Cardoso 1997 Computers & Chemical Engineering

8/11/2019 Cardoso 1997 Computers & Chemical Engineering

1/16

ergamonCompu ters chem. EngngVol. 21. No. 12. pp. 1349-1364. 1997

Copyright 1997 ElsevierScienceLtd.All rights reservedPrinted n GreatBritain

PII: S0 098 -13 54( 97) 000 15.X 0098-1354/97 $17.00+0.00

simu lated anneal ing app roach to thesolut ion of minlp problem s

M. E Cardoso, R. L. Salcedo , S. Feyo de Azevedo and D. Barbosa

Depar tam ento de Engen har ia Qufmica , Faculda de de Engenhar ia da Univers idade do Por to , Ruados B ragas , 4099 Por to , Por tugal

(Received 29 Septem ber 1995; revised 3 March 1997)

A b s t r a c t

An a lgor i thm (M-SIM PSA ) su i tab le fo r the op t imiza t ion o f mixed in t ege r non- linea r p rogramming (MINLP )problem s is presented. A re cent ly proposed cont inuous non- l inear solver (SIM PSA ) is used to update the cont inuousparameters , and the Metrop ol is a lgor i thm is used to update the comp lete solution vector of decis ion var iables . TheM -SIM PS A algor i thm, which does not require feas ible in i t ia l points or any problem decom posi t ion, was tes ted wi thsevera l funct ions publ ished in the l i tera ture , and resul ts were comp ared wi th those obta ined wi th a rob ust adapt iverandom search method. Fo r i l l-condi t ione d problem s, the proposed approach is shown to be m ore re l iable and m oreeff ic ient as regards the overco ming of d i ff icul t ies associa ted wi th local opt ima and in the ab i l i ty to reach feas ibi l ity.The resul ts obta ined reveal i t s adequacy for the opt imizat ion of MIN LP prob lems encountered in chem icalengineer ing pract ice . 1997 Elsevier Science Ltd

K e y w o r d s :Mixed in t eger non- l inea r p rogramm ing ; S imula t ed annea ling ; S implex

1 I n t r o d u c t i o n

The op t imiza t ion o f mixed in t ege r non- l inea r p ro -gramming (MINLP) problems const i tu tes an act ive areaof research (Grossmann and Daichendt , 1994; Floudas ,1995) . Problems in process synthes is and des ign andschedul ing of batch processes are among appl ica t ionsre levant to chemical engineer ing pract ice (Floudas ,1995) . The ge neral s ta tement i s

min imize F (x ,y ) (1 )

subject to the con st ra in ts

h , (x ,y) =0, k= ,2 . . . . ml (2)

gj(x,y)-->0, = 1,2 ....m2 (3)

ai


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So lu tion o f M INLP 1351

repor ted in th is work, the proposed approach shows asuper ior performance in overcomin g local opt ima and inaccuracy. Thus , the M -SIM PSA algor i thm represents animproved a l ternat ive for the opt imizat ion of MINLPproblem s typica l of chem ical engineer ing prac t ice .

2. The m simpsa algorithm

The S IMPS A a lgo r i thm was deve loped fo r t he g loba lso lu t ion o f NLP p rob lems (Cardosoe t a l . , 1996). Itcomb ines the or ig inal Metropo l is a lgor i thm (Metropol ise t a l . , 1953; K irkpat r icke t a l . , 1983) with the non-linears implex of Nelder and Mead (1965) , based upon aproposal by P ress and Teukolsky (1991) . Th is a lgor i thmshows good robustness and accuracy in ar r iv ing a t theglobal opt imum of d i ff icul t non-convex highly con-s t ra ined funct ions . The ro le of s imula ted anneal ing in the

overa l l approach is to a l low for wrong-way mov ements ,s imul taneously providing (asymptot ic) convergence tothe g lobal opt imum. The main aspects are the accep-tance/ re jec t ion cr i te r ia (Metropol ise t a l . , 1953; Kirkpa-t r ick e t a l . , 1983), the init ial annealing temperature(Aars t and Ko rs t , 1989) and the adopt ion of a su i tablecool ing schedule . Extens ive tes ts wi th combinator ia lproblems (Aars t and Kors t , 1989; Dase t a l . , 1990;Do lan e t a l . , 1989; Patele t a l . , 1991 ; Ca rdosoe t a l . ,1994) and wi th NLP problems (Cardosoe t a l . , 1996)show the super ior i ty of the Aars t and van Laarhoven(1985) scheme as com pared to exponent ia l type cool ing

schedules (Kirkpat r icke t a l . , 1983). Thus, in the presentwork the tem pera ture contro l parameter was a l lowed todecrease fo l lowing the A ars t and van L aarhoven (1985)scheme.

The ro le of the non- l inear s implex i s to genera tecont inuous sys tem conf igura tions . A deta i led descr ip t ionof the es t imat ion of the in i t ia l anneal ing tempera ture ,cool ing schedule , genera t ion of the in i t ia l s implex andcont inuous sys tem conf igura t ions can be found e lse-where (Cardosoe t a l . , 1996).

The most obvious and s t ra ight forward extens ion ofthe SIMPSA algor i thm in order to deal wi th MINLP

problem s would be to update the a lgor i thm to accept andgenera te d iscre te decis ion var iables together wi th thecont inuous var iables . This ap proach was tes ted , but thealgor i thm could not evolve in the cont inuous searchspace s ince the d iscre te var iables are changed a t thesame pace as the cont inuous var iables , i .e . a t eachs implex m ove. Thus , a com plete ly d ifferent scheme wasadopted.

The M-SIM PSA a lgo r ithm i s shown schema t ica l ly inFig. I , Basica lly, som e init ial solution vector (x,y), ei therfeas ib le or infeas ib le , i s fed to the cont inuous SIM PSAopt imizer. Each global i te ra t ion of the M-SIMPSAalgor i thm comprises one cycle of the cont inuousSIMPSA op t imize r, fo l lowed by one compar i sonthrough the Metropol is cr i te r ion for the completesolut ion vectors . This compar ison a l lows the se lec t ion ofdi fferent d iscre te conf igura t ions . To increase the chan ceof escap ing di ff icul t local opt ima an d o f f inding feas ib le

points , each cycle of the SIMPSA algor i thm togglesbetween a search over the g lob al in tervals [g iven by (4)]and a search over com pressed in tervals (Cardosoe t a l . ,1996) , fo l lowing the cool ing schedule . The sam e cool ingschedule i s employed for both the cont inuous anddiscrete variables, i .e . the same temperature is used forboth types of var iables. Each SIMP SA i tera t ion ( two p ercycle) comprises a number of s imp lex i te ra tions , g ivenby (100 p 2 1 n ) , where p i s the number of cont inuousdecis ion var iables and n the to ta l number of degrees off reedom. The genera t ing scheme proposed by Dolane ta l . (1989) i s employed each t ime a d i fferent se t ofdiscre te var iables i s needed, whereby a f i rs t randomnumber choo ses the d iscre te variable to be ch anged anda second one chooses i f the change i s to be upwards o rdownw ards by on e uni t.

I f the Metropol is cr i te r ion accepts the current so lu-t ion , the a lgor i thm proceeds by updat ing the solu t ion

vector and genera t ing a d i fferent d iscre te conf igura t ionbefore re-enter ing the cont inuous SIMPSA opt imizer.Since the cont inuous opt im izer only performs two globali te ra t ions for each se t of d iscre te var iables , the con-t inuous var iables may not be ab le to converge i f the se tof d iscre te var iables i s a lways forced to change. Thus , i fthe current so lu t ion i s re jec ted (according to theMetropol is cr i te r ion) wi th a d iscre te conf igura t ion d i ffer-ent f rom the previous one , the a lgor i thm provides a 50probabi l i ty, i .e. an unbiased chance for the previousconf igura tion to re-enter unchanged a n ew c ycle of theSIMPSA opt imizer. On the o ther hand, i f the d iscre te

configurations are the same, a new discre te configu rationis a lways enforced, thus decreas ing the chance ofconvergence to local opt ima.

I t should be noted tha t the M-SIMPSA algor i thm isappl icable to MIN LP p roblems, as wel l as to NLP and tocombinator ia l minimizat ion problems. With NLP prob-lems, only the SIMPSA s teps become act ive . Forcombinator ia l minimizat ion , the a lgor i thm reduces to as imula ted anneal ing scheme, where enforcement ofdi fferent d iscre te configura t ions becom es mandatory a tevery s tep . Also , by speci fy ing a zero value for thecontro l tempera ture , the a lgor i thm reduces to the non-

l inear s implex of Nelder and Mead for the cont inuousvar iables and to a pure ly random search a lgor i thm forthe discrete variables.

2 .1 . D e a l i n g w i t h n o n - l i n e a r c o n s t r a i n t s

An essent ia l fea ture of op t imizat ion a lgor i thms appl i -cable to chemical engineer ing problems is the capabi l i tyof deal ing effec t ive ly wi th non- l inear const ra in ts ,namely of f inding solu t ions on surfaces descr ibed byactive constraints as well as f inding feasible points withhighly const ra ined non-convex funct ions . In theSIMPSA algor i thm, points produced by the s implexmovement tha t do not obey e i ther bound or impl ic i tcons t ra in ts are subs t i tu ted by rand omly g enera ted pointscentered on the current bes t ver tex , through the use of anappropr ia te genera ting funct ion (Ca rdosoe t a l . , 1996).By repeated appl ica t ion of th is funct ion , i t i s guaranteedthat the bound const ra in ts are obeyed. A large constant


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1352 M. F. CARDOSAe t a l .

value i s then used to penal ize the objec t ive function forpoints which do not obey the inequ al i ty const ra in ts .

As the SIMPSA algor i thm is an infeas ib le pa thmethod, where compar ison of the s implex ver t icesthrough the Me tropol is cr i te r ion i s a lways poss ib le us ingperturb ed function value s Press and Teukolsky, 1991),and as the objec t ive funct ion does not inc lude anyinformat ion about the extent of cons t ra in t v io la t ions ,there i s no g uarantee tha t a feas ib le point wi l l be found.With the MSGA algor i thm, a two-phase re laxat ions t ra tegy was emplo yed to reach feas ib i l ity Salcedo,1992). Basically, pha se I identif ies the set of integercandidates for the g lobal opt imum and phase I I so lvesthe corresponding NLP subproblems, cons ider ing thebes t feas ib le so lu t ion as the g lobal opt imum. R ather thanemploy ing a s imi l a r r e l axa t ion scheme , t he M -SIMPSA

algor i thm employs a penal iz ing scheme given by thefol lowing equ at ions :

F ~ . = F o ld + a b s ( F o sd) ( rlm~x+ rgma~) -- l)r ; abs(fold)>-(r lmax+rgmax) 6)

Fo e .= Fo~+ 1 + abs Fo ,d) ) x(rlma~+rgmaO

X - l ) r ; abs(Fold) inequal i ty con-straints, F,~w and Fo~ are, respectiv ely, the pen alized andunpenal ized objec t ive funct ions and r i s se t to zero forminimizat ion and to one for maxim izat ion .

These d i fferent s t ra tegies affec t the re la t ive perform-ance of both a lgor i thms. For problems w here the g lobalopt imum is an i l l-condi t ioned point wi th respect tofeas ib i l ity) , as the opt imizat ion proceeds the re laxed

, .o I _ l a II x y I

J SIMPSA

N

Curren t So lu t ion

x l , y H

F I

Y

Retain cu rrentsoluUonX o u = X l

F o ~ = F

. enerate newinteg er solutiony l

. J Metropolis I- I c d t e r i u m

Y

Y

N

t

N

Retain currenti n t e g e r s o l u t i o n

y,==y~l

Replace curre ntsolutionXImXold

y~ l=yo .F i = F o .

F i g I. Overall strategy for m ixed integer programming M-SIMPS A algorithm).


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Solut ion

M SG A algo r i thm f inds it more and m ore d i fficul t toremain near the g lobal opt imum, s ince the increas inglyconst ra ined region dr ives the a lgor i thm to regions wherefeas ib i l i ty i s eas ier to reach. Thus , the correc t ident i f ica-t ion of d iscre te candidates (phase I ) i s c rucia l for thesuccessful appl ica t ion of the M SG A algor i thm to g lobal

MIN LP op t imizat ion . On the o ther hand, the inc lus ion ofthe extent of the const ra in t v io la t ions on the objec t ivefunct ion through (6)(7) g ives the M-SIM PSA algor i thman increased f lexib i l i ty in rem aining near the i l l -condi t ioned global opt imum, avoiding the necess i ty todecouple the MINLP formula t ion in to a sequence ofNLP subproblems. As the re la t ive impor tance of theconst ra in t v io la t ions decreases wi th the p rogress of theopt imizat ion , more weight i s g iven to the objec t ivefunct ion , and th is increases the chance of f in ishing theopt imizat ion wi th the decis ion var iables a t the g lobalopt imum. The penal iz ing s t ra tegy descr ibed above can

in pr inc ip le be ap pl ied wi th o ther const ra ined opt imiza-t ion a lgor i thms, inc luding the MSG A algor i thm, but th iswas not tested.

I t i s obvious tha t (6) could be used throughout theent i re opt imizat ion . How ever, i f the ob jec t ive funct ionapproaches zero , the const ra in t v io la t ions may not betaken fu lly into accoun t. To avoid this pitfall , both (6)(7)were thus employed. In the M-SIMPSA algor i thm, theproposed penal iz ing scheme can be used whenever thes implex s teps are ac t ive , i .e . for both NL P and MIN LPproblem s, but not w i th pure ly in teger decis ion variables ,where only the s imula ted anneal ing s tep i s used

(Cardosoe t a l . , 1994).

2 . 2. Te r m i n a t i o n c r i t e r i a

The choice of appropr ia te s topping cr i te r ia seems tobe the crucia l s tep for g lobal search a lgor i thms (Schoen,1991) . The proposed a lgor i thm includes two con-vergence tes ts. One is inherent to the s implex m ethod, asimp lemen ted by P res se t a l . (1986) and Press andTeukolsky (1991) , and is a m easure of the col lapse of thecent ro id . The second cr i te r ion i s based on the averagegradient of the objec t ive funct ion wi th respect to thenumber of funct ion evaluat ions , as used before wi th

combinator ia l minimizat ion (Cardosoe t a l . ,

1994) andwi th NLP op t imiza t ion (Ca rdosoe t a l . , 1996).

3 N u m e r i c a l i m p l e m e n t a t i o n a n d c a se s t u d i e s

The M -SIM PSA algor i thm was wri t ten in For t ran 77,and a l l runs were p erformed wi th dou ble prec is ion on aHP 730 Worksta t ion , running compi ler opt imized For-t ran 77 code. To tes t the qual i ty of the proposedalgor i thm, compar ison wi th another MIN LP opt im izer i sneeded. The M SGA adapt ive random search a lgor i thm(Salcedoe t a l . , 1990; Salcedo, 1992) was used for thispurpose .

Simula ted anneal ing and random search methods areof a s tochas t ic na ture , and as such can be sens i t ive to thesequence of p seudo- random numbers.These sequencescorrespond to s tochas t ic movements , whenever boundconst ra in ts are v io la ted , as wel l as to d i fferent opt imiza-

of MIN LP 1353

t ion paths for the cont inuous S IMP SA op t imizer, s ince arandom per turbat ion i s super imposed on the s implexvertices (Press and Teukolsky, 1991). The generation ofdiscre te conf igura t ions , which uses the s tochas t icscheme p roposed by Do lane t a l . (1989), the applicationof the M etropol is cri te r ion for the accep tance/ re jec t ionof poorer so lu t ions and the prob abi l i ty of m ainta in ingthe previous ly accepted con f igura tion before re-enter ingthe cont inuous opt imizer, i s dependent on the pseudo-random num ber sequence .

The performance of the proposed a lgor i thm must thenbe ev aluated on a s ta t i s t ica l bas is , by running d i fferentproblems wi th d i fferent random sequences . I t i s impor-tant tha t the pseudo-random num ber genera tor employ edin s tochas tic a lgor i thms be uni form an d indep endent , sothat def ic iencies in the opt im izat ion a lgor i thms may notsomehow be compensated by non-uniformi ty in thegenerator. Statist ical evalu ation of three pseud o-ran dom

number genera tors showed tha t a Lehmer l inear con-gruential generator (Shedler, 1983) was found to besa t i sfac tory for both mul t i -d imensional uni formi ty andindependence (Salcedoe t a l . , 1990). Thus, in this wo rk,d i fferent random sequences were genera ted by p rovidingdifferent seeds to this congruen tial generator.

Performance evaluat ion of the a lgor i thm was carr iedout by s ta t i s t ica l evaluat ion of severa l cons t ra inedMIN LP problems (Grossmann and Sargent , 1979; Ko cisand Grossmann, 1987Kocis and Grossmann, 1988; Yuane t a l . , 1989; Floudase t a l . , 1989; Wong, 1990; Salcedo,1992; Berman and Ashrafi , 1993; Ciric and Gu, 1994;

Ryoo and S ahinid is, 1995; Floudas , 1995) . The problemswere run with P different start ing points (varying witheach problem) and S d i fferent seeds , corresponding tothe gen era t ion of S P d i fferent in i tia l s implexes andr u n s .

We examine 12 tes t problems, inc luding two mul t i -product ba tch p lant problem s tha t were d i ff icult to so lvefor the g lobal opt imum with the MS GA algor i thm, takenfrom Grossmann and Sargent (1979) and Kocis andGrossmann (1988), as wel l as a non-equi l ibr ium react ivedisti l lat ion process taken fro m C iric and G u (1994).Al though some of these problems may be eas i ly so lvedby di rec t enumerat ion , th is was no t genera l ly performed,except for the react ive d is t i l la t ion problem, s ince herethe g lobal opt imum was unknown. The tes t problemswere taken f rom independent authors and should providea fa i r bas is for tes t ing the proposed a lgor i thm.

A binary expansion was used to represent the d iscre teconfigurations, since this al lows large variations for theor ig inal d iscre te var iables a t each M-SIMPSA cycle .The expans ion employed i s g iven by

ky i= Yi + j~ = ,j Y j + ~ i -C ~ ) X Y k ,8 )

where yj and ~ are bou nds for the disc rete variable Yl[given by (5)] and k is the largest integer that satisfies thefol lowing equat ions :

~>C, (9 )

and


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1354

Table 1. B rief summary of test problems

Exam ple Decision variablesDiscrete Continuous

l

23456789101112

1 binaryI binary1 binary3 binary2 binary4 binary4 binary8 binary

2 discrete (12 binary)3 discrete (6 binary)

6 discrete (12 binary)1 discrete (6 binary)

1

1

2

2

3

371660

C~= yi+ j~l j (10)

where the Yj a re b inar y var iab les . Other mo re or lessequiva len t t ransformat ions a re pos s ib le (F loudas , 1995).

For a l l MIN LP problem s the e rror c r i te r ion was se t a t10 -s , for both the conv ergenc e of the s im plex a lgor i thm(as g iven by Press and Teukolsky, 1991) and forconve rgence o f t he sm oo thed func t i on va lue s , and t hecool ing schedule parameter was se t a t 10-2 (Cardosoe ta l . , 1996). For p ure ly d iscre te dec is ion var iab les (exam -ples 4 , 6 and 8) , the e r ror c r i te r ion was se t a t 10 -3 andthe cool ing schedule parame ter was se t a t 10 (Cardosoe ta l . , 1994). For a l l p rob lems, the in i t ia l search reg ionswere de t e rmined f rom the bound cons t r a in t s . Tab l e 1g ive s a b r i e f summary o f t he t e s t p rob l ems , whe re t hef i r s t n ine p rob l ems we re so lved w i th t he M-SIMPSAalgor i thm s ta r t ing f rom 100 rando mly genera ted in i t ia ls implexes , i . e . 100 d i ffe rent rando m nu mb er sequences .

As the bounds of the dec is ion var iab les a re au tomat-i c a l l y s ea r ched fo r i n bo t h t he M-SIMPSA and MSGAalgo ri thms , t he g lob a l op t imum may b e f ound i r re spec -t ive of the e ffec t iveness of the random search ors imula ted anne al ing/s im plex approaches . Thus , the tes ton the ex t remes was deac t iva ted in both a lgor i thms fora l l p roblems to provide a cor rec t appra isa l of the i rrel iabil i ty.

4 . R e s u l t s a n d d i s c u s s i o n

E x a m p l e 1 .Th i s examp le i s t aken f rom Koc i s andGros smann (1988 ) and i s a l so g iven i n F loudase t a l .(1989) and R yoo and S ahin id is (1995) :

r a in 2x+y

s.t.

1 . 2 5 x 2 y < 0

x + y ~ l . 6

0--


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So lu t io n o f MINL P

Gros smann (1989 ) and a l so g iven i n D iwe ka re t a l .(1992) and D iweka r and Ru bin (1993), i s a two-reac torp r ob l em, whe re s e l ec t i on i s t o be made among twocandida te reac tors for m inim iz ing the cos t of producinga des i red product :

m in 7 .5y~ +5.5y2+7v~ +6v2 +5x

s.t.

yl+y_,= 1

zt =0.911 - exp ( - 0.5 vl)]x

z2 =0.811 - ex p( - 0.4v,)]x2

X I l - X 2 - - X = 0

z~ + z, = 10 s.t.v,


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1356 M. E CARDOSA e t a l .

Table 2. Brief summary of test results (number of successes inobtaining he global optimum out of 100 runs)

Example M-SIMPSA MSGAWithout penalizing Penalizing

99 100 1002 83 100 200/200*3 0 100 1004 100 - - 800/800*5 100 100 1006 100 - - 800/800*7 60 97 598/600*8 100 - - 1009 87 95 100

* Salcedo (1992).

SIMPSA algorithm for example problems 1-9, where itmay be stated that the algorithm is generally robust andreliable. In some MINLP cases, however, the algorithmonly attains feasible solutions and the global optimumwith the penalizing scheme. This occurs at some expensein CPU time, on the average roughly doubling it, but theincreased reliability in obtaining the global optimumjustifies it. By comparison, the adaptive MSGA algo-rithm is extremely reliable for these small-scale prob-lems, where the global optimum was, in practice, alwaysobtained for all problems. We will see below, however,that for larger scale and more demanding problems theM-SIMPSA algorithm behaves better than the MSGAalgorithm.

Exa m ples 10 /11 (mul t i-p roduc t ba tch p lan t ) .Themulti-product batch plant consists of M processingstages in series where fixed amounts Q~ of N productshave to be manufactured. The objective is to determinefor each stage j the number of parallel units N and theirsizes V and for each product i the corresponding batchsizes B~ and cycle times TL~. The problem data are thehorizon time H, the size factors S 0 and processing timesto of product i in stage j, the required productions Q~, andappropriate cost functions a s and fls' The mathematicalformulation of this problem is as follows (Grossmannand Sargent, 1979; Kocis and Grossmann, 1988):

Mmin E tr/vjv~J

S t

i=1

v j > _ s , ~ ,NjTLi>-tljI ~ N j ~ N y

T L i ~ T L i - - T L~I ~ ~ u

B j - B j - B sNj integer

The upper (u) and lower (I) bounds N~, V~, V~ arespecified by the problem and appropriate bounds for TL~and B~ can be determined as follows:

TL =m ax tUy

~LI= max j t UQ,

The optimization of a multi-product batch plant withdata structure as formulated has 2(M+N) degrees offreedom, 4(M+N) bound constraints, 1 (< ) and 2 M N(->) inequality constraints. The objective function, thehorizon time constraint and several inequalities are non-convex. Thus, depending on the values of M and N, afairly large non-convex MINLP problem may result.Non-convexities in this problem may easily be elimi -nated using logarithmic transformations, and this wasactually done by Kocis and Grossmann (1988) to goodadvantage in testing their MINLP algorithm. Thepossible number of parallel units for each stage ~ aswell as the number of stages M will determine thenumber of NLP subproblems embedded in the originalMINLP formulation. We examine below two cases ofincreasing complexity, where the relevant data is givenin Table 3. Both cases were run with 10 random startingpoints and 100 different seeds, corresponding to 1000runs per case.

Example 10 .This problem, proposed by Grossmann

and Sargent (1979), has 10 degrees of freedom (threeintegers corresponding to six binary variables), 26 boundconstraints (with the binary expansion of the discretevariables), and 13 inequality constraints. The problemsize is equivalent to solving 27 NLP subproblems, andthe global optimum, with respect to feasibility, corre-sponds to an extremely ill-conditioned point. With linearprogramming problems, the optimum, when unique, liesin the intersection of two or more constraints. Hence,moving slightly away from it (in the wrong direction)will certainly lead to infeasibility. In this non-linearexample, variations in any direction (up or down) as

small as 0.01 in any of the seven continuousparameters produces infeasibility, and this is not acharacteristic feature of optimum solutions of non-linearoptimization problems.

By direct application of the MSGA algorithm, theglobal optimum was never obtained, although feasiblepoints were always obtained. However, phase I of therelaxed MSGA algorithm was capable of identifying theglobal optimum in all cases, albeit at a greater expense incomputational effort (Salcedo, 1992).

Table 4 gives the results obtained with the M-SIMPSA algorithm in solving this example. It showsthat the global optimum was reached 919 times out ofI000 runs, starting always from infeasible points (thatobey the bound constraints but not the inequalityconstraints). The average number of function evalua-tions was 257,536, against 35,000 with the MSGAalgorithm. However, the MSGA algorithm without


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S o l u ti o n o f M I N L P

Table 3. Input data for Exam ples 10/11

Example M N N~ V~ W; Sij t i .

1357

10 3 2 3 250 2500

I1 6 5 4 300 3000

2 3 4 8 2 0 84 6 3 1 6 4 4

7.9 2.0 5.2 4.9 6.1 4.2 6.4 4.7 8.3 3.9 2.1 1.20.7 0.8 0.9 3.4 2.1 2.5 6.8 6.4 6.5 4.4 2.3 3.20.7 2.6 1.6 3.6 3.2 2.9 1.0 6.3 5.4 11 .9 5.7 6.24.7 2.3 1.6 2.7 1.2 2.5 3.2 3.0 3.5 3.3 2.8 3.41.2 3.6 2.4 4.5 1.6 2.1 2.1 2.5 4.2 3.6 3.7 2.2

Exam ple QI Q2 Q3 Q4 Q5

10 40000 20000I1 2 5 00 0 0 150000 180000 160000 120000

re laxa t ion ar r ived 95% a t feas ib le poin t A, 4% a t feasib lepoin t C and 1% a t feas ib le poin t D, i . e . i t on ly a r r ived a tt he g loba l op t imum th rough t h e app l i c at i on o f t h e two -phase relaxation strategy (Salcedo, 1992).

E x a m p l e l I .Th i s p rob l em was p roposed by Ko c i s a ndGrossm ann (1988) . I t has 22 degrees of f reedom (s ixin tegers cor responding to 12 b inary var iab les ), 56 boundcons t ra in ts (aga in wi th the b inary expans ion of thediscre te var iab les ) , and 61 inequal i ty cons t ra in ts . Thep r ob l em s i ze i s equ iva l en t t o so lv ing 4096 NLPsubp rob lems , and t he g loba l op t imum co r r esponds oncemore to an i l l -cond i t ioned poin t , s ince var ia tions (up ordown) as smal l as 0 .01% in any of severa l of the 16cont inuous parameters prod uce infeas ib i l i ty. The g lobalop t imum co r r e spon ds t o yT= [2 2 3 2 l l ] , VT= [30001891.64 1974.683 2619.195 2328.1 2109.797],B T= [379.7467 77 0.3054 727.5089 638.2978 525.4531 ] ,T~ =[3 .2 3 .4 6 .2 3 .4 3 .7] wi th an objec t ive funct ion of

285,510/yr. This so lu t ion i s so i l l -condi t ioned tha t thecont inuous vec tor as repor ted t runca ted by Kocis andGrossm ann (1988) v io la tes f ive inequal i ty ( ->) con-s t rain ts by as muc h as 0 .34%.

For t h i s p rob l em, t he MSGA a lgo r i t hm cou ld n o tobta in a s ingle feas ib le so lu t ion out of 100 runs wi thdi ffe rent opt imiza t ion pa ths . Al though the re laxa t ions t ra tegy descr ibed for the MSGA algor i thm is con-ceptua l ly s imple , for th i s case i t was n ot s t ra ight forwardto apply s ince phase I could not readi ly ident i fy theposs ib le candida tes for the g lobal opt imum. Theincreased computa t iona l e ffor t necessary to c lear lyident i fy the cor rec t se t of d iscre te var iab les was verylarge , and a to ta l of 566 runs was n eeded to a r r ive a t thedescr ip t ion of a l l 31 feas ib le poin ts tha t cou ld be found ,inc luding the g lobal opt im um (Saicedo, 1992).

Wi th t he M-SIM PSA a lgo r it hm, more t han 100feas ib le poin ts were obta ined , cor respond ing to d i ffe rentse ts of the d iscre te var iab les . The resu l t s we re thusgrouped in to a f requency d is t r ibu t ion , g iven in F ig . 2 .The g lobal opt imum (wi th in the e r ror acceptance for thecont inuous var iab les ) was reached only 26 t imes out of1000 runs , bu t the resu l ts a re muc h be t te r com pared wi ththose ob t a ined w i th t he MSGA a lgo r i t hm, p ro duc i ngonly 15 infeas ib le so lu t ions . F ig . 2 shows tha t there a rethree d is t inc t reg ions . Region A corresponds to the

Table 4. Global optimum and results for Example 10

Nj Vj B i T u c c u r r e n c e s(M-SIMPSA)

Global optimum 1 480 240 201 720 120 16l 960 38499.8 919

Feasible point A 2 250 120 l02 360 60 81 480 40977.5 71

Feasible point B 1 3 4 6 .7 173.3 102 520 86.7 161 69 3.3 42325.5 10

Feasible point C l 40 7.3 140 l0

2 61 0 . 9 1 01 .8 161 560 43813.3 0

Feasible point D 2 3 7 3 .3 186.7 20I 560 93.3 8I 746 .7 41844.4 0


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Solut ion

be considered as a fa i r ly large highly non- l inear non-convex p roblem , s ince the mater ia l balances conta in b i -and t r i l inear terms and the react ion ra te terms, vapor-l iquid equi l ibr ia evaluat ions and ob ject ive funct ion arehighly non- l inear. This problem can be solved as asuccess ion o f up to 20 NLP subprob lems s imply bydirect enumerat ion of the number of theoret ica l t rays .This w as eventual ly perform ed in order to determine theglobal opt imu m, s ince i t was unknown.

Wi th the M-SIMPSA a lgor i thm, the s t a r t ing s im-plexes were generated f rom random ini t ia l solut ionvectors of both the cont inuous and d iscre te var iables , i .e .mo st ly f rom infea s ible points . Cir ic and Gu (1994) s ta tethat the general ized Benders decomposi t ion s implyrequires se lec t ing an in i t ia l solut ion vector of thediscre te var iables , i .e . the number of theoret ica l t rays .How ever, an in i t ia l feas ible se t of cont inuous var iablesseems to be qui te helpful for solving th is par t icular

problem , and the pro cedure proposed by these authors ismore eff ic ient when s tar ting wi th a s ingle t ray column.Such informat ion does not seem to be helpful wi th theM-SIMPSA algor i thm, s ince th is a lgor i thm ini t ia l lyaccepts poore r solut ions wi th a h igh prob abi l i ty, quicklydepar t ing f rom the in i t ia l point to search the var iablespace.

Cir ic and Gu (1994) have found that the bes t solutioncorresponds to 10 theoret ica l t rays , wi th a to ta l cos t of

o f MINL P 1359

15.69 106/yr. These authors have used addi t ionalconst ra in ts on m aximu m m olar f low ra tes of 1000 kmol /h (Cir ic , 1995) . We have s imulated thei r solut ion andfound very s imi lar resul ts , inc luding temperature andcomposi t ion prof i les . However, opt imizat ion wi th theM-S IMP SA a lgor ithm inc lud ing the above cons t r a in t sproduced an object ive funct ion of 15.40 106/yr, a lsocorresponding to 10 t rays but to to ta l ly d i fferent feedtray locat ions and hold-up ra t ios .

The so lut ion ob ta ined wi th the M -SIM PSA a lgor ithmis s l ight ly bet ter than that g iven by C ir ic and G u (1994)and corresponds to ac t ive const ra in ts on the maximummolar flow rates. Thus, these were relaxed in an attemptto obta in bet ter solut ions whi le re ta in ing feas ible columndesigns (Douglas, 1988). Fig. 4 shows the resultsob ta ined wi th the app l i ca tion o f the M -SIM PSA a lgo-r i thm, where the bes t opt imu m correspon ds to e ight t raysand 15.21 106/yr. Here , 22 ful l MIN LP opt imizat ion s

(s tar t ing f rom random points) were performed. Sincevar ious MINLP solut ions corresponding to d i fferentnumber of t rays were very c lose , indicat ing a very f ia tobject ive funct ion prof i le , sys temat ic NLP searchesus ing the S IM PSA op t imize r were pe r fo rmed by d i r ec tenumera t ion o f the number o f t r ays . The bes t NLPsolut ion corresponds to seven t rays and an object ivefunct ion of 15.12 106/yr, i .e. only 0 .6% bet ter thanthe bes t resul t obta ined wi th the fu l l MINLP formula-

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Fig. 4. Objective function values for M-SIMPS A and SIMP SA algorithms (Example 12).


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1360

t ion. This f igure a lso shows the resul ts obta ined wi th theMSGA a lgor i thm (14 runs ) where the bes t r e su l tcorrespond ing to 15.2 6 106/yr i s very c lose to thebes t ob ta ined wi th the M-S IMP SA a lgori thm. However,most resul ts are poorer than those obta ined wi th theproposed a lgor i thm.

I t can be seen tha t t he M-SIMPSA a lgor i thm wascapable of f inding very g ood solut ions in a lm ost a l l runs ,wi th only four runs above 16 106/yr. Also , a lmost a l lMINLP resul ts are wi thin the range obta ined wi th thesystemat ic search for the corresponding num ber of t rays .Fig . 5 compares the bes t resul ts obta ined f rom thesystemat ic NLP search wi th the bes t resul ts obta inedfrom the M INL P search, where a l l d i fferences are wi thin1%.

Fig . 6(a) and (b) show typical t ra jec tor ies for the bes tobject ive funct ion values and the number of theoret ica lt rays , for two ful l MINLP runs . I t i s apparent that thediscre te space is searched as the opt imizat ion proceeds ,providing opp or tuni t ies for locat ing the glob al opt imum .Fig . 7 shows typical temperature t ra jec tor ies , whichindica te a high insens it ivity to the init ial conditions, i .e.the in i t ia l s implexes generated around the infeas iblerandom s tar t ing solut ion vectors . Also , the f inal anneal-ing temperatures a t ta ined are essent ia l ly independent ofthe f inal solut ion vector, despi te the occurrence o f verydifferent solut ion t ra jec tor ies .

M. E CARDOSAe t a l .

5 C o n c l u s i o n s

An a lgor i thm (M-SIMPSA) i s p re sen ted fo r mixedinteger non- l inear opt imizat ion. A recent ly proposedcon tinuous non- l inea r so lve r (S IMPSA, Cardosoe t a l . ,1996) i s used in an inner loop to update the cont inuousparameters . This i s based on a scheme p roposed by Pressand Teukolsky (1991) that combines the s implex m ethodof Nelder and Mead (1965) wi th s imulated anneal ing.The Metropol is a lgor i thm (Metropol ise t a l . , 1953;Kirkpatr icke t a l . , 1983; Kirkpatrick, 1984) is then usedin an outer loop to update the com plete solut ion vector ofdecis ion var iables . The S IM PS A solv er has the abi l i ty todeal wi th arbi t rary const ra in ts through subst i tu t ion ofinfeas ible points by rando mly generated po ints centeredon the bes t point o f the current s implex.

To ensure f inal convergence of the evolving s implexand a t the same t ime to avoid too fas t convergencetowards a local opt imu m from which i t may not recover,the SIM PSA o pt imizer toggles between a search wi th theglobal in tervals and a search wi th comp ressed in tervals(Cardosoe t a l . , 1996) , af ter which compar ison of thediscre te conf igurat ions i s made wi th the Metropol iscr i ter ion. The compress ion of the search in tervalsfol lows the current decrease in the temperature controlparameter, a long the Aars t and van Laarhoven (1985)cool ing schedule . The s tochast ic genera t ing scheme

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13/16

Solution of MINLP 1361

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Fig . 6 . (a) Typical t ra jec tor ies for M-SI MP SA (E xam ple 12, N f inal= 12) ; (b) typica l t ra jec tor ies for M-S IM PS A (E xam ple 12, Nf inal= 13).


14/16


15/16

Solution of M IN LP 1363

T i

v .

YYaux

Yi - I

Yo t a

cycle time (h) of product i (i= I,N)capa city (i) of parallel u nit in processing stagej (j= I,M)continuous decision v ectordiscrete decision vectordiscrete variable temporarily retained fo r com -parison purposesdiscrete variable input to the SIM PS A algo-rithmcurrent va lue for discrete variablebinary variable

Gree k l e t te r sai lower bound on continuous parameter x

(i = t,p)aj cos t coefficient ( ) of stage j (]= 1,M)fll upp er bou nd on continuous parameter x~

(i = I,p)

flj cos t coefficient (dimensionless) of stage j(j= I,M)

y~ low er bou nd on discrete parameter Yi (i= l,n -P )upper bound on discrete parametery i ( i =l , n -P)

Abbrev ia t ionsMS GA Minlp Salcedo-Go nqalves-Azeved o algo-

rithmM IN LP mixed integer non-linear program mingM -SIM PSA M inlp simplex-simulated annealing algo-

rithmNL P non-linear program mingSIM PS A simplex-simulated annealing algorithm

c k n o w l e d g e m e n t s

This work was partially supported by JNICT (Portu-gue se Junta Nacional de InvestigagAo Cientffica eTecnol6gica), u nde r Co ntract No. BD/1457/91-RM , andPRA XIS X XI, under Contract No. BD/3236/94, employ-ing the computational facilities of Instituto de Sistemas eRobttica (ISR), Porto. The authors thank Dr. J. Bastosfor provid ing crucial parts of the code needed to simulatethe distillation process.

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