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9/14/2015
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OPTIMIZATIONCH403ProcessEconomicsandPlantDesign
Whyweneedtooptimize? Economicoptimization:maximizenetpresentworth or net returns of an investmentworthornetreturnsofaninvestment
Processoptimization: Higherprocessingrates Higheryieldofdesiredproduct Reducedenergyconsumption Longproductivetimebetweenshutdowns
Plantoptimization: Lessersupervisoryandmanagementissues
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BasicprobleminOptimizationProblem:Findtheclosestpointonthecurvef
= 2x2 + 3x + 1 from the origin 5
6
2x 3x 1 fromtheorigin Formulateaobjectivefunction: Distanceofanypoint(xp,f(xp))squaredis
Ifwecanminimize squareofdistance,wegettheclosestpoint
2222 132 ppp xxxdD 222 132DMi
2
3
4
5
Takederivative andputtozero
Solve
22 132 ppp xxxDMin 06283616 23' ppp xxxD
3414.0px Threethingstonote:1. Nonlinear2. Unconstrained3. Realvalued
x-1 0 1
-1
0
1
OptimizationinChemicalProcess ProcessVariables:
t t t ti fl t t temperature,pressure,concentration,flowrate,etc. reactionrate,heattransfercoefficient,etc.and equipmentspecificationssuchassize,surfacearea,no.oftrays,valveposition,etc.
Dependent VariablesDecision Variables DependentVariablesThesearerelatedtodecisionvariablethroughconstraints
DecisionVariablesTheseareeithersetor
determined
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AnotherExample Objectivefunction: 124: 2 xxxfMinimize Constraints:
124: 21 xxxfMinimize
0
013
0341010
025
22
21
222
211
22
21
xxxx
xxxxxx
Threethingstonote:1. Nonlinear2. Constrained
Notethatoneofx1 orx2 canbeeliminatedusingtheequalityconstraint.
Onlyoneindependent/decision variableandotherwillbedependent variable
0, 21 xx 3. Realvalued
OptimizationinChemicalProcess ProcessVariables:
temperature,pressure,concentration,flowrate, reactionrate,heattransfercoefficient,etc.and equipmentspecificationssuchassize,surfacearea, Specificationssuchasno.oftrays,valveposition,etc.
ContinuousVariablesThesecantakenoninteger
values.
DiscreteVariablesThesetakeintegervalues
suchason/off,closed/open,/
ObjectiveFunction:Expressthequantity thatistobeoptimized(eitherminimizedormaximized)intermsofdecisionvariablessuchascostofconstructionandoperation.
values. yes/on,etc.
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TypesofOptimizationConstrainedOptimization
UnconstrainedOptimizationOptimization Optimization
IntegerLinearProgramming
(ILP)
MixedIntegerLinear
Programming(MILP)
LinearProgrammingProblem(LPP)
NonLinearProgrammingProblem(NLPP)
TypesofOptimizationLinear NonLinearLinear
ProgrammingProblem(LPP)
Non LinearProgrammingProblem(NLPP)
GraphicalMethodSuccessiveLinearProgramming(SLP)SuccessiveQuadraticProgramming (SQP)
SimplexMethod
Programming(SQP)Augmented
Lagrangian Method(ALM)
GeneralizedReducedGradientMethod
(GRG)
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ClassicalOptimization Classicalsinglevariableoptimization
f f( ) d f d h
However,x=x*couldalsobeamaximumorpointofinflection
NECESSARYCONDITION:Forafunctionf(x),definedintheintervalaxb,hasaminimumatx=x*,whereax*b,ifthederivatived(f(x))/dx =f(x)existsasafinitenumberatx=x*,thenf(x)=0.
SUFFICIENTCONDITION:Ifhigherorderderivativesexists,suchthatf(x*)=f(x*)==f(n1)(x*)=0,butf(n)(x*)0,then(i) f(x*)isaminimum,iff(n)(x*)>0andniseven.(ii) f(x*)isamaximum,iff(n)(x*)
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MultivariableOptimization Consider,f(x,y)=x2y2
f0,0
00
22
yxfy
x
yfxf
f
20
02, 22
2
2
2
ffyxf
xf
yxH
Tofindoutwhether(x,y)=(0,0) isminimum,maximumorpointofinflection,weneedtotestthesufficientcondition.
202yf
xyf
MultivariableOptimization ChecksignoffordifferentXo
T k X (0 1 0) 0 02 0 ooTo XXHX T XXHX
1
0.5
TakeXo =(0.1,0),=0.02>0 TakeXo =(0,0.1),=0.02
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MultivariableOptimization Checkforyourselfthat
)( 2233fhasaminimumat(0,0).
120
40
60
80
100
642),( 2221
32
3121 xxxxxxf
42
0-2
-4-4-2
02
0
-20
20
40
4
Threethingstonote:1. Nonlinear2. Unconstrained3. Realvalued
UnconstrainedOptimization MethodofSteepestDescent(Minimization)orAscent (Maximization)Ascent(Maximization)
21222121 coscos, xxxxxxfMin GlobalMaximum
LocalMaximum
LocalMinimum
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UnconstrainedOptimization
22exp xxxxxfMin MethodofSteepestDescent
21121 exp, xxxxxfMin
30
35
40
5 10 15 20 25 30 35 40
5
10
15
20
25
),( *2*1
* xxX
UnconstrainedOptimization Startwithsomeinitialguess Find a direction to proceed (search direction)
ooo xxX 21 ,oS
MethodofSteepestDescent
Findadirectiontoproceed(searchdirection) Determinehowfartoproceed Reachanewlocation Repeatuntilconvergence
So
oX
ooo SXxxX
12111 ,
kk XfXf 1
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UnconstrainedOptimization Searchdirection,then Determine how far to proceed
kk XfS kkkX
kkkk XfXSXXk
1MethodofSteepestDescent
Determinehowfartoproceed suchthatisminimumi.e.k kkk SXf 0 kkkk SXfdd
00
0
0
1
1
kTkkkkTk
k
ki
kki
i
kkkki
k
ki
i
kkkki
XfXfSXfS
dsxdSXf
x
ddxSXf
x
Innerproductiszeromeansorthogonality
UnconstrainedOptimization Minimize Initial guess:
MethodofSteepestDescent )exp(, 222121 xxxxf Initialguess:
Determine:
SearchDirection:
2,2, 21 ooo xxX 2221
2
1 exp2 xxxx
f
1
Distancetomove: 8exp
11
40
fS o
2222111 exp0 oooo sxsxf
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UnconstrainedOptimization Solve,
MethodofSteepestDescent
1 oooo Nextstep,
21
-0.2
-0.4
-0.6
-0.8
-12
8exp212221 2211
oo
oooo
sssxsx
112
1 oo SXX 2
10
-1-2-2
-10
1
00
8exp11
48exp21
22
UnconstrainedOptimization Minimize
i i l
MethodofSteepestDescent )4exp(, 222121 xxxxf 22oooX Initialguess:
Tryyourself!
-0.2
2,2, 21 ooo xxX
0 5
1
1.5
2
21
0-1
-2-2
-1
0
1
-1
-0.8
-0.6
-0.4
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
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UnconstrainedOptimization Methodofconjugategradient:Insteadoforthogonality we use conjugacy betweenorthogonality,weuseconjugacy betweensuccessivesearchdirection Twodirectionssi andsj aresaidtobeconjugatewithrespecttoapositivedefinitematrixQ if(si)TQ(sj) =0
IfQ=I (identitymatrix),(si)T(sj)=0.Searchesarealongvectors in orthogonal coordinate spacevectorsinorthogonalco ordinatespace.
IfQ=H(Hessianmatrix),searchesarealongvectorsintransformedspacedefinedbyeigenvectorsofH.
UnconstrainedOptimization FirstsearchdirectionisS b l
MethodofConjugateGradient oo XfS
kkTkkk XfXf 1111 Subsequently, Determinehowfartomove, suchthatisminimumi.e. Supposeapproximationtosecondorderisused,i.e.
kkTkkk XfXf XfXfSXfS 11k kkk SXf 0 kkkk SXfdd kkTkkTkk1k SXHS21fSXfXf then, 2 kkTk
kTk
optSXHS
Sf
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UnconstrainedOptimization Minimize
i i l
MethodofConjugateGradient 32 2221 xxXf 22oooX Initialguess: 2,2, 21 ooo xxX
4
6
8
1
1.5
2
21
0-1
-2-2
-1
0
1
-4
-2
0
2
4
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
ConstraintOptimization Consideroptimizationproblemwithonlyequalityconstraintsconstraints,
),...,,()(,...,2,10)(
:
21 n
i
xxxXnmmiXg
XfMinimize
Di t S b tit ti L M lti liDirectSubstitutionEliminatethemvariablesbysubstitutionandsolvethe
unconstraintoptimizationprobleminnmvariables
LagrangeMultipliersUseLagrangemultiplierstoaugment
theobjectivefunctionwithconstraintsandthensolvethe
unconstrainedoptimizationproblem
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ConstrainedOptimization Minimize nxxxxXXf ,,,,; 321
mjbXh 21 0 jjj bXhXgMethodofLagrangeMultipliers
AugmenttheobjectivefunctionusingLagrangemultipliers(i)
Foroptimalpoint:
mjbXh jj ,,2,1
mmj
jjj bXhXfXL ,,,, 211
hfL m 0,XL
0jjj bXhXg
mjbXhLni
xh
xf
xL
jjj
i
jm
jj
ii
,,2,10
,2,101
Thereare(n+m)variablesand
(n+m)unknowns
ConstrainedOptimization Minimize: 2221 xxXf
MethodofLagrangeMultipliers
AugmentedLagrangeFunction
Foroptimalpoint(necessarycondition)is
022 21 xxXh 22),( 212221 xxxxXL
0LL
52,5
4
54022
202
022
*2
*1
21
222
111
xx
xxL
xxxL
xxxL
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ConstrainedOptimization Minimize nxxxxXXf ,,,,; 321 kX 21
MethodofLagrangeMultipliers
AugmenttheobjectivefunctionusingLagrangemultipliers
OptimalpointbyKarushKuhnTucker(KKT)Condition:
rkcXg kk ,,2,1 rkkkr
kk cXgXfXL ,,,, 21
1
XXf r ** rkcXgcXg
cXgnixXg
xXf
kkkkkk
kki
kr
kk
i
,,2,1;00
0;,,2,10
****
**
1
**
Thereare(n+r)variablesand(n+r)unknowns
Foractiveconstraint,gk(X*)=ckForinactiveconstraint,k*=0
ConstrainedOptimization Minimize 221 1ln1ln xxXf 02211 xxXg
MethodofLagrangeMultipliers
Augmenttheobjectivefunction
Optimal condition
0
002
23
12
211
xXgxXgxxXg
0,,21ln1ln, 2121221 xxxxxxXL
AtX*=(1/3,5/3)g1 isabindingoractive
constraint,whileg2 andg3 arenonbindingorinactiveconstraints
Optimalcondition
43,
35,
31
02
01
2;01
1
**2
*1
21
2211
xx
xxLxx
Lxx
L
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ConstraintOptimization Minimize
MethodofLagrangeMultipliers 22 12 yxXf
02or2;02 yxyxXgxyXg Augmenttheobjectivefunctionwithconstraints
Findoptimalpoint
02or 2;0 21 yxyxXgxyXg
0,,,212,
21
22
122
yxyxxyyxMXL
LL
3/2 point, optimal theis 1,1
1,4;1,202;0
012;0222
*2
*1
**
2
2
1
2121
yx
yxyxLyxL
yyLxx
xL
ConstrainedOptimization Minimize
and nxxxxXXf ,,,,; 321
jbXh 21 rkcxg 21 MethodofLagrangeMultipliers
and AugmentedLagrangefunctionis
Optimal point is
mjbXh jj ,,2,1 rkcxg kk ,,2,1
rmkk
r
kk
m
jjjj
M
cXgbXhXfMXL
,,,,,,,
,,
2121
11
Thereare(n+m+r)variablesand Optimalpointis
rkcXgcXgrkcXgmjbXh
nixXg
xXh
xXf
kkkkkk
kkjj
i
kr
kk
m
j i
jj
i
,,2,1;00
,,2,10;,,2,10
,,2,10
****
**
*
1
*
1
**
*
(n+m+r)unknowns
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ConstrainedOptimization Secondordersufficientcondition
MethodofLagrangeMultipliers
SUFFICIENTCONDITION:If(X*,*,*)isanoptimalpoint,thenif
y is vector orthogonal to gradients active at the optimal point
0
vector nonzero 0,,*
***2
yXJyyMXLy x
T
yisvectororthogonaltogradientsactive attheoptimalpointJismatrixwhoserowsaregradientsoftheconstraintsthatareactiveatX*.
ConstrainedOptimization Minimize:
MethodofLagrangeMultipliers 2221 12 xxXf
2x
AugmentedLagrangeFunctionis
Optimal point
012;014 21
22
1 xxXhxxXg
1
41212,, 22
21
212
22
1 xxxxxxXL
Optimalpoint
471,
271
014
;012
2212;2122
*2
*1
22
21
21
222
111
xxxxLxxL
xxxLxx
xL
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ConstrainedOptimization Bothconstraintsareactive;
MethodofLagrangeMultipliers
37523
Forsecondordercondition23
287;
25
7223 **
37
230
037
2341
)1(20
02
2,, ***2
XLx
7
212
714
71
2122 *2
*1* xxXJ
0212
714
710
2
1*
yy
yXJ
ConstrainedOptimization Therefore,
1221a
yyayy*
2323
)1(22
2
1)1(20
02
21
2
**
222
2*
*
22
ay
ayayyLy x
T
minima
037
2337
234
222
ay
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ConstrainedOptimization Minimize:
MethodofLagrangeMultipliers 2221 1 xxXf
AugmentedLagrangeFunctionis:
ByKKTcondition:
0221 xx
2212221 1, xxxxXL 012 1 xL
;2
1,21,1
;0,0,2
0,0
0
022
012
*2
*1
*
*2
*1
*
221
221
222
11
xx
xx
xx
xxL
xxxL
xx
Wehavethreesolutionsbutonlytwoarevalid:(0,0)and(1/2,1/2)
ConstrainedOptimization CheckSecondOrderSufficientCondition:
Fi d H i t i
MethodofLagrangeMultipliers
FindHessianmatrix
Forthepoint(0,0), =2,
12002
22
2
21
221
2
21
2
2
xL
xxL
xxL
xL
Lx
20022Lx
Theconstraintgivenisactive
Check
20 21
2
1*2
* any ,001)(0121)( yyyy
yXJxXJ
02020
020 22
22
2
yyyyLy x
T
Foranyy2,itisnegativedefiniteimpliesmaxima
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ConstrainedOptimization CheckSecondOrderSufficientCondition:
Fi d H i t i
MethodofLagrangeMultipliers
FindHessianmatrix
Forthepoint(1/2,1/2), =1,
12002
22
2
21
221
2
21
2
2
xL
xxL
xxL
xL
Lx
00022Lx
Theconstraintgivenisactive
Check
00 0221)(2121)( 21
2
1*2
*
yyyy
yXJxXJ
0412
0002
12 22222
yyyyLy xTForanyy2,itispositivedefiniteimpliesminima
LinearProgrammingProblem Minimize 2121 54, xxxxf
GraphicalMethod
M i @ (2 33 1 33)
2
3
4
024;1;52;62
21
21
21
21
xxxx
xxxxxx
52 21 xx
1 xx
Maxima@(2.33,1.33)=16.00
-3 -2 -1 0 1 2 3 4-1
0
10, 21 xx 62 21 xx121 xx
24 21 xxMinima@(0.67,0.33)
=4.33
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LinearProgrammingProblem Simplexmethodwithslack,surplus andartificial variables
SimplexMethod
;6262 xxxxx
Slack variables x x
2424;11
;5252;6262
862121
752121
42121
32121
xxxxxxxxxxxx
xxxxxxxxxx
Slack variables x3,x4 Surplusvariables x5,x6 Artificialvariables x7,x8
LinearProgrammingProblem Formulatetheartificialproblem
SimplexMethod
:Minimize xxxxg
Substitute the artificial variables in terms of independent
24;1
;52;62
,:Minimize
8621
7521
421
321
8787
xxxxxxxx
xxxxxx
xxxxg
Substitutetheartificialvariablesintermsofindependent,slackandsurplusvariables
3)0()0()0()0(520421 :Minimize
87654321
216215
xxxxxxxxxxxxxxXg
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Formthesimplextable:choosebasisBasis x1 x2 x3 x4 x5 x6 x7 x8 b
LinearProgrammingProblemSimplexMethod
Basis x1 x2 x3 x4 x5 x6 x7 x8 bx3 2 1 1 0 0 0 0 0 6x4 1 2 0 1 0 0 0 0 5x7 1 1 0 0 -1 0 1 0 1x8 1 4 0 0 0 -1 0 1 2g -2 -5 0 0 1 1 0 0 -3f 4 5 0 0 0 0 0 0 0
6/1=6
5/2=2.5
1/1=1
2/4=0.5
Pickthevariablewithmostnegativecoefficientintheobjectivefunctionthisisthevariabletoenterbasis
Modifyrighthandsidebbydividingbycoefficientsofthisvariable. Theequationwithleastr.h.s.aftermodificationistobereformulated Inthiscase,variablex2 intoenterthebasisandx8 istoleave.
f
Formthesimplextable:changebasisBasis x1 x2 x3 x4 x5 x6 x7 x8 b
LinearProgrammingProblemSimplexMethod
Basis x1 x2 x3 x4 x5 x6 x7 x8 bx3 1 0 0x4 0 1 0x7 0 0 1x2 1/4 1 0 0 0 -1/4 0 1/4 2/4g 0 0 1f 0 0 0
Leavingotherbasisvectorsassuchchangex2 tobasisvector. Coefficientofx2 ismade1bydividingtheequationentireequationby4.
f
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Formthesimplextable:withnewbasisBasis x1 x2 x3 x4 x5 x6 x7 x8 b
LinearProgrammingProblemSimplexMethod
Basis x1 x2 x3 x4 x5 x6 x7 x8 bx3 7/4 0 1 0 0 0 -1/4 11/2x4 0 0 1 0 0 -1/2 4x7 0 0 0 -1 1 -1/4 x2 1 0 0 0 -1/4 0 g -3/4 0 0 0 1 -1/4 1 5/4 -1/2f 11/4 0 0 0 0 5/4 0 -5/4 -5/2
Leavingotherbasisvectorsassuchchangex2 tobasisvector. Coefficientofx2 ismade1bydividingtheequationentireequationby4. Coefficientofx2 ismade0inotherequationsbyrowoperations.
Nowwerepeattheprocesstogetthenewbasis
f
Formthesimplextable:changebasisBasis x1 x2 x3 x4 x5 x6 x7 x8 b
LinearProgrammingProblemSimplexMethod
1 2 3 4 5 6 7 8x3 7/4 0 1 0 0 0 -1/4 11/2x4 0 0 1 0 0 -1/2 4x7 0 0 0 -1 1 -1/4 x2 1 0 0 0 -1/4 0 g -3/4 0 0 0 1 -1/4 1 5/4 -1/2f 11/4 0 0 0 0 5/4 0 -5/4 -5/2
22/7
8
2/3
2
Pickthevariablewithmostnegativecoefficientintheobjectivefunction thisisthevariabletoenterbasis
Modifyrighthandsidebbydividingbycoefficientsofthisvariable. Theequationwithleastr.h.s.aftermodificationistobereformulated Inthiscase,variablex1 intoenterthebasisandx7 istoleave. Continueuntilnomorenegativecoefficientsarelefting(X)
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Formthesimplextable:withnewbasisBasis x1 x2 x3 x4 x5 x6 x7 x8 B
LinearProgrammingProblemSimplexMethod
Basis x1 x2 x3 x4 x5 x6 x7 x8 Bx3 0 0 1 0 7/3 -1/3 -7/3 1/3 13/3x4 0 0 0 1 2/3 1/3 -2/3 -1/3 11/3x1 1 0 0 0 -4/3 1/3 4/3 -1/3 2/3x2 0 1 0 0 1/3 -1/3 -1/3 1/3 1/3g 0 0 0 0 0 0 1 1 0f 0 0 0 0 11/3 1/3 -11/3 -1/3 -13/3
Thevalueofthedecisionatminimumis(2/3,1/3) Theminimumvalueis13/3or4.33
f
References
Edgar,T.F.,Himmelblau,D.M.andLasdon,L.S.,Optimization of Chemical Processes 2nd EditionOptimizationofChemicalProcesses,2nd Edition,McGrawHill,2001.
Peters,M.S.,Timmerhaus,K.andWest,R.E.,PlantDesignandEconomicsforChemicalEngineers,McGrawHillEducation,5th Edition,2002.
h d h Mohan,C.andDeep,K.,OptimizationTechniques,NewAgeInternationalPublishers,1st Edition,2009.