Semi-Infinite Programming With AMPL

15 Pages·2004·0.09 MB·English

SIPAMPL: Semi-Infinite Programming
With AMPL
A.ISMAELF.VAZandEDITEM.G.P.FERNANDES
UniversidadedoMinho
and
M.PAULAS.F.GOMES
ImperialCollegeofScience,TechnologyandMedicine
SIPAMPLisanenvironmentforcodingsemi-infiniteprogramming(SIP)problems.Thisenviron-
ment includes a database containing a set of SIP problems that have been collected from the
literatureandasetofroutines.ItallowsuserstocodetheirownSIPproblemsinAMPL,touseany
problemalreadyinthedatabase,andtodevelopandtestanySIPsolver.TheSIPAMPLroutines
supporttheinterfacebetweenapotentialSIPsolverandtestproblemscodedinAMPL.SIPAMPL
alsoprovidesatoolthatallowstheselectionofproblemsfromthedatabasewithspecifiedchar-
acteristics. As a concept demonstration, we show how MATLAB can use SIPAMPL to solve the
problemsinthedatabase.TheLinuxandMicrosoftWindowsversionstogetherwiththedatabase
ofcodedproblemsarefreelyavailableviatheweb.
Categories and Subject Descriptors: D.2.2 [Software Engineering]: Design Tools and Tech-
niques—Modules and interfaces; G.1.6 [Numerical Analysis]: Optimization—Constrained
optimization
GeneralTerms:Documentation,Languages
AdditionalKeyWordsandPhrases:Evaluationtools,semi-infiniteprogramming,testproblems
1. INTRODUCTION
The existence of software that enables the coding of mathematical program-
ming problems in a modular way is known to be appreciated by users and
developers of mathematical programming solvers. AMPL [Fourer et al. 1990]
is a good example of the finite linear/nonlinear case. It provides an easy-to-
learnandpowerfulmodelinglanguagetogetherwithautomaticdifferentiation
Authors’addresses:A.IsmaelF.VazandEditeM.G.P.Fernandes,DepartamentodeProduc¸a˜o
eSistemas,EscoladeEngenharia,CampusdeGualtar,UniversidadedoMinho,4710-057Braga,
Portugal; email: {aivaz;emgpf}@dps.uminho.pt; M. Paula S. F. Gomes, Mechanical Engineering
Department,MechatronicsinMedicineLaboratory,ImperialCollegeofScience,Technologyand
Medicine,LondonSW72BX,UK;email:[e-mail].
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48 • A.IsmaelF.Vazetal.
ofthefunctionsinvolvedintheproblemdefinition.Solverdevelopersaresup-
ported by a set of AMPL interface routines, which allow communication be-
tweenthesolverandAMPL.Onemayusethoseroutinestoobtainallrequired
information about the problem to be solved. As AMPL does not directly sup-
portsemi-infiniteprogramming(SIP)problems,wedevelopedasetofroutines
that allows a SIP solver to communicate with AMPL. This new environment,
which we have coined SIPAMPL, also contains a database of SIP problems
coded in AMPL, a tool to query the problems database, and another set of
routines that enables the MATLAB [MathWorks 1999] solver (fseminf func-
tion in the Optimization Toolbox [Coleman et al. 1999]) to use the SIPAMPL
routines.
TheoperationofSIPsolversisoutsidethescopeofthispaper.
WestartinSection2bygivingadescriptionofaSIPproblemandwepresent
a short SIP example coded in AMPL. A brief description of AMPL is given in
Section3.ThedevelopedSIPAMPLroutinesaredescribedinSection4,which
hastwosubsections,onerelatedtoMATLABusageandtheothertoanysolver.
InSection5wedescribethecollectionofSIPproblemsalreadyavailableinthe
database. The select tool that allows queries to the database is presented in
Section 6, and some discussion and the conclusions are presented in the last
twosections.
2. SEMI-INFINITEPROGRAMMING
ASIPproblemisamathematicalprogramoftheform
min f(x)
x∈Rn
s.t. g (x,t)≤0, i =1,...,m
i
h(x)≤0, i =1,...,o (1)
i
h(x)=0, i =o+1,...,q
i
∀t ∈T,
where f(x)istheobjectivefunction, g (x,t),i = 1,...,m,aretheinfiniteand
i
h(x), i = 1,...,q, the finite constraint functions. T ⊂ Rp is usually a carte-
i
sian product of intervals ([α ,β ]×[α ,β ]×···×[α ,β ]). Since AMPL does
1 1 2 2 p p
not directly support SIP problems, the following assumptions for coding SIP
problemsinAMPLweremade.
(1) Constraint functions that depend on the infinite variables (t) are coded
withnamesstartingwitht.Conversely,allconstraintfunctionswithnames
startingwithtareassumedtodependontheinfinitevariables.
(2) Allinfinitevariablenamesmuststartwitht.Conversely,allvariableswith
namesstartingwithtareassumedtobeinfinite.
(3) TheAMPL.rowand.colfilesmustbeprovided(optionauxfilessettorc
inAMPL).WefurtherexplainthisinSection3.
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SIPAMPL:Semi-InfiniteProgrammingWithAMPL • 49
Asanexample,considerthefollowingproblem:
min x2+x2
x∈R2 1 2
s.t. x t+x t2 ≤0
1 2
−10≤ x +x ≤10
1 2
∀t ∈[0,1]
ThecorrespondingAMPLcodefollows:
####################################################################
# Objective: Quadratic # Classification for
# Constraints: Linear # the "select" tool
####################################################################
# Example problem in the SIPAMPL user manual.
# This is file userman.mod in the distribution.
# Author: [e-mail] 27/12/99
####################################################################
var x {1..2}; # Finite variables
var t; # Infinite variable starts with t
minimize fx: # Objective function
x[1]^2+x[2]^2;
subject to tcons: # Infinite constraint starts with t
x[1]*t+x[2]*t^2 <= 0;
subject to lincon: # Finite linear constraint
-10 <= x[1]+x[2] <= 10;
subject to boundt: # Infinite index set
0 <= t <= 1; # AMPL will convert this constraint
# into simple bounds on the infinite
# variable
###############################
# End of problem description
###############################
option mysolver_auxfiles rc; # Write .col and .row files
option reset_initial_guesses 1; # No starting point for this
# example
option solver mysolver; # Select SIP solver
solve; # Solve the problem
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50 • A.IsmaelF.Vazetal.
###############################
# Display solver status
###############################
display solve_result_num, solve_result;
###############################
# Solution found
###############################
printf "Solver terminated with solution\n";
display x;
display fx;
mysolveristhenameofthebinarythatwillsolvetheSIPproblemandcan
bereplacedbyanyothersolverforSIPthatisavailable.
TothebestofourknowledgethereisjustonesolveravailableforSIPprob-
lems. Our solver NSIPS [Vaz et al. 2002] is publicly available and uses the
SIPAMPLroutinestoobtainandsolveaSIPproblemcodedinAMPL.Utilizing
theNSIPSsolveravailableontheNEOSserver,1itispossibletosolveproblems
usingSIPAMPLbysubmittingthemviatheInternet.
The MATLAB function fseminf is not a real solver for SIP, since the user
mustprovideafunctionthatevaluatestheinfiniteconstraintsatasetoffinite
points. In providing this function, the user must write the algorithm to select
the desired points, thereby doing most of the work and defining a scheme to
solveSIP.
Lines 2 and 3 give useful information about the SIP problem type. This
informationwillbeusedbytheselecttooldescribedinSection6.
3. AMPL
AMPLisanalgebraicmodelinglanguagethatallowstheformulationofmath-
ematical programming problems. The flexible and natural language used by
AMPLwasthemainreasonforchoosingAMPLforourdevelopment.Thepro-
posed environment could be developed in other modeling languages, such as
CUTE(ConstrainedandUnconstrainedTestingEnvironment[Bongartzetal.
1995]),buttheCUTEStandardInputFormat(SIF)languageisnotasnatural
asAMPL’s.AMPLiscommercialsoftwarebutastudenteditionisavailablefor
evaluation.
WeassumethatthereaderisfamiliarwithAMPLandtheCprogramming
language. For background reading, see Kernighan and Richie [1988]; Fourer
etal.[1990]andGay[1993].
AMPLprovidesawaytocommunicatewithawidevarietyofsolvers.AMPL
runsasolverasaseparateprogramandcommunicateswithitbywritingand
readingfiles.Bydefault,AMPLwritesatemporaryfilewithextension.nl.This
fileonlycontainssufficientinformationtodescribethemodel.Forexample,the
1http://www-neos.mcs.anl.gov/neos/solvers/SIO:NSIPS/.
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SIPAMPL:Semi-InfiniteProgrammingWithAMPL • 51
Fig.1. InteractionbetweenAMPLandsolvers.
parametersdeclaredinthemodelareincludedbutthevariableandconstraint
namesareremoved.AMPLinvokesthesolverwiththe.nlfilenameandexpects
thesolvertowriteafilewithextension.sol,containingaterminationmessage
andthesolutionfound.Tosupportthesolverinreadingthe.nlfileandwriting
the.sol,AMPLhasasetofpubliclyavailableroutines2toprovidetheinterface
betweenthesolverandAMPL.
AMPLwillprocessaSIPmodel(.modfile)inthesamewayasanymodelcoded
inAMPL.TosolveaSIPproblem,asolverneedstoevaluatethefinite/infinite
constraints in an unspecified order. The SIP solver therefore needs to know
whichvariablesandconstraintsarefinite/infinite.Ifthesolveroptionauxfiles
issettothevaluerc,AMPLwillproducethe.rowand.colfiles,whichcontain
the extra needed information about the coded problem. Since AMPL routines
treatallvariablesandconstraintsequally,wedevelopedasetofroutines,called
SIPAMPLroutines,whichidentifythefinite/infinitevariablesandconstraints,
andmaketheinterfacebetweentheSIPsolverandtheAMPLroutines.
The main problem data in the AMPL interface is kept in a data structure
calledasl(whichstandsforAMPL/SolverinterfaceLibrary[Gay1993]).This
datastructure,definedintheasl.hincludefile,isloadedwhenthesolverreads
the.nlfile.SincetheAMPLroutinesmustbeavailabletoSIPAMPL,theasl
datastructureisalsoavailable.
Figure 1 illustrates the interaction between the AMPL binary and solvers.
The SIP solver uses the SIPAMPL and AMPL interface routines to read the
problem from .nl, .row and .col files produced by AMPL. A finite problem
solveronlyusestheAMPLinterfaceroutinestoreadtheproblemfromthe.nl
file.BothsolversusetheAMPLinterfaceroutinestoproducethe.solfileread
byAMPLtodisplaythesolutionfound.
4. USINGTHESIPAMPLROUTINES
The SIPAMPL routines are distributed as two source files: sip.c (the inter-
face routines) and sip.h (functions prototypes, data structures and variables
declaration) together with two makefiles: makefile for the Linux operating
system (gcc compiler) that makes the library libsip.a and makefile.vc for
Microsoft Windows (Microsoft Visual C/C++ compiler) that makes the library
sipampl.lib.
2solvers.tarinhttp://netlib.bell-labs.com/netlib/ampl/.
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52 • A.IsmaelF.Vazetal.
Some variables declarations are macros to the SIP data structure. If one
needs another SIP data structure (for example, if one needs two problems in
memory)oneshouldcreatenewmacrosorusethedatastructuredirectly.
When developing a solver and before using any SIPAMPL routine, the
user must load the problem from the .nl file, using the AMPL routines. The
SIPAMPLroutinesip initmustbecalledbeforeanyotherSIPAMPLroutine.
Later,sip freewillreleasethememoryallocatedbysip init.
TodescribetheSIPAMPLroutinesthefollowingdefinitionsaremade.Since
AMPLdoesnotsupportSIPproblemsandthepresolverhasnospecialconcern
withthevariablesdeclarationorder,thevariableswhosenamestartswithtand
theothersaremixedup.ThenatureofSIPrequiresthatthesecomponentsbe
separated(thesolvermayneedtochangesomevariableswhileothersarekept
fixed). Let the original variables array be defined by x = (x ,x ,...,x ).
NLP 1 2 nvar
We will call x the original x array of variables, where nvar is an AMPL
NLP
variablethatgivesthetotalnumberofvariablesintheproblem.TheSIPxand
t componentsmustbeobtainedfrom x .Letuscallthe x andt components
NLP
x andt respectively.
SIP SIP
AbriefdescriptionoftheSIPAMPLroutinesthatsupporttheSIPevaluation
functions follows. Note that gradients, Hessians and Jacobians are always in
denseformat,definedinaFORTRANway(AMPLdenseformat).
—sip extractxExtractsthe x componentfromtheinitialvariable x .
SIP NLP
—sip extracttAsinsip extractxbutforthet component.
SIP
—sip joinxt Joins the x and t components from SIP back into the x
SIP SIP NLP
arrayofvariables.
—sip init Initializes the variables for the problem, allocates the arrays for
the bounds and copies the bound values to the arrays. This function looks
forvariableandconstraintnamesinordertokeeptrackofthe x andt
SIP SIP
positions in x . (Keeping track of the x and t constraints position is also
NLP
needed in order to be able to compute the original constraint position from
an x ort constraintposition.)
—sip free Frees the memory allocated during the call to sip init. Only the
memoryisfreed;thevariablesinthesipdatastructurearenotreinitialized.
—sip objvalEvaluatestheobjectivefunction.Theobjectivefunctiondepends
only on the x variables. The user must also provide the objective func-
SIP
tionnumber(AMPLsupportsmulti-objectivefunctions).Thisfunctioncalls
AMPLobjvalwitht equalto0.
SIP
—sip objgrdEvaluatestheobjectivefunctiongradientvector.Asinsip objval
only x is needed. sip objgrd sets the array with the derivatives with re-
SIP
specttox alone.(Theothersarezero,sincetheobjectivefunctiondoesnot
SIP
dependonthet variables.)
—sip objhesEvaluatestheobjectivefunctionHessianmatrix.Againtheobjec-
tivefunctiondependsonlyonx ,sotheconsiderationsmadeinsip objval
SIP
andsip objgrdarealsovalid.
—sip convalEvaluatestheconstraints.Constraintsdependonx andsome
SIP
on t , so both must be provided. sip conval fills two arrays, one with the
SIP
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SIPAMPL:Semi-InfiniteProgrammingWithAMPL • 53
values of constraints that do not depend on t and the other with the con-
straintsthatdodependont.sip convalgeneratesx fromx andt and
NLP SIP SIP
callstheAMPLconvalfunction.Itsplitstheconvalresultintotwoarrays.
—sip jacvalEvaluatestheconstraintJacobian.Needsx andt .Fillstwo
SIP SIP
matrices(for x andt constraints).
—sip conxvalEvaluatesanxconstraint.Theconstraintnumbermustbesup-
plied.Theconstraintnumberreferstoxconstraintsonly.Onlyx isneeded.
SIP
—sip contval Same as sip conxval but for t constraints. x and t are
SIP SIP
needed.
—sip conxgrd Evaluates an x constraint gradient. Only needs x and fills
SIP
anarraywiththederivativeswithrespectto x components.
SIP
—sip contgrdEvaluatesat constraintgradient.Needsx andt .Fillstwo
SIP SIP
arraysasinsip conval.
—sip conxhesEvaluatesanx constraintHessianmatrix.Onlyx isneeded.
SIP
Fillsonematrix.
—sip conthes Evaluates a t constraint Hessian matrix. Needs x and t .
SIP SIP
Fillstwomatricesasinsip jacval.
sip extractx,sip extracttandsip joinxtaremostlyusedassupportrou-
tinestotheotherSIPAMPLroutines.
Sincetheasldatastructureisalsoavailable,ifoneneedsotherinformation
abouttheproblemtheAMPLroutinescanbeused.Itistheuser’sresponsibility
toreadthe.nlfileandtoprovidethecorrespondingasldatastructure.
4.1 UsingMATLABwithSIPAMPL
This section describes how MATLAB [MathWorks 1999] can use SIPAMPL to
obtain the problem to be solved. Our MATLAB function sipampl allows prob-
lemscodedinSIPAMPLtobesolvedwithMATLAB.
MATLABinitsOptimizationToolbox[Colemanetal.1999]providesanOpti-
mizationFunction(fseminf)forsemi-infiniteprogramming.Therangeofprob-
lems that can be solved by fseminf is limited to two infinite variables in each
infinite constraint. The function sipampl is therefore limited to two infinite
variables,becausetheproblemformulationpresentedhereinsupposesthatall
theinfinitevariablesappearinalltheinfiniteconstraints.
Thesipamplsyntaxis:
[x0,ntc,xbl,xbu] = sipampl(’userman’)
f = sipampl(x)
[f,Grad] = sipampl(x)
[c,ceq,K1,...,Kntc,s] = sipampl(x,s)
sipampl(’msg’,x)
Thebehaviouroffunctionsipampldependsonthenumberofinputandout-
putarguments,thus:
—[x0,ntc,xbl,xbu] = sipampl(’userman’)readstheproblemnameduserman
(userman.nlfile)andreturnsx0theinitialguess,ntcthenumberofinfinite
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54 • A.IsmaelF.Vazetal.
constraints,andxblandxbuthelowerandupperboundsonthex variables,
respectively.
—f = sipampl(x)returnstheobjectivefunctionvalueatx.
—[f,Grad] = sipampl(x)returnstheobjectivevalueandthegradientvector
atx.
—[c,ceq,K1,...,Kntc,s] = sipampl(x,s)evaluatestheconstraintsatx.sis
the step size for the grid where the infinite constraints are evaluated (see
Coleman et al. [1999] for more details). c and ceq are vectors for the val-
uesofthefiniteconstraints,cforinequalityconstraintsandceqforequality
constraints. K1, ..., Kntc are ntc vectors (or matrices) with the infinite con-
straintsevaluatedonthegrid.
—sipampl(’msg’,x)writestheAMPLsolutionxwiththetextmessagemsg.
Asimpleexampleusingthesefunctionsis:
>> [x0,ntc,xbl,xbu] = sipampl(’userman’);
>> options = optimset(’GradObj’, ’on’);
>> x = fseminf(’sipampl’,x0,ntc,’sipampl’,[],[],[],[],...
xbl,xbu,options);
>> sipampl(’Solution found by MATLAB’,x);
The interface developed here includes two files. sipampl.c is the main pro-
gramthatprovidestheMEX[MathWorks1996]functionsipampl.sipsolve.m
isaMATLABfilewithashortexampleofhowtouseMATLABtosolveproblems
codedwithSIPAMPL.
Thesefilesarealsoavailableviatheinternetwiththeproblemdatabaseand
SIPAMPL(seeSection5).
The intermediate files provided by AMPL are not directly available. The
AMPL write command can be used to provide the .nl, .col and .row files
neededbythesipamplfunction.(SeeFoureretal.[1990]fordetailsonhowto
producethesefiles.)
4.2 UsinganySIPSolverwithSIPAMPL
WhenbuildingaSIPsolvertouseSIPAMPL,theprogrammershouldbefamil-
iar with AMPL [Gay 1993]. The SIPAMPL routines work in a similar way to
theAMPLroutines.
The user must start by using the AMPL and SIPAMPL routines to load
the problem and the data structures needed for the subsequent calls to the
SIPAMPL routines. He can then call any AMPL or SIPAMPL routines as
required.
The directory s solver (see Figure 2 in Section 7) includes an example on
how to build a SIP solver. The example reads the problem and prints some
informationusingtheAMPLandSIPAMPLroutines.
5. THETESTPROBLEMDATABASE
Theproblemsinthedatabasewereobtainedfromseveralpapersandbookson
SIP and were coded in AMPL. They comprise linear and nonlinear, academic
ACMTransactionsonMathematicalSoftware,Vol.30,No.1,March2004.
SIPAMPL:Semi-InfiniteProgrammingWithAMPL • 55
andreal-lifeproblems.Severalproblemshaveparametersthatcanbechanged
bytheusertogeneratenewproblems.Mostareofsmalldimension(lessthan
50finiteand10infinitevariables).
Table I lists the problems in the database. In the table, “Problem” is the
problemfilename(with.modextension),“nx”isthenumberofx variables,“nt”
isthenumberoft variables,“nxc”isthenumberoffiniteconstraintsand“ntc”
isthenumberofinfiniteconstraints.
Sixteen of the problems correspond to the Watson set [Watson 1983; Coope
and Watson 1985]. Seven other problems, quadratic and nonlinear, are de-
scribed by Price [1992]. One problem described by Price has an error. In
problem S, Price considered T to be the set [0,1]p, but the solutions pre-
sentedareintheset[0,2]p.Thelattersetwascoded.ThreeC1 problemsfrom
Coope and Watson [1985] and later presented in Price and Coope [1996] were
coded.
Twelve Chebyshev approximation problems have been taken from Hettich
[Hettich1979a;Hettich1986]andReemtsen[1991].TwomoreChebyshevprob-
lems from Hettich and Gramlich [1990] were coded. In the first one, Hettich
classifies the problem as quadratic, but no quadratic terms in the objective
function appear (problem hettich8). In problem hettich9 the objective function
dependsonasetofpointsthatareunknowntous.
Three quadratic problems were taken from Liu et al. [1999] and another
linear problem (lin1) from Lin et al. [1998, Section 5.2]. In this paper the so-
lutiontotheproblemistheonereportedinCoopeandWatson[1985],butthe
problem is not the same (a coefficient in the objective function, the constraint
and the variable bounds differ from the original one). The problem was coded
asdescribedinLinetal.[1998].
Three linear problems from Fang and Wu [1994] and two from Ferris and
Philpott[1989]werecoded.
Eighteen linear problems were taken from Leo´n et al. [2000]. Problems
leon13 and leon18 correspond to problem 1 and 2 in Panier and Tits [1989]
respectively.
Two problems from production planning were coded as described by Li and
Wang[1997]andWangandFang[1996].
One problem was coded from Tanaka [1999], two more from Polak et al.
[1999] and problem hettich10 was coded as described in Lawrence and Tits
[1998].
ThetwoexampleproblemsfromTeoandGoh[1987]werealsocoded.Problem
teo2 has appeared in a previous paper from Gonzaga et al. [1980] and also
appears in Jennings and Teo [1990] and Teo et al. [1993]. The description in
TeoandGoh[1987]isincorrect.
ThreeproblemsfromBlankenshipandFalk[1976]werecoded.Inproblems
blankenship2andblankenship3theinfinitesetT isasetoftype[0,+∞[.
SevenFIRfilterdesignproblemsfromPotchinkov[1997]werealsoincluded
inthedatabase.
TheexampleusedbyPowelltoshowthathisKarmarkar[Adleretal.1989]
generalized algorithm can converge to a non-optimal point was also coded as
describedinTodd[1994].
ACMTransactionsonMathematicalSoftware,Vol.30,No.1,March2004.
56 • A.IsmaelF.Vazetal.
TableI. ProblemsintheSIPAMPLDatabase
Problem nx nt nxc ntc Problem nx nt nxc ntc
anderson1 3 2 0 1 blankenship1 2 1 0 1
blankenship2 2 4 0 4 blankenship3 3 2 3 3
coopeL 2 1 0 1 coopeM 2 1 1 1
coopeN 2 1 0 1 elke1 9 1 0 10
elke2 9 1 0 10 elke3 9 1 0 10
elke4 9 1 0 4 elke5 9 1 0 10
elke6 9 1 0 10 elke7 9 1 0 10
elke8 9 1 0 7 elke9 9 1 0 7
elke10 9 1 0 7 fang1 50 1 0 1
fang2 50 1 0 1 fang3 50 1 0 1
ferris1 7 1 0 2 ferris2 7 1 0 1
gockenbach1 33 1 120 16 gockenbach2 33 1 120 16
gockenbach3 33 1 120 16 gockenbach4 33 1 120 16
gockenbach5 33 1 120 16 gockenbach6 33 1 120 16
gockenbach7 33 1 120 16 gockenbach8 33 1 120 16
gockenbach9 33 1 120 16 gockenbach10 33 1 120 16
goerner1 4 1 0 2 goerner2 5 1 0 2
goerner3 7 1 0 2 goerner4 7 2 0 2
goerner5 7 2 0 2 goerner6 16 2 0 2
goerner7 8 2 0 2 gugat1 9 1 0 4
gugat2 9 1 0 4 gugat3 7 1 2 2
gugat4a 9 1 0 4 gugat4b 9 1 0 4
gugat4c 9 1 0 4 gugat4d 9 1 0 4
gugat4e 9 1 0 4 gugat4f 9 1 0 4
gugat5a 7 1 0 4 gugat5b 7 1 0 4
gugat5c 7 1 0 4 gugat5d 7 1 0 4
gugat5e 7 1 0 4 gugat5f 7 1 0 4
gugat6 6 1 0 4 gugat7 4 1 0 4
hettich1 9 2 0 2 hettich2 3 1 0 2
hettich3 5 1 0 2 hettich4 2 1 0 2
hettich5 3 2 0 2 hettich6 7 2 0 2
hettich7 7 2 0 2 hettich8 5 1 0 2
hettich9 11 2 0 2 hettich10 2 1 0 2
hettich10c 2 1 0 2 kortanek1 2 1 0 1
kortanek2 2 2 0 1 kortanek3 7 1 0 1
kortanek4 8 1 0 1 leon1 4 1 0 2
leon2 6 1 0 2 leon3 6 1 0 2
leon4 7 1 0 2 leon5 8 1 0 2
leon6 5 1 0 2 leon7 5 1 0 2
leon8 7 1 0 2 leon9 7 1 0 2
leon10 3 1 0 2 leon11 3 1 0 2
leon12 2 1 0 1 leon13 2 1 0 1
leon14 2 1 0 1 leon15 2 1 0 1
leon16 3 1 0 1 leon17 3 1 0 1
leon18 2 1 0 1 leon19 5 1 0 1
li1 10 1 0 1 li2 6 1 0 1
lin1 6 2 0 1 liu1 2 1 0 1
liu2 2 1 0 1 liu3 16 1 0 2
polak1 4 2 0 2 polak2 4 2 0 2
potchinkov1 298 2 0 4 potchinkov2 65 3 0 6
potchinkov3 66 2 0 4 potchinkov4a 67 1 19 2
potchinkov4b 65 1 19 1 potchinkovPL 122 2 0 4
Continues
ACMTransactionsonMathematicalSoftware,Vol.30,No.1,March2004.