Title: Optima
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Philip Wole. .....

The Ellipsoid Algorithm

Who would have thought that what
must be the only front-page article about
mathematics ever to appear in the New
York Times ('A Soviet discovery rocks
world of mathematics', November 7,
1979) would be about mathematical pro-
gramming? Our Society was keeping up
with I i, I, I. before the Times made
his work so famous. Some of us first
heard of him at a Conference on \I., I.-
matical 'n" ."ii lnII in Oberwolfaeh,
West Germany in May, 1979, where his
i.. -..1. paper [1], really a long abstract,
was circulated, but all we understood was
its intriguing title. Eugene Lawler stir-
red up interest in it on his return to
Berkeley, and by July Gacs and Lovasz,
visiting Stanford University, had recon-
structed the derivations of assertions rel-
ated to Khachian's. At our invitation
they presented their work to some 300
people at a special session of the Tenth
International Symposium on Mathemati-
cal Programming held in Montreal in Au-
gust. The science press then took notice,
especially in an article [2] in Science, the
version of the story which the Times fur-
ther garbled, creating the impression that
P = NP and that all the hard problems
had been solved. Lawler has written an
engaging account [31 of that part of the
story. A flood of technical papers fol-
lowed, which I have been trying to col-
lect. Our bibliography [4] lists 46, as
well as background literature, mostly So-
viet, and some of the popular commen-
tary. Now the phenomenon has peaked.
The numbers of technical papers in each
month (date on paper, if given; otherwise
of receipt) are:


Jan 1

This February the Society held a
'Workshop on polynomial-time algor-
ithms for linear programming' attended
by 80 people with an active interest in
the subject. The 17 reports presented
and lively discussions brought us pretty
well up to date, and it appears that few
new ideas have arisen since then. There
are no written proceedings, but most of
the short presentations were extracted
from authors' longer written papers re-
viewed in [4j.
I I. algorithm has a history. Most of
it is actually due to other Soviet mathe-
maticians: D.B. Yudin and A.S. '. n,..
sky of Moscow, and N.Z. Shor of Kiev,
whose article 15] states the algorithm in
its clearest form, and also in its best
form for practical computation -- not
only for linear Ilr.':r.,,ilimln._ but for the
much more general convex programming
problem. Because of the multiple au-
thorship and its basic idea -- the genera-
tion of numbers which, geometrically,
describe a sequence of ellipsoids which
must all contain a solution and which
shrink, so that it is eventually identified
-- we refer to it and its near relatives as
the 'ellipsoid algorithm'.
Actually, at least three different al-
gorithms have been presented:

1. That to be attributed to Shor, Yu-
din, and Nemirovsky (above).
2. Khachian's [1]: This is Version 1,
slightly modified so as to be provably im-
plementable in polynomial time on a pos-
sible computer, but hardly practical.

3. That of Gacs and Lovasz [6]. While
inspired by Khachian's, it uses different
formulas. It is better viewed as a revi-
sion of Shor's algorithm, to which it is
mathematically equivalent, although it is
numerically ill-behaved. The ellipsoid
becomes an hyperboloid after a few hun-
dred iterations, destroying the computa-
tion. Many authors use the term
See page 3

ABOUT. ......

Irately tlil Mathematical Progranming
ociety hias been trying tto broa(de its
acitvit e in a number of wav'. 'lI Wolfe s
report tto the meniberslhp distrillbutled i
Miontreal lasAugustlll testifies to this fact. In
order to continue and enhance these efforts
many of the officers and imemlber-s of ithe
society felt that it would be desirable to
havl an il orlnial lmeanlls of 'Olll'inlllicatiol)
available and that tl- cul) 1w b' attainel
by establishing a Newsletter. In addition it
has been pointed out that a Newsletter could
also be used as a vehicle for ani outreaeli
program on llte part of thl(e society by, for
example, permitting a new class of imember-
ship for students.
Consequently thle iPu'ilications Conm-
imittee began to explore several possibilities.
HRecoognizing that the SIG(;1AP Newsletter
(now Huilletin) served the mnalhema/ical
prograining coiniui nilty well in the past,
one proposal was to lound a joint News-
letter. Both lI ;NMAl' and the A( were
somiewliat receptive but not overly enthusi-
astic about such a venture. However, of
much greater significance was the lact that
when such a proposal was aired ill Montreal
ill m meetings of tIhe ,,iil ,i ..... C()o llit-
tee and the (Council, it became evident that a
significant sentiment existed in favor of
establishing our own Newsletter. Another
proposal that was explored was tlhe possibil-
ity of expanding the (OA()L\ Newsletter.
Ilowever, it quickly became evident that this
would not bet feasible because of the highly
specialized interest of COAL and the ad
hoc nature of its publication.
Thus the decision was made to estab-
lish a new Newsletter---(I' I I I'V. The name
was -i. -.I 1 by Phil Wolfe since it trans-
lates so well into many I.,,,.iI. A success-
ful search to find an Editor culminated in
the appointment of Don Hlearn to this
important position. We are: indeed fortunate
that such an able and energetic person as
Donl has agreed to undertake this task. 1
know that I speak for all members of the
Society in wishing hinl every success in ihe
launching of' OTIMA. However (1'1 I',I
See page 2

JUNE, 1980

Cr PTl .... ..from page

will only succeed if YOU!, the members,
support it through your submissions andi
- .. lions. Elsewhere in this first issue you
will finl a column by D)on stating the kind
of material lie is seeking.
Many pe'olex worked hard to bring
OPTIMA into being. In particular I would
like to thank G(;orge Nenhmauser who was
always there when needed and who also
nominated the successful candlidate for
editor. Thanks are also due to 1.1 Powell
whol dtebatled most persuasively during the
many discussions whicl w ere held.
-Michael Heldt
Publications Conuniittee

1)EAl)LINFS Fo'( l ()PT1\IA

Fall Issle S tept:nmbr I

Winter Issue December I


'ITh IV Honn Workshop oni Combi-
natorial ()lptimiizatioln .1 take place August
28 .1) I :;.1, at the Institute fI'r Okono-
im:etrie und (Operations lResea('.rc, IUniv:ersity

As with thie preceding ones, the
workshop is devoted primarily to recent
research ill the area of discrete and combi-i
natorial optimization and related topics suchl
as graph theory, matroids and indepmldencel
systems, polyhe(dral conbinatorics, anatlsis
of comblinatorial algorithms, etc. In addition
to the nemibers of thel Institute and its
visitors for the academic year 191' .1,
leading experts of these fields will partic-
'The' structure of this workshop will he
very informal. There IIl he no program ini
advance. The time schedule for the lectures
I11 he made on tIhe spot, butt participants
are expected to give their most recent and
very best research papers in the above
mentlionedil areas.
If there is enough interest we might
arrange a hike in thie Eifel for Sunday.
Augusti. 3 1, or a boat trip on thl Itiver Rhinc.
I' iticipants are kindly requested to
notify us (of their intention to attend at their
earliest convenience. IFurther information
call be obtained front thIe Institute.
-hernhard korit\

Colmputational results are o)ft'en usedil
in evaluating mathematical programming
algoritllms. To date, our professional has not
developed a clear understanding of how
comtiputational testing should I(e carried out.
The Committee on Algorithms

tives ol (1) developing procedures for
appraising tIhe usefulness of computational
results; (2) n ... ...1.. those who distribute
lprogranis to imeet certain standards of
portability, "' I;,'_ ease of use, and docu-
ienltaioin; and (3) acting as a focal point for
iloiati('o l aLion about clllputr programs that
are available for general calculations and for
test probleIms and test problem generators.
Since 1973, when C()AL was forlmed,t
its i lmembers have been active iln many areas
;,, 1 tI.i. ._ llte collection of test problems s, the
investigation of techniques for removing
Wtiiing., comlpiler, and computer variability
from computational results. the establish-
ment of reliable performance evaluation
criteria for compari g ilathematic:al pIrogralm-
ming software, and tile develolpmentl of a
sound '. il,,. ..l... for comparing ,ii'
software. Guidelines for reporting results of
computational experiments were developed
yv three members of C(AL and published
in the ,ii.. Hi. journals: Matthemiatical
"-,., ,.;,. J() IIOSA, and Transactilons oni
Mathematical Soflware.
CO()A has organized sessions presenlt-
ing results of software testing research at
conferences sponsored by the Mathematical
..- .1111 1 Society, IFAC, I ,,I ,r)li-opeall
O(;ogreSS on operationss Research, thWe
Institute for M anagemenlt Science. aind the
(Operations Research Society of Amenrica. In
1978, a NATO Advanced Stud)y Research
Institute entitled "Design and Inplell enlta-
tion of Optimization Software" was orga-
nized ib ((COA and a text of the prolced-i
ings, edited lb Dr. IlHavey (;reelnberg, is
available through Sijthoff and ...n.d !l..11
Our work continues. In the near

1,ii l il ,i ,i1 1 ,, i i ; l I . p l l.i ilr ii ,
,, n,, i li ,r,, I l b . I, ,, ,,, I, -I. I. I
, ,,,[ l ,d I .,,I ,, i ,I .,

..11.....I i.1 .

.,. I 1 1 I 'II .I i II , ,, I ,,I i- i i I li -I
1 1 .....11 i, i 1 , 1 ,I .. i, .I . I ,hII ,

talks will lbe presented.
1 '. -,I RI I Sy.... ,s ,, h l

I i. Relationship Bielween 0L.. and
softwaree," M. J efifreys
S I I ..i ii- i Evaluation o i Nonilineair
*\... .In n. Codes via Multi-criteria Deci-
sion Analysis," iF. I oolLsma
"The Testing and Evtial nation of'
Mathematical Prograim ing Software," K.
Ilofhnan and I. Jackson.
Tl ere will also he a session sponsored
by (OAL at the 01RSANi I -n meeting in
Colorado Springs in November 1980.
)On January 5-6, 1981, a conference on
'T T. aind Validating MP Algorithmis and
Software will be held in Boulder, Colorado.
This conference will bring together research-
ers from a variety of dIisciplines whio have
performed computational analysis (cithier in
mnatl programming or general computations)
in or der to further develop ilmet ologies
for conducting software evaluation.
o Finally, tIe com im it'ee on AlgorihtIui,
ipblislies a Nlewsletter twice' a year. Cll
purposes off this news letter are to gene rate
an international interchange of' ideas, high-
light research being perforll'ed in software
testing, announce thle availab ility of new
software, and highlight international meet-
ings where rcsearc oil software evaluation
will be presented.
The committee ont Algoritihmr is, a
rather ill group with extremely ambitious
golts. Only witl tlle assistance of muer oTl
tlie MP community can we succeed 'I. I
help us by answering our surveys, contrib-
uting to our newsletter, ad providing rus
witli your suggestions as to how this Coi-
Illittee can be tter serve youtl.
Karila Hofrtfiant ditor
Comniittee oni Algoritions Newsletter

GO()RIT '-. ... o page The Newsletter

1'Kihachians4 algorithmI while actually working with this version. I I..... who have
bee, n able to ( inputll ..... ll have usd some forill of Version 1.

On(e obtains the versions above by setting d = 0 in equations (3,4)
below. The ellipsoid (1) is then 'cut' through its center by intersecting it
with thle halfspace y : a*(y x) < 0(, whose boundary is parallel to the
chosen violated constraint, and the new ellipsoid, which is that of smallest
volume circumscribing the intersection, is given by the updated quantities.
Using (2) instead to calculate d gives a cut whose boundary is the violat-
ed constraint; the volume of the resulting 11'l.'....I may he much smaller.
(It is remarkable that none of the Soviet authors gave the formula for the
"deep cut'. which is not hard to work out. It was discovered independent-
ly by authors of 18 of our 46 technical papers.)

Recommended implementation of the 'ellipsoid' algorithm for solving
Ax < b, where A is an m by n matrix and xeR":

Start: Have x and a nonsingular matrix J such that the ellipsoid
(1) x: I J-l(y x) I <
will contain a solution of the system, if any exists.

Recursion (x, J are replaced by x+,J+ below):
Find a violated inequality a x- [ > 0.

(2) Set d


1 + nd Jrja
(3) Set x = x ---+ J-
1+n al

(4) 1 = n'V I 2 i 1) 1 -
n2_ (n+I) (I+d)

I al2

It seems to be good practice to choose a substantially violated inequality,
but not to do all the work required to choose one which maximizes d. If
d > 1, then there is no solution. jT is proportional to Ak of Shor and
serves as the Q of Khachian. jJJ is the Ak of Gacs and Lovasz.
The basic theory of the method is that each ellipsoid contains all the
feasible points of its predecessor, while the ratio of its volume to that of
the predecessor is

(5) n ( -d2 ) 2

/ (n-1) (1-d)
(n+l) (l+d)

For d = 0 this ratio never exceeds e-1/2n (and is asymptotically equal
to that quantity for large n) so the ellipsoids steadily shrink. If the
feasible set has a positive volume then the algorithm must stop (with a
new x violating no inequality), while if it has zero volume then the cen-
ters x must converge toward the feasible set (although not with any
particular monotonicity). If the feasible region is empty, then one should
eventually discover d > 1.
All that is miatlheimatics, of course, a(nd we know that direct transcription
of formulas into computer code, in the hope. that the computer's representation
of mathematics' 'real numbers will be accurate enough to make thl theory
work, hIas *.,i I il.
Khachian's formulas differ froin those above inl only onei important
respect: lite furttlhr multiplies .1 byI a factor slightly greater than one to ensure
that the inew ellipsoid will indeed enclose l ti old allf-ellipsoid. In order to do
See page 4

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Im-111 l t. t I11 1 1 (114)14 br-'l I 41('4114('
i 4 I I orillat4 o I I4 4 a 1 I I t c 44I rr (4414' (II 14(1i i
of' ille cf rl~ctinlgs rciews c~ilb u Illelidwrsr s

III addl~itioll to these~sc re(lul r itt(irlls we

44144~re'-[ ill ('((d1 iss, ('. V(I4441ple'. (of '(olrc1 c('
MT p ((Ia (14111 I'4 ar4' (III 414 about4 1 [ c
C (1~l da.4 of, In4 (4th ('444 ,lti(l pr'o4(4'raill III 4iu,
alid a ;rtit"lIcs abou~rt spec Mi~l researcil~l ael t k I tICS
ii a~ raiv t lcclinical article appe~l~ar- thtat
irfll be called "444'wsXV it eo)141(1 a 414'(41(44
aIrticl e. I llo IIr Iwe(w('\ 1. X will 144' 1(40Itillo
(otr atrticle's~ whic h offel'cr pecrspectl,e o ili tll('
l'i (l d fII Iatt ell( 4 I I aticl ( 1 ) rop4 '4(4a 44I(I.III itI

440t ll' 4 I ((4(1

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t:It rr auticl'e1 shoul (.4)1 act llo)that1\Y1c
ca ll t t4ke toau 'ii'i' I wl e it 444 I
-iz( ('ron) 8 to (1) w II ia

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((lSltrii t l a rltic tlle (1 4(14''4'llt i

11 I' I. tll epartlu(Ills whel l wll appcar oil

ai41 (1 4'.. I 1ii' ('4'. 1(44 tirllcle basis14444
ChairrluaIT1- l's Cotlj unill i rt j.C

;11(11((1 1 114('I' ill lii (l44(411 41 1)( ; ( 11411 144lli.(

1441 'X411I' ('()1441(:11,1) 144 (14'4'('141('(I)ll
(t ,, lo14C]111

1l etsI terI 1s1 to114 \latll'l4,l'til',l lro4a14 il
II', 1' I ews4r Items
Illectiligs, plus lly other articici of' Iliterest

a d vI I t i i Irt I a 1) o s I o I I al lioun Iweognl ('11 t, all

that using finite Ipereision,. he uses thie kawtor
2'l n- and suppose. that hlsi computer can)
acioiodate fix'ed-point data having 231 bits
mo tihe left and I l, hbitslt to te right of' the
billnarv point, where 1, is the total number of
bits rnq'ired to encode tlihe data of tihe
particular problem, 11i thein notes lhat (i) if
thie systeil has a solution, tIhe it has one in
a sphere of radius 21 (so that \ 0., 21'1
works in (I ) above), and (ii) if ltre system
has a solution, and its inei palitie- arc
relaxil d bI the quantity 2 L. lihenl lle
feasible set has vaoumn e at leas 2 hi-
appro\ximations he can work down fronm thie
startii volume o tllhis lintl volunw in noI
Imore li an Im Mseps. resolving qmue-
lion as to r .whethr nllot ia te ,system ia
feasible. Since this number, as i als tlie
number of arithlli tic operations per step)
,r luired (()(n )) is polynomial in the length
1, of til input ili ; soi if t ihe im e fl or thi(
w hole algorithnl.
O()f course. Khawhian s short paper
doe's not give the proofs thal all this
works; indeed, it does not show just how
to find a feasible point if there is one. let
alone how to solve a linear programming
problem. A number of authors have
filled in t e i latter gap, andl I .lii ; andi
7 [g7]ive the full proofs.

S j- *Fr F ia ni I
'Th'ie Society, j illtly with ltle i 'ocictyii
o Industrial and Applied Matl(hl aiti'ic ill
tpol sor The (, eo'(rge i)allltzi Prize inl
honor of Professor I)antzig's roltrihutiols
to iinatheinmatical progranllilng. lir-st award
of the prize I- ailticipated at thlle Ileventh
\latheim atical i ........n ... 111' y iposium in
l ill i I' Thi ( pri e is sch(ldul lor
po -sible a ad i\ery thllree vy.ars thelireaft'r.
with every third award being at a 51\IA1
national ni'eting.
The award, in ie lonnii of a icerilificat
containing the citation and a cash prize, is to
be made for original work. which by its
breadth al(nd s(()cope. constitutes ain outstand-
inlg contribution to tihe field. \ominilationl
will be made to thtle Societie" lxecuiive
(oiitll iil tv by an ad hoc col itteei of iMP
and SItAM iiicmiIerts. Coninittei (Chairlman
for tihe first award is Professor Ipoger J.
H,. W'ct o( the university \ o(l Keniuckv. who.
aloni| with Winhard (.otle. E lis ]ohlson and
ich d van Silykc, suggested (the l1)antzigl

In practice the formulas (1-4) do
work on a real computer. I computer us-
ing APL on an i. 370/168. In, arith-
nmetic is 370 double precision: 54-56 bits
mantissa, exponent range 10I 1 have
had no trouble on various problems, tak-
ing up to as many as 30,(100 steps: the
mathenaics seems to work. and when I
compute tlhe volume of an ellipsoid. it
has (in the d = 0 case) very nearly tihe
predicted volume. I have also tried some
small problems in which the feasible re-
a'on was a single point and had trouble
(signalled by a false indication of infeasi-
iility) only after the procedure had
produced a point that the computer
could hardly distinguish from the right
'. I one reason the I .... ,', au-
thors have wisely not offered the ellip-
soid algorithm as an alternative to the
simplex method is that the theory. at
least for the d = 0 ease. predicts dismal-
ly slow convergence. In order to add one
decimal place to the accuracy of a solu-
tion i.e. to reduce the distance to a
solution by a factor of 10 in each dimen-
sion -- we can expect to have to reduce tlhe
volume of th (ll rrnt llipsolid b1> a lac-
tor of 10) Asking the( estimate above
for thel ratio (5). in k steps the volume
will be reduced by the factor e

Prize. \All are f(wller students of Professor
I )antzi/g.
iTh prize per ification i li at that tine
i.1. I! I.. -. ) for which 1 ll at ward is m ade
iimut tlbe l blich al ailable and that thev mail
belong lto ally aspec of mathematical
prograininn;g in its broadest sense. The
colitri)ulito1 (s) eli ihle forn consideration ar
it re ricted i with respect to tlie ag or
number of' their authors although fprelerecc
will bIe given ll o si i ly-aulhored work of
"younger people. Professor \\Ils has stated
that the com nittec ill seek work in llt l
spirit of lhantzig' own in that, ideally, it
should contribute to both application and
Support for thle l)antzig prizu will
conc from individual, corporate., and
institutional contributio i. I . 'wishling to
contribute should sendl a cheWk payable to
thie (;. )a .tziig luIlnd to SIA.\\l. l: I Tth
Street. i .: .. Il .1,, Pa. 19103. Individual
contribution, are tax deductible.

which gives k = 4.6nt as the required
number. \' e would expect the simplex
method to solve such a problem nearly to
machine accuracy in less than 2n steps.
The following data are typical. I I.
illustrative problem in thet user's manual
for the IBM linear programming routine
MPSX/370 has 7 variables, 14 inequali-
ties. apd I equation (which I write as 2
inequalities),. i; t the primal and
dual together (admittedly not the best
way to handle the problem). I get a ss-
tem of 47 inequalities in 23 variable,. I
chose the initial ellipsoid as a sphere of
radius 1000 about the origin. My pro-
gram monitors the worst violation of any
inequality: the table below gives the
number of the first step on which the
worst violation fell below the indicated
power of ten (after being only 200 ini-

I. alteration
Version I
15 2

3 12
2 17
1 934
0 2941()

-2 72 18
3 9280
4I, 114903
, 13820
0 16130
.7 1 : .7

Deep Cut

5 7

I l. number of steps to reduce b' 0.1 i:.
close to the predicted value 2433 for
Version 1. while the deep cut spends
things utp bv a factor of 3 -- amounltilln to
a trifling reduction in tle ellipsoid \1o-
i Our o\il \ 1, v\rion o( the Osu'-
plex method rcquir"d 21 iterations to 0ohc
the( 47-inequialityl problem (reduced tot a
24-inequalit Prohblem in noinn gati\lc ,ari-
ablel ) w\illt a m illaxii m i laltioin o1' I1
It take.- 10 o()r 15 iteration- to i-)ke the
original lin air Ipro ra nmil no |ill prolmll i., (IeI)ei-
ding on whether )1or not tile usua11 ( l speciall
device for ... i.ri ui'pp r- utnded larialic,
is Used.

'licre are still. o)' conIrcI a i reait l nit -
i"T of o!"rtunitiAW tor furtller himpto-
Ilinls. and iliantv lia l iieen t wi' -ted in
the papers received, lH id fi l tile Ieep cut.
imnlprovement- )r-opol -d for te hI I aiu
coitillue i'(t page

LG(ORITHiM.....frolr page 4

algorithm include: general methods for
getting a smaller starting ellipsoid; dis-
carding unneeded inequalities and early
determination of infeasibility; combining
inequalities for improved cuts; and ways
for removing equations from the system.
Further, possibly better ways to do linear
programming than by the expensive si-
multaneous primal-dual formulation are
proposed: hybridization with simplex or
related methods; the construction of cuts
using the objective function; and using
complementary slackness to predict irrel-
evant variables and constraints so that
the size of the system can be progressive-
ly reduced. (The references 8-10 cover
most of these points.)
despitee these opportunities, we have
yet to hear of evidence that the ellipsoid
algorithm can compete with the simplex
method. Since the invention of the lat-
ter in 1947, mathematical improvements
in the algorithm itself have probably
speeded it up by no more than a factor
of 10. The most important development
has been in data-handling -- the exploita-
tion of sparseness, which accounts for
several orders of magnitude in computing
time for problems of serious size. We
think the ellipsoid algorithm would not.
have competed with the simplex method
if it had been available in 1947; further,
no one seems to have found a solid way
to exploit sparseness in it.
However, whatever its demerits for
linear "...-' iii ,;": we can say this for
the ellipsoid algorithm: it solved a signif-
icant theoretical problem, and can be
used to solve others; it may still be prac-
tically useful for the difficult nonlinear
problems that Shor proposed it for; and
it certainly brought a new kind of excite-
ment to our area of applied mathematics.


1. I. C. Khachian, 'A polynomial algor-
ithm in linear programming'. Dokla-
dy Akademniia Nauk iJ: 1' 244 (No.
>. February 1979) 109:-96 (translated
as Soviet i ii. i.n i. s Dokladv 20,
2. G . K olata. '''i Ii I.. 1....... am azed
bv Russian's discovery'. Science 206(
(2 .... *.. r 1979), 54V5-46.
3. E.I. Lawyer, 'The great mathematical
sputnik of 1970'. University of Cali-

fornia, I,. I i. y, Calif., U.S.A., Feb-
ruary 1'::11.
4. P. Wolfe, 'A bibliography for the el-
lipsoid algorithm'. I !1 Research
Center Report No. 8237, April 1980.
Available on request.
5. N Z. .... 'Cut-off I. I ....1 witli
Space Extension in Convex Program-
ming Problems'. Kibernetike 13 (No.
1, Jan.-Feb. 1977), (' -.", Translated
as Cybernetics 13, 94-96).
6. P. Gacs and L. Lovasz, 'Khaehian's
Algorithm for Linear Programming'.
Distributed in preliminary form at
the X International Symposium on
Mathenatical Programming in Mont-
real, Canada, 27-31 August 1979, and
later as Report CS 750, Computer Sci-
ence Department, Stanford Universi-
ty, Stanford, California 94305. U.S.A.
7. M.W. Padberg and IX I,. Rao. 'The
Russian method for inequalities II:
approximate arithmetic', January
1980. Graduate Schlool of Business
Administration, New York Universi-
ty, New York, NY 10006, U.S.A.
8. D. Goldfarb and M.J. Todd,
ti.. .l ii ..ii, and implementation of
the Shor-Khachian algorithm for line-
ar programming', January 1980. D)e-
partment of Computer Science, Cor-
nell University, Ithaca, NY 14853,
9. P.C. Jones and E.S. Marwil, 'A dimen-
sional reduction variant of
Khachian's algorithm for linear pro-
gramming problems', January 1980.
EG & G Idaho, Inc., lPO.B. 1625.
Idaho Falls, Idaho 83415, U.S.A..
10. iP.. 'ickel' 'So ime improve ienits t
Khachiyan s algorithm in linear pro-
grammning, December 1979. Polytech-
nic hinsilutute of New York, BHoute I10.
Fariningdale. N.Y. 117:35. U.S.A.
* @

International Summer School in
Optinization Techniques and Applicalions

\ sulluller school entitled "Optiniza-
tion: Techniques and Applications" will Ibe
held June II July 11 1'" at tile Balls
Park Site of ith latfield Polytechnic.
1)irector for tile course is Dr. I.C.W'. I)ixon.
(Course f ees are 200 for registration
and X I .I for accoiodationls. For further
information, contact Mrs. P. Ingraim, School
of Information Sciences, I I- Hatfield
Polytechnic, P.O. Box 109, llatf'icL Herts.
/\1,10 9 ,\ United Kingdom.

1980 N.A.T.O. Advanced Research Institute
on Generalized Concavity

A NAT() Advanced Study Institute oni
-"( neralized Coneavily in Optimization and
cI'conollics" Il be held in Vancouver/
(tnadla on August 41-15, 1980. The school is
directed by M. Avriel, laifa: S. Schaibhc,
,dmoltol; and W.T. Zienliba. VaIncouver.
Topics .11 include characterizations of
various concepts of1' generalized concav-
it special functional forms, optirnality and
duality fractional ., _. ........ i m ulti-
critcria optimization, numerical solution
methods, applications in management sci-

wishing to participate sl would contact I i.1. -
soa Siegfried Schailath., Faculty of Blusiness
Administration, iUniversit\ of Albecrta.
Ednonton, Alberta TGM 2(Gl, (anada, as
soon as possible.
S. Schaibhle

1981 N.A.T.O. Advanced Researeh Institute
on Nonlinear Optimization

A\ NATO Advanced Iescarcli Institute
on Nonlinear Optimization (co-sponsored by
the Nlatlienatical I'rogramming Society) ill
le held in (Camlridge, eI'ngland from July
13lh 24lh, 1981. Because the main purpose
of these Institutes is to make a critical
assessment of the currently knowledge of a
subject, and to publish conclusions for the
benelit of a wider coninundily, tie main part
of the program will I)e discussion sessions oil
different subjects ill Nonlinear O()ptimiz-
ation, including the theory and development
of algorithms and their software. Key
addrc:sses at the discussions will hie iven Iby
K.M.I lH ale, JE is .. IFlehtcher, PI'.I .
(;ill, '. t a, Loo. More, anld i.H.
Schnabel. Also there b I.l he many opportu-
nities for informal discussions and "or
participants to present seminars on their
recent research. Because the numllber of
participants is limited to about fifty, attend-
ance at the Institute is b1 invitation only.
and the organizing commlnittee ,I imeet in
January y '1,'. to select participants. In
order that the comunilhe canll give proper
consideration to those people who wishl to
attend, you are invited to submit an applica-
tion if you would like to participate. A,\ppli-
pation forms and further information are
available froil Professor M.J.1). Powel,
D)A I 1'S silverr Street, Caimbridgey CH3
91W\, lKigland,
M.J.1). P'owell

Technical ports &Working Paprs

I, I i -.1, I' 01' BONN
Department of Operations Research
D)-5300 Bonn, West Germany

11. Giles, "AIdjacency on the postman polyhedron," WP 79128-OK.
M. Grotschel, F. HIaarry, "The graphs for which all strong orientations
,re allmiltonian, WP' 79129-01t.
A. Bacheli, I. Korte, "'linimnum norm problems over transportation
polytopes," WP 79130-OI .
A. Bachetl, "Theorems of the alternative in combinational program-
ring, WP 79131-01t.
1). Ilausmann, It KIa an., 1I. Korte, "' .i. U ,,.- untere Komplex-
ilatss:chranklcn fiir eine Klasse von Knapsack-l'roblernn," WP 79132-OR.
1). laiusinann. "l'arbklassen und Pseudoknoten ifur lAdjazenzch-
rirakterisierugen, W P 79133-0 t.
I. W. Cottll, Obser nations on a class of nasty linear complemen-
tarity problems, W P 79134-O01t.
M. Grotschel, C. .Thomassen, Y. Wakabayashi, "llypotraceable ldi -
raphs, WP 79135-0 Ii.
II. Giles, 1L. Trotter, "On stable set polyhedra for KI 13 -free graphs,
WP 79136-(l01t
A. lHachem, 1. Schirader, "Minimal inequalities and subadditive
duality, "WP 79137-011.
1i. C(ottle, I. von Randow, "On ()-matrices, centroids and simplo-
topes," WP 79138-01l.
C.L. Monm1a, L.E. Trotter, Jr., "On perfect graphs and polyhedra
with (0.1) valued extreme points," WP 79139-OR.
I). PallasIchlke "/In algebraic representation of classical systems, WP
79140-0 t.
I). Ilatsmiann, 1. Korte, ........ versus axiomnatic definitions of
matroids," WP 79141-OR.
A. Bachem, M. Grotschel, "Adjacency relations on polyhedra," WP
1). Ilausmlann, II. Korle, "The. relative strength of oracles for indepen-
ltence systems, WP 79143-OR( .
I). Hlausmann, IB. Korct, "Computational relations between various
,definitions of maltroids and independence systems, WIP 79144-0 I.
"'Sommerschule iUher Optimierung und Operations Research. Zusainmen-
fassung der Vortrage, "WP 7914501-t.
A. Haclhem, "(Concepts of algorithmic computations." WP79146-OR.
M. Grotsciel, "Approaches to hard combinatorial optimization
problems." WP 79147-01l.
l. Korlc, "Maltroids and independence systems," WP 79148-01{.
A. Iachei, M. (Grotschel, "'roof techniques in polyhedral theory."
WP 79149-0 1.

Department of Applied Mathernatics and Theoretical Physics
Silver Street, Cambridge CB3 9ElW, England

MI. J. 1). Powell, "Ouasi-NewIon formulae for sparse second derivative
matrices," 1) A MTPI 1979/NA 7.
M. J. I). Powcll, "Optimization algorithms in 1979," )AMTP
1I. M. Chamberlain, C. Lemarecliud, 11. C. Pederson and M. J. I).
Powell, "The watchdog technique for forcing convergence in algorithms for
constrained optimization, D)AMTP i980/NAl.
Computer Science Department
Cleveland, Ohio 44115

.S. Ladsdon and A.I). Warren, "A Survey of Nonlinear Programming
Applications, Wp 10-01.

Industrial and Systems aEngineering
303 IVeil lHall
Gainesville, Fl 32610

Russell R. larton and Donald W. Hear, "l)Dconmpositioin ITechniques
for Nonlinear Cost Multiconimodity Flow Problems, 79-2.
Bi. C. Tanscl. .L,. Francis, and T.J. Lowc, "Duality and the Nonlinear
p-Center Problem and Covering Problem on a Tree Network, 79-3.
l)onald W. Learn and Jamses js inathadas, "Analysisand E(xtensions
of Algorithms for Sylvester's Minimnax Location 1 problem ''lit 80-1.
lue G. Chailnct "'t i .... in Minisum Rectilinear Distance Loca-
tion Problems,"'TH 80-2.
Donald W. learn and Jaimel ibera, On )aganzo's .'.. I, 1' rankii
Iolft/( Method for Certain lBounded Variablre .1. Assignment P'roob
lerns, "'TI'i 80-13.
Donald W. learn, "I ...... ,. I'lows in Traffic Assignment Models,"
'LR 80-4.
lIarbaros C. Tansel, Richard L. Francis and Timothy J. Lowe, "A Bi-
( ,I ,,, II.,1,i... i Miinimax location Problem on a Tree Network,"

M.L. Chen, It. 1. Francis, T.J. Lowe and 1,.(:. Tansel, "Distance Con-
straints lfor Tree Network Versions of the Nonlinear p-Center and Covering
Problems," 'I'T 80-7.

1M/1 1 1 I1 TICS Rl I IR(CllCU I I.
University of Wisconsin
610 Walnut Street
Madison, Wisconsin 99164

0(. L. Mangasarian, "Optimal simplex tableau characterization of
unique and bounded solutions of linear programs," 'I' l 2034, 1980.
S. M,. Ilobillsoll, "Some continuity properties of polyhedral multi-
fulnctions, "T' 2014, 1979.

School of( Computer Science
805 Sherbrooke Street West
Montreal, Quebec f13A 2K6

V. C ivatal, "lHard Knapsack Problems," 79.1
1). Avis. "On the Complexity of :,t ...1... the Convex Hull of a Set of
Points, 79.2
V. (Chvatal. "Cheap, Middling and Dear, 79.3
1). McCallum and I). Avis. "Finding the Convex Hull of a Simple
IPolygon,"l 79.5
(;. T. Toussaint. "1The Relative Nieighbourhlood Graph o(f a Finite'
Planar Set," 79.7
L.. Devrovc, "A .i, ... R aio on Random Search, 79.9
V. Chva'tal. -.... n t. tersection Patterns," 79.10
1). Avis, "A Note on Some ,,..,... T .. iv? !',tn .I. Set Covering
Problems, 79. 11
I.. -. I orspool, "Constrctling the Voronoi Diagram in the I'la e, "
11.T. Lau, "' ,, i., '.E S- G, raphIs, 70.15
S. Kourouklis, "Comiputinig 'l. i,,1r..I Linear Least Squares," 79.16
V. Chlvatal and EIl Szemiiircdi, "On thie Erdos. -I .... Theorem, 79 .17
J. Akiyaina, 1). Avis. V. Chvatal and II. Era. "Balancing ,'...I
(Craphs," 79.18
M. Ajtai, V. Clivatal, IM.M. Newblorn and E.. Szemercdi, "(Crossing-.free
hanmiltonian circuits, 79.2 1 MID 1

WORKIN(; P,'i -l; . ..

Schllool of Engineering and lApplied Science
Princeton, New. Jersey 085,14

Richard 11. 1F. jIackslonI oltJin M. lMulvev. "1 critical review of co isons of tmatliematical progrTamming algorilhms and software
(1 9,;3- 1977), ES 78-8.
John M. Mulvey, ..... .... homogeneity for multivariale .i, I /
sampling." ES 79-1.
John lM. Mlvey, ...I. .. micro-data files by optimization models
or by i ,1.1, ..I ..t.. ..." E1 S 79-2.
Hired t lovcrt. John M. IMulvey. "l'qulitvalonce of lthl 0-1 integer pro-
gramming problem to discrete generalized and pure networks, EES 79-5.
J onathan S. I1. lornbluth. ]valph E. Stcrr. l aittli t.I o' ljectcive linear
i'a 'tion l ; .. ...I ;,-,_ I 79-7.
Fred (;lovcr, ,ohn M. \Mulvey. ,. ,. rhlanalio.ns and f'nallies for
mulitiple choice problemsi" l',E'S 79-12.
Jlonathan S.I1. Kornbhlth. lIalph E. Scnr. "An comping thesowl q[
all weakly effic'inl vrerties in naultiplt oijlrtive linar firactiiionl progral-
miny ,"l';S 7<)-1 3.

Departmntiil of Operations Research
S. I l university
Stanford, C I -4305

Eric lioenlerg. "GilobAIly CIonv'erg..ent lg. orithms for t('tonx Pro
grm m ing, SO ) 79-1.
(eorge H. i )aniz"g and I eter L,. jlacksro I ... linlteremployed
Capacity ina linear lconoimic lModel. SO(, 79-2,
S. A. Parikh. "L1 1, .. ,Equilibriumn Model (Wl'E1) of energy Supply,
'nergy 1Demrand, and Economic (Growth, SOL, 79-3.
Nathan huras. "Determining lthie leasibility of Inhmegrting aler,
Resource Constraints into energy y Moodels," SOI, 79-1.L
Mic AI comparisono n Preconditioning Methods," SOI, 79-5,
FrcdriMk S. Hillier amnd naoy Eilcn Jac(min. "A Hounding Tech-
nique for Integer Linear Irogramming with liinary I variables SOL 7 9-.
Nancy Eihl.n jac(lillin. "I)orumenltation of a Comntputer Program forN
the loulutand-Scan ilgorithmo for Integer Linwar Programming," SOL

Nallv I'ilecll Jacqmiin. "]Doculinntalion of a ('.Computer Program or "
Ilillier's Heuristic Procedure in Integer linear ......... SOL, 79-8.
A.I. Sillion and N. kira. I I ..... I 'of later Resource (Constlraints
in IEnergy Models, SOl. 79-9
Richard W\. Cottle and lahbe von landow, "Oil Q-Malrices. Co.n-
troids and itimplotlopes, SO 1, 79- 10.
Eric liosenhebr. 1 IGlobally C(onvergeni Condensation hllhod ,lor
Geometric ; ... .... .. S()O L 79-11.
RIichard W. Cottlec. "Completely (.-Iatrices. SOL 7-12.
1hukund Thapa. "1 \ole on Sparse Quasi-Newtlon Methods." SOI.
l o\w-Y irh (Chang. "Ieast-Index Resolution of Dogeneracy in Linear
Complemenlvilarity Problemns." SOL 79-1 1.
Philip 1,E. ill and halter Murru. "C.onjugate-C ratdint lethodsfor
L.argeScale Nonlinear Optimization," SOI, 79- 15.
heniamin \vi-itlzhlak and Al'frecdo lseni. "-I Consumers Energy
Services lodel. SO L. 79-10.
ioh)crt Fourer. "Sparse G(aussialn Plilmination of S.laricase Linear
Systemss" SOL 79-17.,
Hobliert F(urcr. "Sohling Staricase Lineur Programns by the Simpler
lMthod, I : Inversion. andI 2: Pricing, ,SO 79-1 i8 and SOL, 79-19.
Richard K. McCord. "inimizalion withll One Linear Eiquality Con-
straint and i]oundson the I arialles,. Ol, 79-20.

Walter Murray and Michael 1. Overton, "A Projected LaIorangiant
I gorithm for Nonlinear Minimax Optimization,'" SO 79-21.
(IGorrge i. I)ant/zig "Commentts on Kiachian's t .ll ,ian for Linear
i .... .. SOL 79-22.
Jerome H. Friedman and Margarct II. ...._, "A Nested IPrtitioning
Procedure for Numerical Multiple Integration." SOL. 79-23,.
"I i .. i, Wright, "Algorithms for Nonlinearly constrainedd Opti-
mization, SOL 79-24.
P.E. Gill, W. Murray, M.A. Sautnders. and M.IL .i,, "Two Step-
Length I.. i..-.. for Numerical Optimization," SO1. 79-25.
/Zachary F. LanIsdowne, Survey off Research on Model Aggregation
and ..., .., ...... SOL 79-20.
li. Avi-ltzhiak. T.J. Connollv, and A. lusem. "ThI'lIe Coal Module of thi,
Pilol System, SOL. 79-27.
I.A. Murtagh and M.A. Saunders, "The Implementationl ofa ltagrang-
ian Based Algorithm for Sparse Nonalinear (Constraints SOL, 1O-I.
tallter Murray and Margaret II ,.1~ I "Computational of the
Search Direction in Constrained Optimization Algorithms," S01, 80-2.

I/NIVIER/.S/Y7'Y 01' TI'L\ IS AT it It
Department of generall lHusiness
E11I1 608
Austin. TX 78712

Fred (;lover. Genc Jones, David karncy, Darwin hii -.... andj ohn
Mote. ",,In Inlegratled Production, Distribution, and Inventory I .......
Richard ,arr. Fred (Glover. and Darwin Klingmian. ".1 A'IIv Optimiza
tion liethod for LarpgeScal Ftlixed Charge Transportation Problems."
Frcd Glover. D)arwin I n....... john \ioleo, and David Whitman.
"Comprehensive Computer Evaluation and Enhancement of Maximum
Flow Algoritims. "
1ov Crum. Darwin ...... and Lce Travis. ,I ,1 Management
of' Mulltinational Companies: Network-liased ,t I......... Systems. "
Fred (;lovcr. Darwint i ....... John Mol and David Whitman. "A
Primal Sitmpllex Variiant for the Maximum iFlow Iroblem. "
Darwin i ... '.. alnd j ohn Motle "Solution approaches for Network
Flow PIroblems with Multiple ('riterit "
lanrwill In....... 1 and John Mote. "G(eneralized Neliwork .Approaches
for Sol ing Least Absolute Value and T1 '1, i, I Regrlession Problems."
I .S. Lasdon and A.I). Warln, "Cenralized Rleduced Gradient Soft
ware for I,inearly and Nonlinearly Constrained Problemsi."

i t tSVIRGiTO' ,S'7' 1T/ttI 77: ttI RiSIT7'
Department of llathernatics
'ulmaoin, ii I ..... .. .. 9916I

Si. iffli -I. NI. i and extension of Iearnlrechal's algorithmlN
for nostnooth minimization, "TH 80-1,

Ui\AIVESITY OF f 0 INCOSA\-,.l I/150\
Comiiputer Sciences )epartmient
1210 Rtest IMyton Street
Madison, Wisconsin 53700

C, ". Kao. R. lI. Mcver. "Secant approximation methods for conrev
optimization, 1TR 352. 1979.
aarian. aigasarian "leratlive solution of linear programs, TH 327.
(). 1. 1antasariain. "Locally unique solutions ofquadratic programs.
linear and nonlinear complementarity problems," TIl 315. 10979,
0. L. Mangasarian. "(Charlacterizations of bounded solutions of linear
completmenta'rity problems. T'Il 359. 1979.
H. H. Mover. "'Continuity properties of linear programls.," TI 373.
{. H. lever. "C(onmpulationau aspects of tIwo-sejment separable
programming, 'T'1 3:I2. 19N0.

IThe John von Neumann Prize was awarded at the May, 1980 O(SA/TIMS meeting in
\\ ashington, I).(., to Profs. David Gale, larold W. Kuhn and A.W. Tucker for contributions
to e e theory of i'athei atical prIogramming. D)ick Coltic, new Editor-lI-( In. I of the
journal and Studies has announced three new (o-Editors-L.C.W. Dixon (latfield Poly-
technic), Bernhard Korle (Bonn), and M.J. Todd ((ornell. There are 10 new Associate
Editors-E.L. Allgower ((olorado State), R.G. Jeroslow ((eorgia Tech), D.S. Johnson (Bell
Labs.) L. Lovasz (Szeged). M.W. Padherg ( York University), W.R. Pulleybank (Calgary),
Ritter (Stuttgart), R.W.II. ': i. ii (linperrial College), D.F. Shanno (Arizona), and L.E.
'Irotler, Jr. (( *... 11). An article on thl stnructural changes and Editlorial Hoard operation will
hi in Vol. 19, No. 1. .S.M. Robinson ( ... .....) has accepted tei editorship of Mlathel tics
il' Operations lResearch. .Tli Iniviers it of Ilorida Centi'r for E'conometrics & )Decision
'-icl:lces initiated a distinguished scminiar liries in the past year. Initial lectures were give
iy .. Christofides (Inlerial ( ..11 .,). George Nemlhauser ((.. .. IIl) and Stanl in' I .I
( N Y, Biuffalo). Jerome Kreuser is the new Editor of thlle SItMAPi Bullein. .The most
recent issue of SI>11 \P (\o. 24. April 1 1', contains 8 papers (froil a special session at the'
\pril 1979. O(ISA/ I I I meeting) on -Recent aid Future Developnment of Math Program-
rIing Syslcnsem presented Ib hardware & software representatives. .Society membership
rose _'11 from March 1979 to March Ia :1i due largely to the membership drive at the
Montreal lmee:ting. There are now I .-, total members. 279 from North America. 131 from
I. 1. i Europe. 2-1f from Eastern Eurol(, 26 I'rom Asia and Australia, 17 rom altin \Amer-
i'a and 5 fromn Africa.
I I. ,Sprilng Prof. Biela Martos visited Purdle's irannert School . .J. Stoer (.' ,, .-
Ilirg) is visiting the National iHureau of Stamndards. .J. Telgren (Erasmus) has been visiting
ihe Inlivcr.sit\ of\l'eni lssee and is to visit SUN H 1ffl'fao...K. Schitllkowski ( i hlas
recently visited \rgo lnne, University of Texas alnd N S. . ..T. Polyak ( I ) will lie
siting the lU.S. in earl suimmllner. Ilis schedulee is being arranged by D.P. Bertsekas (M. I.T.)..
.Ego Bailas (Carnegie- 1. 11...) ,i be visiting the Mathlematischel Institut, IUniversitet
KohLn. beginning Sep iltember 1.

303 \ eil Hall
College of Engineering
University of' lorida
Gainesville, Florida :.' 11

Technion Announces 1980-81
',ilr., Naor )istinguished Fellowship
The Faculty of Industrial Engineering
and Management at the Technion Israel
Institute of T liI ... invites applications
for the 1980-81 '.i, NNaor Distinguished
Fellowship. PIinlas (Paul) Naor was tel
founder of tie Faculty and Professor of
Operations Research until his death in
an airplane accident in Decemberr 1970.
I 1. E II.. -Il,1 will be granted to a
person with .;:.,if];. .I accomplishments in
one of the I..'1.. i, fields of activities of
the Faculty: Applied Probability, Behavioral
Sciences, Economics, Industrial ,,; ;,i
'.1 .. ... 1. .. i Science, (Operations Research,
Statistics, and related areas. The recipient
will be invited to stay at tlhe Techlnio as
a distinguished guest of tlie Faculty, for the
Fall or Spring Semester, and deliver the
Annual Pinhlas Naor Memorial ,ectur:.

at thle Tlchnion, round trip travel to Israel
and assistance in housing.
Applications and nominations should
be sent, by July 11. 1' *... to 'rof'essor
Michael 4ubinovitch, Dean, I'aultv of
Industrial Engineering and Management,
Technlion, lHaila. Israel.

Nonprofit Org.
U.S. Postage Paid
Gainesville, Florida
Permit 94



Maintained by the Mathematical Programming Society (MPS)

This Calendar lists meetings specializing in mathematical programming or one of its subfields in the
general area of optimization and applications, whether or not the Society is involved in the meeting.
(These meetings are not necessarily "open".) Any one knowing of a forthcoming meeting not listed here
is urged to inform the Chairman of the Executive Committee of the Society, Dr. A. C. Williams,
Computer Science Department, Mobil Oil Co. Technical Center, Box 1025, Princeton, New Jersey
08540, USA; telephone 609-737-3000, extension 2342.
Substantial portions of regular meetings of other societies such as SIAM, TIMS, and the many
national OR societies are devoted to mathematical programming, and their schedules should be


June 2-6: "Workshop on Large-scale Linear Programming", International Institute for Applied Systems
Analysis, Laxenburg, Austria. Contact: Dr. Markku Kallio, IIASA, 22361 Laxenburg, Austria.
Cosponsored by Systems Optimization Laboratory, Stanford University, and the MPS.
June 16-20: "Workshop in Numerical Methods for System Engineering Problems" in Lexington,
Kentucky. Contact: Professor Roger J.-B. Wets, Department of Mathematics, University of
Kentucky, Lexington, Kentucky 40506; telephone 606-257-2836. Sponsored by the MPS.
July 7-11: "Tutorial Conference on Practical Optimization" in Stanford, California. Contact: Systems
Optimization Laboratory, Department of Operations Research, Stanford University, Stanford,
California 94305, U.S.A.
July 14-16: "Nonlinear Programming Symposium 4" in Madison, Wisconsin. Contact: Professor Olvi
Mangasarian, Computer Sciences Department, University of Wisconsin, 1210 West Dayton
Street, Madison, Wisconsin 53706; telephone 608-262-1204. Sponsored by the MPS.
July 22-25: "Fourth European Congress on Operations Research" in Cambridge, England. At least one
session on software testing will be sponsored by the Committee on Algorithms of the MPS.
For general information on the Congress contact Prof. J. P. Brans, University of Brussels,
VUB/CSOO, Pleinlaan 2, B-1050 Brussels, Belgium; reading the Committee contact Dr.
Susan Powell, Datalogisk Institut, Kobenhavns Universitet, DK-2200 Kobenhavn N, Denmark.
July 28-August 1: "International Workshop on Advances in Linear Optimization Algorithms and
Software", Pisa, Italy. Contact: Dott. Claudio Sandi, IBM Italia Centro Scientifico, via
Santa Maria 67, 56100 Pisa, Italy; telephone 50 (= Pisa) 47383.
August 4-15: "Generalized concavity in Optimization and Economics", NATO Advanced Study
Institute, University of British Columbia, Vancouver, B.B., Canada. Contact: Prof. Siegfried
Schaible, Faculty of Business Administration and Commerce, University of Alberta, Edmon-
ton, Alberta, Canada T6G 2G ; Telex 037-2979, Telephone 403-432-5027.
August 28-30: "Bonn Workshop on Combinatorial Optimization", Institute of Operations Research,
Bonn, Federal Republic of Germany. Contact: See 1982, late summer.
September 15-17: "2nd IFAC Workshop on Control Applications of Nonlinear Programming and
Optimization", Oberpfaffenhofen, Federal Republic of Germany. Contact: Dr. Klaus Weil,
Institute fur Dynamik der Flugsysteme, DFVLR, Oberpfaffenhofen, D-8031 Wessling, F.R.G.;
telephone (0 81 53) 2 81.
See other side. please....

September 16-18: "6th International Seminar on Algorithms for Production Control and Scheduling",
Karlovy Vary, Czechoslovakia. Contact: Ing. Jifi Kral, House of Technology, Gorkeho nam.
23, 1: Praha 1, Czechoslovakia.

December 10-12: "19th IEEE Conference on Decision and Control", Albuquerque, New Mexico,
U.S.A. Submission deadline 31 March 1980. Contact: Prof. Michael K. Sain, Dept. Electrical
i ,,._.i., i iii, Notre Dame University, South Bend, IN 46556, U.S.A.


January 5-6: 1l.11ii. 1 m. .11 Pl.',: .,nimrmi;. Testing and Validating Algorithms and Software". U. S.
National Bureau of Standards, Boulder, Colorado. Organized by the Committee on Algor-
ithms of the MPS, the Bureau of Standards, and the Department of Energy. Contact: Dr.
Richard H. F. Jackson, Center for Applied Mathematics, National Bureau of Standards,
Washington, D.C. 20234; telephone 301-921-3855.

January 26-31: ''l.iil cli,. i..l_--. Optimierung", Mathematisches Forschungsinstitut Oberwolfach,
Oberwolfach, Federal Republic of Germany. Contact: Institut fiur Okonometrie und Opera-
tions Research (see 1982, late summer).

April 6-8: "International Congress on Mathematical Programming", Rio de Janeiro, Brazil. Contacts:
Professor R.W. Cottle, Dept. Operations Research, Stanford University, Stanford, CA94305,
U.S.A.; Professor Milton Kelmanson, Pontificia Univesidade Catolica, Departamento de
Engenharia Eletrica, Rua Marques de Sao Vicente, 225, Rio de Janeiro, R.J., Brazil; Professor
B. Korte, Bonn (see 1982, late summer).

July 13-24: "NATO Advanced Research Institute on Nonlinear Optimization", Cambridge, England.
Contact: Professor M.J.D. Powell, Department of Applied Mathematics and Theoretical
I'. .i,-. University of Cambridge, Silver Street, Cambridge CB3 9EW, England. Sponsored
by the MPS.


Late summer: Eleventh International Symposium on Mathematical Programming in Bonn, Federal
Republic of Germany. Contact: Institut fur Okonometrie und Operations Research Universitat
Bonn, Nassestrasse 2, 5300 Bonn 1, Federal Republic of Germany; Telex 886657 unibo b,
Telephone (02221) 739285. Official triennial meeting of the MPS.



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The dues for 1980 are: Please send this application with your dues to:
42 Dollars (U.S.A.) The Mathematical Programming Society
20 Pounds (U.K.) % The International Statistical Institute
68 Francs (Switzerland) 428 Prinses Beatrixlaan
178 Francs (France) 2270 AZ Voorburg, Netherlands
76 Marks (Fed. Rep. Germany)
84 Guilders (Netherlands)

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