of Industrial
Administration in Carnegie Mellon University.
There are several groups at Carnegie Mellon University that have excep
tional expertise in the topic of Optimization. The most important and
undoubtedly better known is the Operations Research Group at the
Graduate School of Industrial Administration (GSIA). Within GSIA, facul
ty members at the Operations Management Group keep in close contact
with their colleagues in Operations Research (OR), and often share teach
ing responsibilities and research projects. Other researchers within Carnegie
Mellon include faculty at the Chemical Engineering Department, the
Department of Mathematics, the School of Computer Science, and the
Heinz School of Urban and Public Affairs. Collectively, Carnegie Mellon
has one of the best groups of researchers in Optimization in the whole
The group at GSIA was the place of origin of the first industrial applica
tions of OR in the early 1950s. It was the kind of atmosphere where
research that crossed conventional boundaries was very much encouraged,
and unconventional approaches were enthusiastically supported. In those
days the Operations Research Faculty consisted mainly of Bill Cooper and
Gerry Thompson. Bill Cooper was one of the leading authorities in Linear
Programming in the fifties, and had significant influence on the first appli
cations of OR to solving applied problems. Among other achievements,
Cooper is credited with the invention of Goal Programming and Data
Envelopment Analysis. His book with Charnes is still a classic in the early
applications of Linear Programming. Gerry Thompson, who is still among
the faculty at GSIA, pioneered the use of computers and quantitative meth
ods in business education. He often used classical methods to apply OR
technology to problems in economics, marketing and other general disci
plines of business.
In the late sixties the group welcomed a Romanian immigrant, who
would later become one of the most influential leading researchers in the
area of Optimization. When Egon Balas arrived to GSIA, he had already
published what would be later recognized as "the most frequently cited
paper in OR between 1954 and 1981." The paper described an additive
algorithm for solving zero-one problems and it was one of the early proto
types of implicit enumeration or branch-and-bound algorithms, which used
logical tests not unlike those underlying the constraint propagation
approach of our days. Egon Balas has one of the most interesting life stories
I have ever heard (or read), a rather stormy life that included action in the

S P T A 61

Hungarian anti-Nazi underground movement, followed by arrest, prison,
escape and hiding, a brief tenure as a Romanian diplomat in London, soon
followed by arrest and solitary confinement in the horrible prisons of
Bucharest. He had managed to emigrate to the West in 1966, with a brief
stay in Rome, a summer in Stanford's OR group, and the final arrival to
GSIA. His life story is the subject of a new book, which should reach the
press later this year.
Balas helped build a prosperous group with an important concentration
in integer programming. He attracted the late Bob Jeroslow, and strength
ened the doctoral program that had already fostered notorious Ph.D. stu
dents such as Fred Glover and M.R. Rao. During his first years at GSIA,
Balas worked with a brilliant Ph.D. student, Manfred Padberg. Padberg's
work on the knapsack polyhedron and set-partitioning problems would
later become one of the seminal studies in modern Integer Programming.
Reading Padberg's Ph.D. thesis 25 years later is an interesting experience
since it is an excellent compendium of ideas to come in the late seventies
and early eighties. It is during this time that Balas completed his first paper
on Disjunctive Programming. Surprisingly, this paper was never published,
because Balas refused to write it according to the referee's suggestion. The
paper is being published now, 25 years later, as an invited paper with an
introductory note appraising its (original) merits. During the seventies the
group at GSIA became a power house in Integer Programming, as Egon
Balas and others completed several papers on the structure of basic polyhe
dra, disjunctive programming, intersection cuts, and general cutting plane
Jong Shi Pang joined the faculty in the mid-seventies, and concentrated
his research on the solution of linear-complementarity problems and the
study of variational inequalities. In the late seventies two young and prom
ising stars joined the faculty: Gerard Cornuejols and Dorit Hochbaum.
Gerard Cornuejols joined the group after completing his dissertation in
Cornell under the supervision of George Nemhauser. His joint work with
Nemhauser and Fisher on the solution of facility location problem had been
recognized with the Lanchester Prize, and is now considered one of the
seminal papers in theory and computation for Facility Location. As a grad
uate student, Cornuejols had also completed several papers on graph theo
retical problems. Dorit Hochbaum was a rising star in the area of approxi
nation theory. She had completed her dissertation under the supervision of
Marshall Fisher at Wharton. During the late seventies and early eighties,
the structure of the group was solidified under the leadership of Balas and
the influence of the young researchers that had joined the group, either as
faculty or as Ph.D. students. Balas was extremely supportive of the young
faculty and interceded on many occasions to shield them from the ongoing
debates on OR education within a business school.

In the eighties, the interests of the group in other areas increased. Balas
was involved in several projects that would leave a mark in other areas of
optimization. He worked on heuristics for general 0-1 programs (pivot
and-complement), heuristics for scheduling problems (the shifting-bottle
neck procedure), the structure of set-partitioning and set-covering polyhe
dra, representation and projection of general polyhedra, and the traveling
salesman problem. Thompson developed his pivot-and-probe version of the
simplex algorithm, and Cornuejols wrote a number of papers on a variety
of topics, like the structure of the traveling salesman polytope, a compari
son of Lagrangian relaxations for Facility Location Problems, and the first
steps in the structure of 0-1 matrices. The group lost Pang and Hochbaum
in the early eighties, but received two very interesting and different new
comers in the eighties, John Hooker and Michael Trick. Hooker had com-
pleted two doctorates, one in philosophy and another one in OR. He had
an interest in logic and operations research, and a diverse range of interests
in several philosophical aspects of the discipline. Michael Trick joined the
group after graduating from Georgia Tech under the supervision of John
Bartholdi, and spending a year as a postdoc in Bonn. The two newest hires
are R. Ravi and Javier Pena, from Brown University and Cornell University,
respectively. Ravi specializes in approximation algorithms for combinatori
al optimization problems, and Pena in nonlinear programming.
The academic record of the Operations Research Group at GSIA is more
than impressive. Gerald Thompson is the IBM Professor of Systems and
Operations Research and a Senior Fellow at the IC2 Institute at the
University of Texas. Egon Balas is University Professor and the Thomas
Lord Professor of Operations Research. He was recently recognized with the
John von Neumann Theory Prize of INFORMS. Gerard Cornuejols is a
recipient of the Lanchester Prize and Editor-in-Chief of Mathematics of
Operations Research. John Hooker recently received the INFORMS Award
for the Best Paper in the OR/Computer Science Interface, and Michael
Trick is the Founding Editor of INFORMS Online and the Director of the
Bosch Institute for International Management. His web site
(mat.gsia.cmu.edu) is a must-see destination for academics and practition
ers with an interest in OR. Finally, R. Ravi is an NSF Career Awardee.
The Operations Research Group is in charge of the Ph.D. program in
Operations Research at GSIA, and hasjoint responsibilities for the program
in Algorithms, Combinatorics and Optimization (ACO). The ACO pro
gram was created in 1989, in collaboration with the Discrete Math Group
in Mathematics and the Theory Group at Computer Science. This program
has been so successful that it has been used as a model for other leading
institutions throughout the world.

MARCH 1999



We invite OPTIMA readers to submit

solutions to the problems to Robert

Bosch (bobb@cs.oberlin.edu). The

most attractive solutions will be pre-

sented in a forthcoming issue.

* ;,/FI77/

Maximizing Vitality

Robert A. Bosch

February 26, 1999

rule set

Figure I gives the rule set of a simple one
dimensional cellular automaton, first investigate
ed by Stephen Wolfram, that when properly
initialized produces evolution patterns that
look remarkably like the natural patterns found
on certain mollusk shells. (See [2, p. 71-73].)


mollusk pattern
ii l 'I I 9



periodic pattern
i l W Ii'

"Aff 00 M "
ME Mim , M 1
11N. - ff -- -I
K ift 00 ,
-0p - : E!
7 Id
laff JO --

This particular cellular automaton, or CA, consists
of n cells arranged in a horizontal line. We refer to the
leftmost cell as cell 0 and the rightmost cell as cell
n-1. We consider each cell i to have two neighbors: a
left neighbor ](i) and a right neighbor r(i). For each
0 r(i) = i+. To allow the cells to "wrap around," we set
1(0) n-1 and r(n-l) = 0.
At each point in time, each cell is either alive or
dead. To start the CA, we must decide which cells will
be alive at time 0 and which ones will be dead then.
(To generate the mollusk pattern, we decided that cells

0, 4, 5, 8, 9, 10, 12, 13, 14, 17, and 19 would be alive
at time 0.) To run the CA, we simply apply the rule
set over and over again. The first application of the
rule set determines the states of the cells at time -1.
The second application determines the states of the
cells at time t-2. And so on. (In the mollusk pattern,
cell 0 is dead at time 1 because of rule 5, which states
that if cells ](i) and i are alive at time t and cell r(i) is
dead at time t, then cell must be dead at time t+1.
Cell 1 is alive at time 1 because of rule 2. Cell 2 is
dead at time 1 because of rule 1.)

S T A 6 1

The Maximum Average Vitality Problem

We define the vitality of a cell over a time interval to be
the fraction of the time the cell is alive during that time
interval. (In the mollusk pattern, cell 0 has a vitality of
0.45 over the displayed time interval.) In addition, we
define the average vitality of the CA over a time interval
to be the average of its cells' vitalities over that time
interval. (In the mollusk pattern, the average vitality
over the displayed time interval works out to be
0.4975.) And finally, we define the maximum average
vitality problem to be the problem of finding an initial
assignment of states to cells that maximizes the average
vitality of the CA over a given time interval [a,b].

An IP Formulation

It is easy to model the maximum average vitality prob
lem as an IP. For each 0 i n-1 and each 0 t b,

Note that as the length of [a,b] increases, the upper
bound gets closer and closer to 3/5. As a consequence, it
follows that the periodic evolution pattern displayed in
Figure 1 is the "most vital" of all periodic evolution pat


Interested readers may enjoy trying to solve the follow
ing problems:
1. Find constraints that enforce rules 2-5 and 7.
2. Prove that the upper bound on v(a,b) is correct.
Hint: Prove that the following inequality is valid
for the IP formulation:

1 if cell i is alive at time t,
x -
xt 0 if not.

Clearly, our objective is to maximize

n-1 b

i=0 t=a
To enforce rule 1 for cell i during the transition from
time tto time t+1, we can impose the constraint
(i) -1 (i)

),t xi,t xr(i),t + xi,t+1 0, t
t+1 1

which works by prohibiting the only "configuration"
that violates rule 1 (i.e., the configuration that has cells
1(i), i, and ri) dead at time t, and cell i alive at time
t+l). Similarly, to enforce rule 6 for cell i during the
transition from time tto time t+1, we can impose the

Xl),t + xr(,t + xi,t+1 2,

3. The IP formulation has n(btl) binary variables.
But only n of them -the variables for time 0 -are
really decision variables in the truest sense. (Once
we have the values of the time-0 variables, the val
ues of the remaining ones are completely deter
mined. In fact, we could run the CA to obtain
their values.) Devise a solution strategy for the
maximum average vitality problem that exploits
this "property."
We will present solutions in a future issue of
OPTIMA. Please send solutions or comments to
bobb@cs.oberlin.edu. See [1] for a detailed discussion of
how integer programming can be used to find interest
ing patterns in another cellular automaton, Conway's
game of Life.

(i) i r)

t+1 1

which prohibits the only configuration that violates rule
6 (and, in addition, the only configuration that violates
rule 8). To enforce the remaining rules, we can impose
similar constraints.

An Upper Bound on Vitality

Using the constraints of the formulation described
above, it is relatively easy to prove that the maximum
average vitality v(a,b) of the CA over the time interval
[a,b] satisfies

3 2
v ,b) + .
5 5)-a+1)

Cymbiola imperialism

MARCH 1999



[1] R.A. Bosch, Integer
Programming and Conways
Game of Life, preprint (to
appear in SIAM Review).
[2] S. Levy, Artificial Life,
Random House, New York,

t (l ) 1 r()
t 1 1 1

1 PT M A 11

MARCH 1999 PAGE 12

) Canadian Operational Research Society Conference CORS '99
June 7-9, 1999, Windsor, Ontario, Canada
URL: http://www.cors.calwindsor/
) Seventh Conference on Integer Programming and Combinatorial Optimization IPCO '99
June 9-11, 1999,TU Graz, Graz, Austria
URL: http://www.opt.math.tu-graz.ac.atlipco99
) Fourth Workshop on Models and Algorithms for Planning and Scheduling Problems MAPSP'99
June 14-18, 1999, Renesse, The Netherlands
URL: http://www.win.tue.nl/~mapsp99/index.html (general information);
http://www.win.tue.nl/~mapsp99/regis.html (registration)
) Workshop on Continuous Optimization
June 21-26, 1999, Rio de Janeiro
URL: http://www.impa.br/~opt/
) Computational Mathematics Driven by Industrial Applications
June 21-27, 1999, Martina Franca, Apulia, Italy
URL: http://www.math.unifi.it/CIME/
) Paul Erdos and His Mathematics
July 4-11, 1999, Hungarian Academy of Sciences, Budapest, Hungary
E-mail: erdos99@math-inst.hu
URL: http://www.math-inst.hu/~erdos99
) Fourth International Conference on Industrial and Applied Mathematics
July 5-9, 1999, Edinburgh, Scotland
URL: http://www.ma.hw.ac.uk/iciam99
) First ASMO UK/ISSMO Conference on Engineering Design Optimization
July 8-9, 1999, Ilkley, West Yorkshire, UK
URL: http://www.brad.ac.uk/staff/vtoropov/asmo_uklasmoukc.htm
) 19th IFIP TC7 Conference on System Modeling and Optimization
July 12-16, 1999, Cambridge, England
URL: http://www.damtp.cam.ac.uk/user/na/tc7con
) Third Workshop on Algorithm Engineering W A E '99
July 19-21, 1999, London, UK
E-mail: wae99@dcs.kcl.ac.uk
URL: http://www.dcs.kcl.ac.uk/events/wae99/
) Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems,
Third International Workshop on Randomization and Approximation Techniques in Computer Science
August 8-11, 1999, Berkeley, CA, USA
URL: http://cuiwww.unige.ch/-rolim/approx; http://cuiwww.unige.ch/-rolim/random
) Sixth International Conference on Parametric Optimization and Related Topics
October 4-8, 1999, Dubrovnik, Croatia
URL: http://www.math.hr/dubrovnik/index.htm
) INFORMS National Meeting
November 7-10, 1999, Philadelphia, PA, USA
URL: http://www.informs.org/Conf/Philadelphia99/
7th INFORMS Computing Society Conference on Computing and Optimization: Tools for the New Millenium
January 5-7, 2000, Cancun, Mexico
URL: http://www-bus.colorado.edu/Faculty/Laguna/cancun2000.html

S T A 6 1

MARCH 1999

International Symposium on Mathematical Programming

The 17th International Symposium on Mathematical

Programming will be held August 7-11, 2000, on the cam-

S pus of Georgia Institute of Technology in Atlanta, Georgia,

USA. A brochure with more information and a call for

papers will be issued soon. The official web page of the

symposium is currently under construction, but watch for

it at http://www.isye.gatech.edu/ismp2000.

First Announcement and Call for Papers
Seventh International Workshop on Project Management and Scheduling (PMS 2000)
April 17-19, 2000, University of Osnabrueck, Germany


Call for Proposals to Host

IS M P 20 0 3

The time has come for all interested parties to
make proposals for hosting the 2003
International Symposium on Mathematical
Programming. Following tradition, a university
site outside the US will host the 2003
All proposals are welcome and will be examined
by the Symposium Advisory Committee, com-
posed of Karen Aardal, John Dennis, Martin
Grotschel and Thomas Liebling (Chair). It will
make its recommendation based on criteria such
as professional reputation of the local organizers,
facilities, accommodations, accessibility and fund
ing. Based on the recommendations of the
Advisory Committee, the final decision will be
made and announced by the MPS Council dur
ing the 2000 Symposium in Atlanta.
Detailed proposal letters should be addressed to:
Prof. Thomas M. Liebling, DMA-EPFL, CH
1015 Lausanne, Switzerland (E-mail:

Following the six successful work
shops in Lisbon (Portugal), Como
(Italy), Compiegne (France),
Leuven (Belgium), Poznan
(Poland), and in Istanbul (Turkey),
the Seventh International
Workshop on Project Management
and Scheduling is to be held in
Osnabruck, a small, charming city
located halfway between Cologne
and Hamburg.
The main objectives of PMS 2000
are to bring together researchers in
the area of project management and
scheduling in order to provide a
medium for discussions of research
results and research ideas and to
create an opportunity for
researchers and practitioners to get
involved in joint research.
Another objective is to attract new
recruits to the field of project man
agement and scheduling to make
them feel a part of a larger network.
For this aim there will be special
sessions on railway scheduling,
timetabling, batch scheduling in the
chemical industry, and robot

Program Committee Peter Brucker,
Chair (University of Osnabrueck),
Lucio Bianco (IASI, Rome), Jacek
Blazewicz (Poznan University of
Technology), Fayez Boctor (Laval
University), Jacques Carlier
(Universite de Technologie
Compiegne), Eric Demeulemeester
(Katholieke Universiteit Leuven),
Andreas Drexl (Christian-Albrechts
Universitat zu Kiel), Salak E.
Elmaghraby (North Carolina State
University), Selcuk Erenguc
(University of Florida), Willy
Herroelen (Katholieke Universiteit
Leuven), Wieslaw Kubiak
(Memorial University of
Newfoundland), Chung-Yee Lee
(Texas A & M University), Klaus
Neumann (University of
Karlsruhe), Linet Ozdamar
(Istanbul Kultur University), James
Patterson (Indiana University),
Erwin Pesch (University of Bonn),
Marie-Claude Portmann (Ecole des
Mines de Nancy INPL), Avraham
Shtub (Technion Israel Institute of
Technology), Roman Slowinski
(Poznan University of Technology),

Luis Valadares Tavares (Instituto
Superior Technico, Lisbon),
Gunduz Ulusoy (Bogazici
University, Istanbul), Vicente Vails
(University of Valencia), Jan
Weglarz (Poznan University of
Technology), Robert J. Willis
(Monash University).
Preregistration If you are interested
in participating, please visit our web
site (http://www.mathematik.
2000/) and complete the pre-regis
tration form, or contact us by e
m ail IIuIIiu-' i, .,n l ,iI
Pre-registration does not involve
any obligations, but helps us to
plan the schedule and keep you
informed. In your e-mail please
include your surname, first name(s),
affiliation and e-mail address, and
whether or not you intend to give a
talk. Presentations will be selected
on the basis of a three-page extend
ed abstract to be submitted no later
than September 15, 1999.

Important Dates Abstract submis
sion: September 15, 1999;
Notification of acceptance:
November 1, 1999; Workshop reg
istration deadline: December 15,
Registration costs include the con
ference fee, a welcoming party, cof
fee breaks, and three lunches. The
following prices are provisional:
Early registration fee, DM 300;
Late registration fee, DM 350;
Excursion and dinner, to be
The deadline for early registration is
December 15, 1999. Please consult
the conference web site to register.
Information Sources For up to
date information, including infor
nation on hotels and the city of
Osnabruck, please visit our web site

S0 P TIM A 1*1

MARCH 1999

First Announcement
6th International Symposium on Generalized Convexity/Monotonicity
Karlovassi, Samos, Greece, August 30 -September 3, 1999
Summer School on Generalized Convexity/Monotonicity
Karlovassi, Samos, Greece, August 25-28, 1999

Scope Various generalizations of convex func
tions have been introduced in areas such as
mathematical programming, economics, man
agement science, engineering, stochastics and
applied sciences. Such functions preserve one or
more properties of convex functions and give
rise to models that are more adaptable to real
world situations than convex models. Similarly,
generalizations of monotone maps have been
studied recently. A growing literature in this
interdisciplinary field has appeared, including
the proceedings of the five preceding internal
tional symposia since 1980. The Symposium is
organized by the Working Group on
Generalized Convexity and aims to review the
latest developments in the field.
Invited Speakers J. Jahn (University of
Erlangen, Germany), H. Konno (Tokyo
Institute of Technology, Japan), P. Pardalos
(University of Florida, USA), A. Prekopa
(Rutgers University, USA)
Program Committee
S. Komlosi (Pecs, Hungary), Chair; C.R. Bector
(Winnipeg, Canada); R. Cambini (Pisa, Italy);
B.D. Craven (Melbourne, Australia); J.P.
Crouzeix (Clermont-Ferrand, France); J.B.G.
Frenk (Rotterdam, The Netherlands); N.
Hadjisavvas (Samos, Greece); D.T. Luc (Hanoi,
Vietnam); J.E. Martinez-Legaz (Barcelona,
Spain); P. Mazzoleni (Milan, Italy); J.P
Penot(Pau, France); S. Schaible (Riverside,
Organizing Committee N. Hadjisavvas (Samos,
Greece), Chair; R. Cambini (Pisa, Italy); A.
Daniilidis (Pau, France); J.B.G. Frenk
(Rotterdam, The Netherlands); S. Schaible
(Riverside, USA)
Symposium Information
General information The Symposium will be
hosted by the Department of Mathematics at
the University of the Aegean, located in
Karlovassi on the island of Samos. Samos, the
birthplace of Pythagoras and Aristarchus, is one
of the biggest and most picturesque islands in
the Aegean Archipelago. Its unique natural and

archeological sites make it a distinct resort and a
historical treasure.
Registration The Symposium fee is USD 150
(USD 75 for students submitting verification of
their status) for registration until March 31,
1999 and USD 200 (USD 100 for students)
after this date and until June 30, 1999. The fee
includes: admission to all sessions, transport
tion from and to the airport for those partici
pants who will come at the beginning and leave
at the end of the Symposium or the Summer
School, a one-day excursion around the island, a
banquet, and a copy of the proceedings when
published. For more information, please send an
e-mail to gc6@math.aegean.gr.
Important dates
Please note the following deadlines: Early reg
istration: March 31, 1999; Late registration
(after March 31, 1999): June 30, 1999; Final
manuscripts of invited papers: At the
Symposium; Titles and abstracts of talks: June
30, 1999; Submission of manuscripts for public
cation in the Symposium proceedings:
September 30, 1999
Important addresses/More information
Additional information about the Symposium
may be obtained by writing to Mrs. Thea Vigli
Papadaki, Department of Mathematics,
University of the Aegean, Karlovassi 83200,
Samos, Greece. Phone (+30-273-33914, 34750;
Fax: +30-273-33896; e-mail:
gc6@math.aegean.gr) or by visiting the web page
of the Symposium: http://kerkis.math.aegean.
More information on the activities of the
Working Group on Generalized Convexity can
be obtained at the URL address
Proceedings The symposium proceedings will
be published by Springer-Verlag in the series
"Lecture Notes in Economics and Mathematical
Systems." The editors of this volume, response
ble for the refereeing process, are: N.
Hadjisavvas (nhad@aegean.gr);
J.E. Martinez-Legaz

and J.P Penot
(jean-paul.penot@univ-pau.fr). Manuscripts
should be written in plain LaTeX (with AMS
symbols if necessary). Those not familiar with
LaTeX may use an interface such as Scientific
Word (choose style: standard LaTeX article).
Contributors must send the electronic file and
one hard copy to one of the editors by
September 30, 1999.
Summer School
A Summer School will precede the Symposium.
It will be held from August 25 to August 28,
1999, at the same place as the Symposium
(Karlovassi, Samos, Greece). The Summer
School aims to introduce graduate and Ph.D.
students, as well as scientists from other fields,
to the subject of Generalized Convexity and
Generalized Monotonicity. Topics include:
Introduction to convexity and generalized con
vexity (J.B.G. Frenk), Uses of generalized con
vexity in economics (J.E. Martinez-Legaz),
Fractional programming (S. Schaible), 1st and
2nd order characterizations (J.P. Crouzeix),
Generalized convexity and nonsmooth analysis
(J.P. Penot), Duality and application to economy
ics (J.E. Martinez-Legaz), Algorithmical aspects
(J.B.G. Frenk), Vector optimization (D.T. Luc),
Introduction to global optimization and its
applications (P. Pardalos), Generalized monoto
nicity (S. Schaible), Variational inequalities and
equilibrium problems (N. Hadjisavvas).
No fees are required for participation, but grad
uate and Ph.D. students wishing to attend
should send a brief CV and a letter of recom-
mendation to: Professor Nicolas Hadjisavvas,
Department of Mathematics, University of the
Aegean, 83200 Karlovassi, Samos, Greece. You
may also e-mail the above to
gc6@math.aegean.gr (CV and recommendation
letters are not required for established scientists).
An effort will be made to cover, at least in part,
local expenses of a number of participants.

From the Nordic Section

The next meeting of the Nordic Section of the Mathematical Programming Society will
be held at Malardalen University in Vasteras, Sweden, on September 25-26, 1999. For
more information, please look on the web (http://www.ima.mdh.se/tom).
See you there.


S *0 A M 1 P 1

Workshop on Continuous
Rio De Janeiro, June 21-26, 1999
Special Issue of Annals of
Operations Research

An agreement has been reached with the editors
of Annals of Operations Research concerning
the publication of a special issue of this journal
devoted to papers submitted to our workshop.
Papers submitted to the special issue will be sub
ject to the standard refereeing process, and the
issue will be a regular one (i.e., it will not be the
proceedings of the workshop, but rather it will
consist of those papers submitted to the work
shop which the referees assess as deserving public
We request that all potential participants let us
know if they intend to submit a paper to be
considered for this special issue. This does not
represent a full commitment to submit the
paper, or even to attend the workshop, but it
will allow us to estimate an upper bound on the
number of papers to be processed.
We would appreciate it if you would send a
message to optim@impa.br announcing your
intention to submit, preferably including a ten
tative title of the paper. If you do so, you will
receive specific directions and deadlines for sub
Additional Information For further information
about the workshop, please consult the
Optimization home page at IMPA

Advanced Design Problems in
Aerospace Engineering
July 11-18, 1999, Erice, Italy
The meeting will be coordinated by Prof. Aldo
Frediani (Pisa University, Italy) and Prof.
Angelo Miele (Rice University, Houston, USA).
There will be a set of lectures on the design of
new generation aircraft (subsonic, transonic,
supersonic and hypersonic) and space vehicles
(orbital, interplanetar) with a particular empha
sis on the interactions of mathematical methods
and numerical applications, including optimiza
tion, on the design of aerospace vehicles. A lim
ited number of people can attend the meeting.
Details can be found on the web page of Ettore
Majorana Center of Erice

International Conference on Optimization and Numerical Algebra
September 27-30, 1999, Nanjing Normal University, Nanjing, China

First Announcement

Objectives The conference aims to review and
discuss recent advances and promising research
trends in some areas of Optimization and
Numerical Algebra.
Topics Include Linear Programming and
Nonlinear Programming, Convex Programming
and Nonconvex Programming, Nonsmooth
Optimization, Global Optimization, Stochastic
Programming, Multiobjective Optimization,
Network Programming, Variational Inequalities,
Linear and Nonlinear Systems of Equations, Least
Squares Problems, Computation of Eigenvalue
Problems, Matrix Computation and Generalized
Inverses, Applications of Optimization and
Numerical Algebra
Conference Organizers Z. Bai,
bzz@lsec.cc.ac.cn (Chinese Academy of Sciences,
Beijing); Q. Ni, nifqs@dns.nuaa.edu.cn
(Nanjing University of Aero and Astronautics,
Nanjing); L. Qi, 1.qi@unsw.edu.au (University
of New South Wales, Sydney, Australia); Y.
Song, yzsong@pine.njnu.edu.cn (Nanjing
Normal University, Nanjing); W. Sun,
wysun@pine.njnu.edu.cn (Nanjing Normal
University, Nanjing); Y. Yuan,
yyx@indyl.cc.ac.cn (Chinese Academy of
Sciences, Beijing).
Sponsorship National Natural Science
Foundation of China, Chinese Academy of
Sciences, Chinese Mathematical Society,
Chinese Computational Mathematics Society,
Chinese Society of Mathematical Programming,
Institute of Computational Mathematics and
Sci-Eng Computing, Nanjing Normal
University, Nanjing University of Aeronautics
and Astronautics
Scientific Program Committee X. Chang
(McGill University, Canada), Z. Cao (Fudan
University, Shanghai), C. Dang (City University
of Hong Kong, Hong Kong), J. Ding
(University of Southern Mississippi, USA), M.
Fukushima (Kyoto University, Japan), B. He
(Nanjing University, Nanjing), C. Kanzow
(University of Hamburg, Germany), A. Rubinov
(University of Ballarat, Australia), J. Shi (Science
University of Tokyo, Japan), T. Tanaka (Hirosaki
University, Japan), Ph.L. Toint (University of
Namur, Belgium), C. Xu (Xian Jiaotong
University, Xian), X. Yang (University of
Western Australia, Australia), J. Zhang (City

University of Hong Kong, Hong Kong), S.
Zhang (Erasmus University, The Netherlands),
S. Wu (National Cheng Kung University,
Taiwan), J. Yuan (Federal University of Parana,
Invited Speakers O. Axelsson (Nijmegen
University, The Netherlands), J.R. Birge
(University of Michigan, USA), T. Coleman
(Cornell University, USA), D.Y. Cai (Tsinghua
University, Beijing), S.C. Fang (North Carolina
State University, USA), M. Ferris (University of
Wisconsin, USA), M. Fukushima (Kyoto
University, Japan), W. Gander (ETH,
Switzerland), J.Y. Han (Chinese Academy of
Sciences, Beijing), E.X. Jiang (Fudan University,
Shanghai, Shanghai), P. Kall (University of
Zurich, Switzerland), WW Lin (National
Tsinghua University, Taiwan), W Niethammer
(Karlsruhe University, Germany), L. Qi
(University of New South Wales, Australia), D.
Ralph (University of Melbourne, Australia), Z.
Shen (Nanjing University, Nanjing), E.
Spedicato (University of Bergamo, Italy), J.
Stoer (Wuerzburg University, Germany), J. Sun
(National University of Singapore, Singapore),
J.G. Sun (University of Umea, Sweden), K.
Tanabe (The Institute of Statistical
Mathematics, Japan), Ph.L. Toint (University of
Namur, Belgium), K.L. Teo (The Hong Kong
Polytechnic University, Hong Kong), J.-Ph. Vial
(University of Geneva, Switzerland), A.J.
Wathen (Oxford University, UK), T Yamamoto
(Ehime University, Japan), Y. Yuan (Chinese
Academy of Sciences, Beijing), J. Zhang (City
University of Hong Kong, Hong Kong), X.
Zhang (Chinese Academy of Sciences, Beijing),
Y. Zhang (Rice University, USA)
Call for Papers Titles and abstracts of invited
and contributed papers must be received by July
10, 1999. The abstracts should be typed in
LaTeX, not exceed one page, and be sent by
e-mail (niqfs@dns.nuaa.edu.cn or


MARCH 1999

S0 P TIM A 1*1

MARCH 1999

Fourth Workshop on Models and Algorithms for Planning and
Scheduling Problems
Second Announcement

June 14-18, 1999, Renesse, The Netherlands

Conference Approach The work
shop aims to provide a forum for
scientific exchange and cooperation
in the field of planning, scheduling,
and related areas. To maintain the
informality of the previous work
shops and to encourage discussion
and cooperation, there will be a
limit of 100 participants and a sin
gle stream of presentations.
Invited Speakers Michel Goemans,
CORE, Louvain-la-Neuve,
Belgium; Martin Gr6etschel, ZIB,
Berlin, Germany; Michael Pinedo,
New York University, New York,
USA; Lex Schrijver, CWI,
Amsterdam, The Netherlands; Eric
Taillard, IDSIA, Lugano,
Switzerland; Richard Weber,
University of Cambridge, England;
Joel Wein, Polytechnic University,
Brooklyn, USA; Gerhard
Woeginger, Technische Universitaet
Graz, Austria
The invited speakers will present a
one-hour lecture. Abstracts of these
talks can be found on the web
Contact Address Cor Hurkens,
Department of Mathematics and
Computing Science, Eindhoven
University of Technology, P.O. Box
513, 5600 MB Eindhoven, The
Netherlands; Fax: (0031)
402465995; E-mail:
Organizers/Program Committee
Emile Aarts, Eindhoven University
of Technology, Philips Research;
Han Hoogeveen, Eindhoven
University of Technology; Cor
Hurkens, Eindhoven University of
Technology; Jan Karel Lenstra,
Eindhoven University of
Technology; Leen Stougie,
Eindhoven University of
Technology; Steef van de Velde,
Erasmus University, Rotterdam
Instructions for Participants
Persons interested in participating
are encouraged to send an e-mail to:

mapsp99@win.tue.nl. Details about
dates, accommodation, fees, travel,
etc., can be obtained at the web site
address listed above.
Important Dates May 1, 1999
Last date for early registration; June
9-11, 1999 IPCO '99, Graz; June
14-18, 1999 MAPSP '99
Conference Sponsors This confer
ence is supported by: Eindhoven
University of Technology; Dutch
Technology Foundation STW;
EIDMA Euler Institute in Discrete
Mathematics and its Applications;
BETA Research Institute for
Business Engineering and
Technology Application; IPA
Institute for Programming Research
and Algorithmics; Baan Company,
Ede, The Netherlands; TNO
Physics and Electronics Laboratory,
The Hague, The Netherlands;
Centre for Quantitative Methods
CQM B.V., Eindhoven, The
Netherlands; OM Partners, Capelle
a/d IJssel, The Netherlands;
Numetrix, Brussels, Belgium;
Philips Research Laboratories,
Eindhoven, The Netherlands; and
ORTEC Consultants BV, Gouda,
The Netherlands.


Call for Applications
1999 Nanjing Award in Optimization and Numerical
Algebra for Young Researcher

The applicant should be either
a current student, or a
researcher who obtained his/her
last degree on or after January
1, 1994, or any person born on
or after January 1, 1968. Only
one paper can be submitted for
consideration and it must also
be submitted for presentation at
the conference. The paper can
be co-authored but the appli
cant must be the major contrib
utor (significantly more than
The award selection committee
consists of: T.F. Coleman
(USA), coleman@tc.cornell.edu;
P. Kall (Switzerland),
kall@ior.unizh.ch; and J. Zhang
(Hong Kong), Committee
Chair, mazhang@cityu.edu.hk.
Applicants should send their
application to all three of the
award selection committee
members by e-mail on or before
July 10. The application (an
ASCII file) should include a
short curriculum vitae and a
detailed abstract of the paper
(no more than three pages).
The technical part of the appli
cation should be in plain TeX,

LaTeX or should be a separate
postscript file, as should be the
full paper which may be
requested if the selection com-
mittee feels it to be necessary in
making its judgment.
The committee will shortlist
about three papers for final
competition and notify the can
didates around September 10.
The three selected papers will
be presented in a special session
in the conference. The award
will be announced at the con
ference banquet.
Special Arrangements
Conference proceedings, special
issues of some journals, tours
and accommodations arrange
ments will be indicated in the
Second Announcement.
Further information can be
obtained online
http://www.usm.edu) or by con
acting the conference organize
ers or any members of the sci
entific program committee.

International Conference on Complementarity Problems

June 9-12, 1999, Madison, Wisconsin, USA

The contemporary applications and algorithms that will be emphasized at the meeting will reflect the 35 years
that have passed since complementarity was formally introduced and employed as a powerful mathematical
model for a wide spectrum of problems in diverse fields. The conference is intended to bring together engi
neers, economists, industrialists, and academics from the U.S. and abroad who are involved in pure, applied,
and/or computational research of complementarity and related problems.
The conference will consist of invited presentations, and is limited to 100 participants (including the speak
ers). A refereed volume of proceedings of the conference will be published. There are three major themes of
the conference: engineering and machine learning applications, economic and financial applications, and com-
putational methods. Each theme will be represented by several experts in the area.
Further details on the meeting, including registration deadlines, hotel and travel information can be found
online (http://www.cs.wisc.edu/cpnet/iccp99).

S T A 6 1

MARCH 1999

Workshop on the Theory and Practice of Integer Programming
in Honor of Ralph E. Gomory on the Occasion of his 70th Birthday

We are pleased to
announce a workshop
in celebration of Ralph
Gomory's 70th birthday. The focus
of the workshop will be on integer
linear programming. The workshop
is sponsored by DIMACS, as part
of the 1998-99 Special Year on
Large-Scale Discrete Optimization,
and by IBM. The workshop will be
held August 2-4, 1999, at the IBM
Watson Research Center in
Yorktown Heights, New York. The
workshop will include lectures by
leading international experts cover
ing all aspects of integer program
ming. We hope that the lecture
program will be of particular inter
est to young researchers in the field,
including Ph.D. students and post
doctoral fellows.

Symposium on Operations
Research 1999, SOR '99
During September 1-3, 1999, an
International Symposium, SOR
'99, organized by the German
Operations Research Society
(GOR) will take place in
Magdeburg, Germany. All areas
of Operations Research will be
covered at this conference. For
more information, contact: G.
Schwodiauer (general chair),
University of Magdeburg,
Faculty of Economics and
Management, P.O. Box 41 20,
D-39016 Magdeburg, Germany;
phone +49 391 6718739; fax
+49 391 6711136; E-mail
burg.de. Additional information
about the conference can be
found online (http://www.uni

A conference banquet will be held
with Alan Hoffman (IBM) as the
Master of Ceremonies. The ban
quet speakers will include Paul
Gilmore (University of British
Columbia), Ellis Johnson (Georgia
Tech), and Herb Scarf (Yale).
For more details, please see
Invited Lecturers include: Karen I.
Aardal, Utrecht University; Egon
Balas, Carnegie Mellon University;
Francisco Barahona, IBM Watson
Research Center; Imre Barany,
Hungarian Academy of Sciences;
Daniel Bienstock, Columbia
University; Robert Bixby, Rice
University; Charles E. Blair,
University of Illinois; Vasek
Chvatal, Rutgers University;

Sebastian Ceria, Columbia
University; Gerard Cornuejols,
Carnegie Mellon University;
William H. Cunningham,
University of Waterloo; John J.
Forrest, IBM Watson Research
Center; Michel X. Goemans,
University Catholique de Louvain;
Ralph Gomory, Sloan Foundation;
Peter Hammer, Rutgers University;
T.C. Hu, University of California at
San Diego; Ellis Johnson, Georgia
Tech; Mike Juenger, Universitat zu
Koeln; Berhard Korte, University of
Bonn; Thomas L. Magnanti,
Massachusetts Institute of
Technology; George L. Nemhauser,
Georgia Institute of Technology;
Gerd Reinelt, Universitat
Heidelberg; Martin W.P.
Savelsbergh, Georgia Institute of

Technology; Herbert E. Scarf, Yale
University; Andras Sebo, University
of Grenoble; Bruce Shepherd,
Lucent Bell Laboratories; Bernd
Sturmfels, University of California
at Berkeley; Mike Trick, Carnegie
Mellon University; Leslie Earl
Trotter, Jr., Cornell University;
Robert Weismantel, University of
Magdeburg; David P Williamson,
IBM Watson Research Laboratory;
Laurence Alexander Wolsey,
University Catholique de Louvain;
and Gunter Ziegler, Technische
Universitat Berlin.
Conference Organizers: William
Cook, Rice University; and William
Pulleyblank, IBM Watson Research

Optimization Prize for Young Researchers

Prize for Young Researchers, established in 1998
and administered by the Optimization Section
(OS) within the Institute for Operations
Research and Management Science
(INFORMS), is awarded annually at the
INFORMS Fall National Meeting to one (or
more) young researchers for the most outstand
ing paper in optimization that is submitted to
or published in a refereed professional journal.
The prize serves as an esteemed recognition of
promising colleagues who are at the beginning
of their academic or industrial career.
Optimization award includes a cash amount of
US$1,000 and a citation certificate. The award
winners will be invited to give a one-hour lec
ture of the winning paper at the INFORMS
Fall National Meeting in the year the award is
made. It is expected that the winners will be
responsible for the travel expenses to present
the paper at the INFORMS meeting.
ELIGIBILITY: The authors and paper must
satisfy the following three conditions to be eli
gible for the prize:

(a) the paper must either be published in a
refereed professional journal no more than
three years before the closing date of nom-
ination, or be submitted to and received
by a refereed professional journal no more
than three years before the closing date of
(b) all authors must have been awarded their
terminal degree within five years of the
closing date of nomination;
(c) the topic of the paper must belong to the
field of optimization in its broadest sense.
mittee for 1999 consists of John Birge, Gerard
Cornuejols, Michel Goemans, Jong-Shi Pang
and Michael Todd.
NOMINATION: Nominations should be sent
before July 15, 1999 to Gerard Cornuejols
Graduate School of Industrial Administration
Carnegie Mellon University Pittsburgh, PA
15213, or to any other member of the prize
committee. Nominations should be accompa
nied by a supporting letter.



Combinatorial Optimization

W.J. Cook, W.H. Cunningham, W.R. Pulleyblank
and A. Schrijver

Wiley, 1998

ISBN 0-471-55894-X

ombinatorial optimization is by now a mature field, yet few text
books covering the area are available. This new book (sometimes
referred to as the "4 Bill's book," because of the first names of the
authors) is highly welcome, and will certainly become a classic in the
The primary use of this book is as a textbook. It is equally suitable for
undergraduate and graduate courses. The coverage is indeed extensive, start
ing from elementary topics such as the shortest path and minimum span
ning tree problems to more advanced topics more appropriate for (even
advanced) graduate courses (or self-study) such as total dual integrality,
weighted matroid intersection or the maximum cut problem on planar
graphs. The book is self-contained with a brief appendix with the important
concepts/results in linear programming and a chapter on NP completeness.
There are also many exercises (without solutions) throughout the book,
spanning a wide spectrum of difficulty.
The book covers a variety of very recent advances, such as novel approach
es for the minimum cut problem (the deterministic and random contraction
algorithms of Nagamochi and Ibaraki, and Karger, respectively). Elegance
and simplicity were probably the criteria used by the authors in selecting
which developments to present. When appropriate, simple applications are
discussed in the book. For example, the authors show how the search of a
rectilinear planar layout can be formulated as a minimum cost flow problem,
or how the max flow/min cut theorem can be used to decide if sports teams
will be eliminated.
Being primarily a textbook does not mean that researchers cannot benefit
from its reading. Because of the wide array of results covered in this book,
most, or maybe even all of us except maybe the authors themselves, will dis
cover a few gems (both in terms of results and proofs) while reading it.
The exposition is clear and mathematically rigorous. Also, the authors
give intuitive (or informal) explanations whenever possible and sometimes
mention why alternative approaches fail. The proofs are written with great
One aspect that I particularly liked about the book is the fact that several
problems are looked at from different perspectives. Combinatorial opti
mization is at the intersection between combinatorics, linear programming
and algorithms, and we are often reminded of this multi-faceted aspect of
the field in this book. As an illustration, for the non-bipartite matching
problem, the Tutte-Berge formula is given and proved in Section 5.1,
Edmonds' blossom algorithm is presented in Section 5.2, and polyhedral
results are discussed and proved in several ways in Chapter 6 (including solv
ing matching problems using a cutting plane algorithm based on a mini
mum odd cut separation routine).
In summary, this book should definitely be considered by instructors of
combinatorial optimization courses and can also be invaluable to any
researcher in the field.

S10 A 6 1

MARCH 1999

Handbook of Discrete and Computational Geometry

edited by Jacob E. Goodman and Joseph O'Rourke

CRC Press, 1997

ISBN 0-8493-8524-5

his extensive handbook sums up the knowledge in discrete geometry
and the newer field of computational geometry, two fields that have
flourished in the last decade due to the collaboration between them.
The book contains 52 chapters written by different leading researchers
in their respective areas. The chapters are arranged in six major sec
tions, ranging from more theoretical aspects of combinatorial and discrete
geometry and the theory of polytopes and polyhedra, over algorithms and com-
plexity of fundamental geometric objects and a summary of the most impor
tant geometric data structures and searching and computational techniques, to
applications of discrete and computational geometry.
Although some chapters are probably of interest mainly for researchers in
specialized areas (like a chapter on polyominoes), much of the covered mate
rial is relevant for mathematical programmers: on the one hand, geometric
notions are often prevalent in optimization with several variables, like the
obvious relation between linear programming theory and convex polytopes,
to which six chapters of the book are devoted; but as spatial and geometric
computation is becoming more and more important in applications, it is
also indispensable to have some knowledge of the basic techniques and
results of computational geometry, as they are given in the later parts of the
The book is very clearly structured. After a short introduction, each chap
ter (or subsection) starts with a glossary, giving concise definitions of the
main concepts and technical terms. The main text contains explanations of
the concepts and a summary of the main results; sometimes the main ideas
of proofs are sketched, and open problems are mentioned at the end. Very
often, comparisons between different algorithms or results are summarized
in tables. Each chapter concludes with references to other sources like
monographs, textbooks, or survey articles, and to related chapters in the
book. The cited literature is listed with each chapter.
There is only little overlap between different chapters, like recurring def
initions of the affine span or what a subdivision is. Each chapter can be read
independently of the other chapters, and the material is presented in such a
way that a novice can quickly grasp the main concepts and results of a sub
ject. The layout and visual appearance are very appealing and help to
emphasize the structure. This, as well as the extensive index of terms, makes
the book also well accessible as a reference work.

I will list only those chapters that are probably most interesting from the
point of view of mathematical programming. The chapters about polytopes
has already been mentioned. The algorithmic problems that are treated in
the section about algorithms include convex hull computations, Voronoi dia
grams and Delaunay triangulations, various other sorts of optimal triangula
tions and mesh generation, geometric reconstruction problems of objects about
which only partial or indirect information (like projections or cross-sec
tions) is available, and shortest paths and networks in geometric settings like
a surface or room with obstacles.
The fundamental geometric problems that are dealt with in the section
about data structures are point location (locating a point among a previously
given set of regions); range searching (the problem of preprocessing a set of
objects so that one can quickly report or count the objects contained in a
query region; for example, in a rectangle or half-space); ray shooting andlines
in space, and geometric intersection problems.
The section about computational techniques consists of four chapters cov
ering the geometric aspects and applications of techniques which are rele
vant for algorithm design in general: randomized algorithms, robust geomet
ric computation, dealing with problems of numerics and degeneracy, paral
lel algorithms in geometry, and the technique of parametric search.
The section about applications starts with a chapter on linear program-
ming in low dimensions. Then there is even a chapter on mathematical pro
gramming (by Mike Todd). The other practical chapters algorithmic motion
planning, robotics, computer graphics, pattern recognition, graph drawing,
splines and geometric modeling, geometric problems in automated design and
manufacturinglike molding, milling, and inspection of parts, solid modeling,
geometric applications of the Grassmann-Cayley algebra to mechanical prob
lems of bar frameworks in robotics, rigidity and scene analysis, sphere packing
and coding theory, and crystals and quasicrystals. The final chapter on compu
national geometry software (by Nina Amenta) provides a valuable starting
point for people that are looking for ready-made computer programs for
their geometric problems, and gives the sources for many publicly available
codes, complete with internet addresses.


MARCH 1999

Professor Sigfried Schaible, University of California,

Riverside, was elected AAAS Fellow in September 1998

by the American Association for the Advancement of

Science "for pioneering studies in optimization and oper

nations research, particularly in fractional programming

and generalizations of convexity and monotonicity for

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reprinted with permission
from the book Invitation
to the Traveling Salesman
Problem by Yoshitsugu
Yamamoto and Mikio
Kubo, published by
Asakura, 1996. The
authors cite a passage from
the article 7397-City
Traveling Salesman Instance
Solved Another Layer of
Icing on the Cake," which
appeared in OPTIMA No.
45, 1995. The Japanese
illustration is clearly
inspired by the subtitle!

10 P T I M A 6 1

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pres~xmt Tm r is to describe some of the icing we have added on top of the
previous ]yIer. --

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Vaherg, RinalcU I(2392 cl ies
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llantzig, Flkerson, Johnson (49 cities)

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I MARCH 1999 PAGE 21

S0 *P M I 199 122

Application for Membership

I wish to enroll as a member of the Society.

My subscription is for my personal use and not for the benefit of any library or institution.

E I will pay my membership dues on receipt of your invoice.

H I wish to pay by credit card (Master/Euro or Visa).








Mail to:

Mathematical Programming Society
3600 University City Sciences Center
Philadelphia PA 19104-2688 USA

Cheques or money orders should be made
payable to The Mathematical Programming
Society, Inc. Dues for 1999, including sub
scription to the journal Mathematical
Programming, are US $75.
Student applications: Dues are one-half the
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w LINK Service
Sfor Mathematical Programming Society Members

e are pleased to inform you that Mathematical Programming
S1 is now in the LINK Information Service (accessible through
http://link.springer.de). As a member of the Society, we
would like to offer you the opportunity to use this service free of charge.
Please note that the use of the LINK Information Service is only for your
own purposes.
Please register online (http://link.springer.de/society_access.htm) for
access to the electronic version of the journal in LINK. Important: In the
next two issues of Mathematical Programming (Vol. 84/3 and 85/1), we
will include a title number and registration code which you will need to
use in completing your registration. After registering, you will receive
your personal user name, your password and your LINK-Number via e
mail. For support or questions regarding this online access, please get in
touch with our help desk (helpdesk@link.springer.de).
If you would like to automatically receive the Tables of Contents, include
ing a link to the abstracts, please register for our e-mail alerting service,
LINK Alert (http://link.springer.de/alert/).

MARCH 1999


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Elsa Drake, DESIGNER
University of Florida

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