Dscrete Optmzato 5 (2008) 08 20 www.elsever.com/locate/dsopt Global soluto of optmzato problems wth sgomal parts ay Pör a, Kaj-Mkael Björk b,, Tapo Westerlud c a Sector for Techology ad Commucato at Swedsh Polytechc, Wollfskaväge 33, PB 6, FIN-6520 Vasa, Flad b IAMS at Åbo Akadem Uversty, Lemmkäsek. 4, FIN-20540 Turku, Flad c Process Desg Laboratory at Åbo Akadem Uversty, Bskopsgata 8, FIN-20500 Åbo, Flad eceved 8 October 2004; receved revsed form 5 November 2007; accepted 25 November 2007 Avalable ole 3 December 2007 Abstract I ths paper a ew approach for the global soluto of ocovex MINLP (Mxed Iteger NoLear Programmg) problems that cota sgomal (geeralzed geometrc) expressos s proposed ad llustrated. By applyg dfferet varable trasformato techques ad a dscretzato scheme a lower boudg covex MINLP problem ca be derved. The covexfed MINLP problem ca be solved wth stadard methods. The key elemet ths approach s that all trasformatos are appled termwse. I ths way all covex parts of the problem are left uaffected by the trasformatos. The method s llustrated by four example problems. c 2007 Elsever B.V. All rghts reserved. Keywords: Covexfcato; Global optmzato; Mxed teger olear programmg; Sgomals; Varable trasformatos. Itroducto Optmzato problems that cota sgomal expressos occur frequetly may areas of egeerg ad process sythess. A sgomal cossts of a sum of terms of the form ax r, a, r,..., r. The defto set s usually +. A term wth postve sg (a > 0) s called a posyomal term ad a fucto that cossts of a sum of postvely sged terms s called a posyomal. Thus, by groupg together terms wth detcal sg a sgomal fucto ca be wrtte as a dfferece betwee two posyomals. The type of optmzato problem cosdered ths paper has the property that the objectve fucto ad all equalty costrats ca be decomposed to a covex part ad a sgomal part. All equalty costrats should be lear. Techcally, ths s ot a strget requremet sce a equalty costrat wth sgomal parts ca be rewrtte as two sgomal equalty costrats. The focus ths paper s o the costructo of trasformatos, whch covexfes the orgal ocovex problem wthout troducg ay addtoal ocovexty to the problem. Formally, the proposed trasformatos are of the Correspodg author. E-mal addresses: ray.por@syh.f (. Pör), kbjork@abo.f (K.-M. Björk), twesterl@abo.f (T. Westerlud). 572-5286/$ - see frot matter c 2007 Elsever B.V. All rghts reserved. do:0.06/j.dsopt.2007..005
form (P) MIN f (z) + S 0 (z) s.t. g (z) + S (z) 0 Az = a l z u {, 2,..., N}. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 09 MIN s.t. ( P cov) f (z) + S0 cov (z, z) g (z) + S cov (z, z) 0 Az = a T [ z, z ] T = t l z u, l z ũ {, 2,..., N}, where S (z) represet the sgomals. The fuctos f (z), g (z) are assumed to be oce dfferetable ad covex o the hyper-rectagle defed by the bouds l z u. A ad T are matrces ad a ad t colum vectors of approprate dmesos. The varables z ca both be cotuous ad teger. If the fuctos f (z) ad g (z) have o terms, the equaltes Az = a are ot preset ad all fuctos S (z) are posyomals, problem (P) correspods to a classcal geometrc program. The addtoal varables z are lked to the orgal varables z through the lear dscretzato costrats T [ z, z ] T = t. The trasformatos should be chose so that all ocovex fuctos (P) wll be covex (P cov ). The covexfed sgomals are called S cov (z, z). Note that the covex ad lear parts (P) are left uaffected by the covexfcato, whch s dfferet from basc Geometrc Programmg (GP) techques. For pure teger problems the covexfcato results a equvalet covex problem. Some results for pure teger problems are gve, for example, [ 3]. I ths paper we wll cocetrate o the covexfcato of cotuous problems. I ths case the covexfcato procedure wll lead to a approxmate ad uderestmatg covex MINLP problem. Optmzato problems that cota oly sgomal expressos are usually called Geeralzed Geometrc Programmg (GGP) problems. GGP problems are a subclass of problem class (P). GGPs was frst studed by [4]. Ulke GP (Geometrc Programmg) problems, that cota posyomals oly, GGP problems rema ocovex both ther prmal ad dual represetatos. May local optmzato approaches have bee developed for the soluto of GGP problems. Some examples from the lterature are [5 7]. A computatoal study of local GGP codes s reported [8]. Specalzed global optmzato approaches for GGP problems are scarce. Falk [9] ad Maraas ad Floudas [0] proposed specal purpose brach- ad boud-based global optmzato methods for the soluto of GGP problems. Both methods use the expoetal varable trasformato ad covex uderestmato. The partto ad brachg scheme used the methods dffer. To the authors kowledge, there exsts o global optmzato method, whch s specally desged to hadle the structure of the mportat problem class (P) cosdered ths paper. 2. Trasformato techques The basc dea the covexfcato scheme s the costructo of varable trasformatos that covexfy sgomal terms. Frst we eed some results from covex aalyss. Property. The fucto ae {r x +r 2 x 2 + +r x } s covex o + f a 0, r. Property 2. The fucto a e{r x +r 2 x 2 + +r x} x s x s 2 2 x s s covex o + f a, s 0, r. Property 3. The fucto ax r s covex o + f a 0, r 0 ad = = r. The frst property s elemetary ad the thrd was orgally show [] a slghtly dfferet form but t s also foud [2,3]. The secod s proved Appedx A. The expoetal term Property s a specal case of the expresso Property 2. The dea s to trasform a geeral sgomal term ax r, a, r,..., r to oe of the types gve above. Postvely sged sgomal terms are trasformed to oe of the frst two types ad sgomal terms wth egatve sg are trasformed to the thrd type. For smplcty we reorder the varables pror to the trasformato so that varables wth postve expoets appear frst the term ad varables wth egatve expoets follows,.e. r,..., r m > 0 ad r m+,..., r < 0. Postvely sged terms (a > 0). New varables X are troduced accordg to x = e X, =, 2,..., m. The followg equvalece s the establshed.
0. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 ax r x = e X,. x r m+ m+ x r a er X + +r m X m =,..., m The latter term o the rght-had sde s ow covex the jot X X m x m+ x -space accordg to Property 2, whereas the trasformato costrat remas ocovex. Ths trasformato s called the expoetal trasformato (ET) ad orgs from GP. I GP all varables a posyomal term are trasformed, regardless of the sg of the expoets. Fewer trasformatos are eeded f the covexty of the expresso Property 2 s used. A alteratve trasformato that covexfes a posyomal term wth postve sg s the verse trasformato(it). New varables are ths case troduced through the equalty x = X, =, 2,..., m ad the resultg equvalece becomes ax r x =, X =,..., m a X r Xr m m x r m+ m+ x r. Also ths case the covexfed term s covex accordg to Property 2 (a specal case where the umerator s a costat). Negatvely sged terms (a < 0). New varables are troduced by x = X, =,..., m ad x = X, = m +,..., where = = r. The trasformato results the equvalece ax r x = X, =,..., m; x = X, = m +,..., r a X X rm r m+ m Xm+ X r (PT). If <, oly the varables wth egatve expoets are trasformed accordg to the suggested scheme. The term o the rght-had sde s ow covex due to Property 3, sce all expoets are postve ad ther sum equals. Ths trasformato s referred to as the potetal trasformato (PT). Ths trasformato dffers from the two gve above, sce t s depedet o the term that s covexfed. Dfferet terms have dfferet -values. (ET) (IT) 3. Dscretzato ad covexfcato If the trasformatos are used ths form a optmzato problem a dlemma becomes evdet. The sgomal term has bee covexfed, but the ocovexty has moved to the trasformato costrats stead. Nolear equalty costrats are lkely to cause multple local optma optmzato problems. To avod ths problem a approxmato scheme s proposed. The oe-dmesoal trasformato costrats are approxmated o closed tervals wth pecewse lear fuctos. The pecewse lear approxmato ca the be modeled learly by usg 0 varables ([4], chapter 9, [5], chapter 7, [6] or [7] for stace). I the followg, a well-kow formulato of the pecewse lear fucto wll be preseted. The formulato may ot always be the best oe, but s wdely used the lterature; for a alteratve formulato, see [8], for stace. Suppose a pecewse lear fucto pl(x) has break pots p, p 2,..., p k. If x [p, p k ] there exsts some j wth p j x p j+. The for some real umber λ j [0, ] x ca be wrtte as x = λ j p j + ( λ j )p j+. The t holds that pl(x) = λ j pl(p j ) + ( λ j )pl(p j+ ). By assocatg a bary varable β j wth each terval [ p j, p j+ ] the pecewse fucto ca be represeted learly as pl(x) = λ pl(p ) + λ 2 pl(p 2 ) + + λ k pl(p k ) x = λ p + λ 2 p 2 + + λ k p k λ β, λ 2 β + β 2,..., λ k β k 2 + β k, λ k β k β + β 2 + + β k = λ + λ 2 + + λ k = β j {0, }, λ j [0, ]
. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 Fg.. Pecewse lear approxmato of a covex ad a cocave fucto. Ths set of varables ad costrats represet a specal ordered set of type 2 (SOS 2). I may moder MILP solvers ( [9 2]) t s possble to declare specal ordered sets explctly. If the uderlyg solver supports explct SOS declaratos the pecewse learzato of the fucto pl(x) ca be rewrtte compact form as pl(x) = λ pl(p ) + λ 2 pl(p 2 ) + + λ k pl(p k ) x = λ p + λ 2 p 2 + + λ k p k λ + λ 2 + + λ k = λ j [0, ] At most two adjacet λ are ozero. I the latter formulato the bary varables β ca be omtted, sce the logc s hadled by the solver usg SOS rules. A cotuous oe-dmesoal fucto f (x) ca be approxmated o a closed terval wth a pecewse lear fucto whch cocdes wth f (x) at least at some gve break pots. Note that the formulato above s ot optmal for performace, but s used for llustrato purpose oly. Ay pecewse lear formulato wll work fe ad SOS 2 varables could preferably be used to expedte the calculatos (see [22,23] for stace). A pecewse lear fucto wll uderestmate a cocave fucto ad overestmate a covex fucto. See Fg.. The olear trasformato costrats for the sgomals are frst rewrtte as (ET): X = l(x )(cocave) (IT): X = x (covex) (PT): X = x, X =, > (covex). x These costrats are the approxmated wth the pecewse lear expresso gve above. A separate set of costrats s eeded for every varable the sgomal term. The varables troduced the covexfcato procedure of a geeral sgomal term ca ow be wrtte as z = (X, λ j ). The varables X ad λ j are called SOS2 varables ad the costrats troduced the approxmato are called pecewse learzato costrats. The ET ad IT techques are llustrated the example below. Example. Ths s a classcal blear problem from [24]. MIN xy x y s.t. 6x + 8y 3 3x y 3 0 x, y.5, x, y. Both varables partcpate the postvely sged posyomal x y. The break pots are set to 0.0, 0.5,.0,.5 for both x ad y. Together wth a traslato (+), thus avodg problems at x = 0 ad y = 0, the expoetal trasformato s { x + = e X, y + = e Y xy e X+Y x y.
2. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 It s also possble to approxmate zero wth a small ε > 0. Followg the gudeles for the pecewse lear approxmato the followg approxmate covex uderestmatg MINLP problem s obtaed MIN e X+Y 2x 2y s.t. 6x + 8y 3, 3x y 3 x = 0.5λ2 x +.0λx 3 +.5λx 4, X = l(.5)λx 2 + l(2.0)λx 3 + l(2.5)λx 4 ( l(x)) y = 0.5λ y 2 +.0λy 3 +.5λy 4, Y = l(.5)λy 2 + l(2.0)λy 3 + l(2.5)λy 4 ( l(y)) λ x + λx 2 + λx 3 + λx 4 =, λy + λy 2 + λy 3 + λy 4 = At most two adjacet λ x are ozero At most two adjacet λ y are ozero x, y [0,.5], λ x j, λy j [0, ]. The trasformed objectve s ow covex the jot X Y x y-space. The olear trasformato costrats that are approxmated are X = l(x) ad Y = l(y). The global optmal soluto reported [24] s (x, y) = (7/6, 0.5) wth optmal objectve value.0833. Whe the approxmate covex MINLP s solved wth the ECP method [25] the soluto (.43, 0.3429) wth objectve value.35 s obtaed. It s also worth otcg that oe of the break pots the pecewse learzato cocdes wth the soluto. If the soluto values were foud oly at some break pots (ad ot betwee), the soluto would be feasble ad optmal the orgal problem (P) as well. Now, sce oe soluto value s foud at the break pot ad oe betwee, the covexfed approxmate problem wll uderestmate the orgal ocovex problem. If IT s used the objectve ad the fourth ad sxth costrats are replaced by XY 2x 2y + ; X =.5 λx 2 + 2.0 λx 3 + 2.5 λx 4 ; Y =.5 λy 2 + 2.0 λy 3 + 2.5 λy 4. I ths case the soluto to the approxmate MINLP s (.2393, 0.780) wth objectve value.98. Note that the soluto to both approxmate covex problems uderestmates the soluto to the orgal blear problem ad that IT gave a looser uderestmate tha ET, whch s, fact, ofte the case. The error both the trasformatos s derved Appedx B. 4. Propertes of the approxmate covex MINLP Some mportat propertes for the approxmate problem are gve the theorem below. All the orgal varables are cotaed the vector z ad all the varables troduced the covexfcato are called z. Each SOS2 varable s assumed to be lked to the orgal varable through a set of pecewse learzato costrats. Theorem. The followg propertes hold for the ocovex problem (P) ad the covexfed problem (P cov ) (whe (P cov ) cludes the pecewse lear reformulatos of the trasformato costrats). () Every covexfed term uderestmates the correspodg sgomal term. That s () a > 0 a er X +r 2 X 2+ +rm Xm (2) a > 0 x r m+ m+ x r a ax r X r Xr 2 2 Xr m m x r m+ m+ x r r m+ X ax r r r2 (3) a < 0 a X X 2 X rm m m+ X r ax r () The covexfed fuctos uderestmate the orgal fuctos. That s. S cov (z, z) S (z), = 0,,..., N, z, z wth l z u, l z ũ. () The feasble rego (P) s a subset of the feasble rego (P cov ). (v) Let z be the global soluto to (P) ad (zcov, z cov ) the soluto to (Pcov ). If zcov s feasble (P) the f (zcov ) + Scov 0 (zcov, z cov ) f (z ) + S 0 (z ) f (zcov ) + S 0(zcov ). If zcov s feasble (P) the oly the frst equalty s vald.
. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 3 Proof. Case. Frst we cosder terms wth a > 0. Case.. I the expoetal trasformato of a posyomal term oly varables wth postve expoets are trasformed. Each factor x r, =,..., m s replaced by the expresso e r X the covexfed term. Due to the cocavty of the l-fucto the pecewse lear approxmato X wll uderestmate l(x ). Sce r 0 we obta e r X e r l(x ) = x r. Ths holds for all factors =,..., m. Sce a, r 0 we obta a e r X + +r m X m x r m+ m+2 m+ x r m+2 x r ax r. Hece, a term covexfed wth ET uderestmates the posyomal term. Case.2. I ths case each factor x r, =,..., m s replaced by the expresso X r. Due to the covexty of the verse fucto the pecewse lear ( ) approxmato X wll overestmate x. Sce r 0 we obta X r r x X r x r. Ths holds for all factors =,..., m. Sce a, r 0 we obta the equalty a X r Xr m m x r m+ m+2 m+ x r m+2 x r ax r. Hece, a term covexfed wth IT uderestmates the posyomal term. Case.3. Negatvely sged terms are ow cosdered (a < 0. I ths case each factor x r, =,..., m s replaced by the expresso X x r r ad each factor, = m +,..., s replaced by X r. Sce both the trasformato costrats PT are covex fuctos the pecewse lear approxmato wll be a overestmate. For =,..., m we have that X = m +,..., we have that X r r ( ) r = X we obta the equaltes r X X rm m r m+ X m+ X r x r x = x r r ax X rm m r (x ) r = x r ad for = x r. Sce ths holds for all factors =,..., r m+ X m+ X r ax r Hece, a term covexfed wth PT uderestmates a egatvely sged sgomal term. Case. Sce every covexfed term uderestmates the correspodg ocovex sgomal term t s clear that each covexfed fucto S cov (z, z) (P cov ) uderestmates the correspodg sgomal fucto S (z) (P). Case. Take a feasble pot z arbtrarly ( P). For every choce of z the trasformato varables z obta uque values from the pecewse learzato costrats. Accordg to case above we get for every =,..., N 0 g (z) + S (z) g (z) + S cov (z, z), whch mples that the feasble rego (P) s a subset of the feasble rego (P cov ). Case v. Assume that zcov s feasble (P). Sce z s optmal (P) we mmedately get that f (z ) + S 0 (z ) f (z cov ) + S 0(z, z cov ). Accordg to the uderestmato property t follows that f (z ) + S cov 0 (z, z ) f (z ) + S 0 (z ) ad sce (z cov, z cov ) was assumed optmal (Pcov ) we obta f (z cov ) + S 0(z cov, z cov ) f (z ) + S cov 0 (z, z ). O the other had, f zcov s feasble (P) the the rght equalty may be vald. The theorem states that all covexfed fuctos (P cov ) are uderestmators ad the feasble rego (P cov ) s a overestmate. Pot v Theorem ca be used to boud the global soluto to the ocovex problem (P). The optmal objectve value to the covex MINLP s a lower boud for the global soluto to (P). A upper boud ca be computed as a evaluato of the objectve (P) at the soluto to the uderestmatg covex MINLP problem..
4. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 After the soluto of the frst approxmate problems Example t ca be cocluded that the sought global optmal objectve value to the blear problem les the terval [.35,.075] f ET s used ad the terval [.98,.0675] f IT s used. 5. Descrpto of the global optmzato algorthm A ovel determstc global optmzato algorthm for problem class ( P) ca easly be costructed by solvg a sequece of uderestmatg covex MINLP problems (P cov ). The qualty of the uderestmato wll prmarly deped o the desty of the grd used. If the gap betwee the upper boud ad lower boud s cosdered usatsfactory ew grd pots are sequetally added to the prevous grd ad the updated tghter uderestmatg covex problem s resolved. The steps of the algorthm are gve below.. Pre-processg Determe tght lower ad upper bouds for all varables that partcpate the sgomal expressos. Let SI be the dex set of all cotuous varables that partcpate the sgomals ad N cov the dex set of all pure covex costrats (P). Tght bouds o all varables ca effcetly be obtaed by solvg a umber of covex boudg problems of the form l k := MIN z k s.t. g (z) 0 Az = a l z u N cov u k := MAX z k s.t. g (z) 0 Az = a l z u N cov, for every k SI. Ths results a tght hyper-rectagle that cotas the feasble rego defed by all lear ad covex costrats. The same boudg techque s also frequetly used the pre-processg step of determstc global optmzato methods, for example [3,27 30]. Tght bouds ca also be obtaed by applyg smple terval aalyss of the costrats or by specto. 2. Covexfcato Defe tal dscretzato grds for all varables that partcpate the sgomals. Covexfy problem ( P). Let z = (X, λ j ). 3. Soluto Solve the covexfed problem (P cov ) by ay MINLP method sutable for ths purpose. Call the obtaed soluto (zcov, z cov ) ad the objectve value Z cov. 4. Boudg Due to the uderestmatg property of (P cov ) Z cov s a lower boud for the global soluto to (P). Set L B = Z cov. A upper boud, UB, ca be costructed several ways. () Evaluate the objectve (P) at the soluto to the covexfed problem (zcov, z cov ). Call ths value Z eval ad set U B = Z eval. () Apply a local NLP/MINLP method usg zcov as startg pot. Call the result Z loc. A vald upper boud ca ow be set to U B = Z loc. 5. Termato Termate f U B L B < tol or f the soluto s suffcetly close to a exstg grd pot. 6. Updatg the grd The dscretzato grd ca be updated several ways. Add the () soluto to the covexfed problem to the grd. (2) soluto to the local search to the grd (f t s ot already cluded). (3) mdpot of all tervals for whch the soluto to the covexfed problem does ot le at a grd pot. New grd pots are added for each varable that s ot suffcetly close to a exstg grd pot. Go back to step 3. To obta a more detaled proof of covergece ad the covergece propertes of the method, the reader s referred to [25]. I classcal brach ad boud methods for global optmzato the partto of the space s doe for oly oe varable at a tme. I ths method several ew grd pots are added each terato,.e. the partto s doe several dmesos. If updatg alteratve s used the grds are updated for all varables that do ot le exactly at a exstg grd pot at the soluto to the covexfed problem. Covergece for the method ca geerally be esured by
. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 5 perodcally usg updatg alteratve 3. Ths follows from the compactess ad cotuty assumpto of problem (P). Oce aga we retur to Example ad the ET approach. If updatg alteratve s used the pot (.43, 0.3429) s added to the grd ad the problem s resolved. The ew costrats are X = l(.5)λ x 2 + l(2.0)λx 3 + l(2.43)λx 4 + l(2.5)λx 5 Y = l(.3429)λ y 2 + l(.5)λy 3 + l(2.0)λy 4 + l(2.5)λy 5. The NLP-heurstc cludes the pot (7/6, 0.5) the grd ad alteratve 3 (mdpot) the pot (2.25,.25). The updated MINLP has four addtoal varables, two bary ad two cotuous, ad two addtoal equalty costrats. 6. Test problems I ths chapter, three examples are gve to llustrate the global optmzato procedure. I all the examples, the Exteded Cuttg Plae algorthm [26] has bee used to solve the covex uderestmatg MINLP problems. Example 2. Ths example s take from [3]. The problem cossts of mmzg the total heat exchager area-cost for a specfc structure. Covergece to the global optmum s ot guarateed for tradtoal NLP solvers sce the objectve fucto cossts of lear fractoal terms. All costrats are lear. q MIN 270 + 720 + 240 q 3 + 900 q 4 0. t 0. t 2 t 3 t 4 s.t. q = 5.555 (t 395); q 2 = 3.25 (t 2 398) q 3 = 4.545 (t 2 365); q 3 = 5.555(575 t ) q 4 = 3.57 (t 4 358); q 4 = 3.25 (78 t 2 ) q + q 2 = 000 q 2 t = t 305 ; t 2 = t 2 302 2 2 t 3 = t t 3 + 20 ; t 4 = t 2 t 4 + 360 2 2 405 t 575, 405 t 2 78, 365 t 3, 358 t 4 q 0, t 5, =,..., 4. If the varables q are trasformed to e Q for =, 2, 3, 4, respectvely, the objectve fucto s covex accordg to Property 2. The approxmate covexfed problem wll be the orgal plus the costrats cotag the pecewse approxmato of Q = l(q ); =, 2, 3, 4 ad the objectve fucto replaced by e Q e Q 2 MIN 270 + 720 + 240 eq3 + 900 eq4. 0. t 0. t 2 t 3 t 4 Pecewse lear fuctos three steps (four grd pots) were used to solve the frst covexfed subproblem. Each subproblem correspods to a covex MINLP program. The soluto from the covexfed problems was subsequetly added as ew grd pots to make the approxmato more accurate (updatg alteratve ). The upper boud was obtaed by smply evaluatg the ocovex objectve fucto at the soluto to the covexfed MINLP problem. The global optmum was foud the sxth subproblem wth a objectve value of 3663. The terato path s llustrated Fg. 2. If IT s used stead of ET Example 2 there s a eed for solvg sgfcatly more covexfed subproblems, sce IT geerally geerates poorer lower bouds tha ET. The terato path for IT s llustrated Fg. 3. A vestgato of the uderestmato qualty of ET ad IT s made Appedx B. Some geeral error estmates are derved [32]. Example 3. Ths example s a ocovex MINLP foud [33]. The ocovextes are located the two equalty costrats. Each of these costrats s frst rewrtte as two equaltes ad the resultg two cocave costrats are fally covexfed. The ocovextes le etrely the cotuous space.
6. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 Fg. 2. Iterato path ad a comparso betwee the szes of the orgal problem ad the last subproblem for Example 2. Fg. 3. Iterato path ad a comparso betwee the szes of the orgal problem ad the last subproblem for Example 2 solved wth IT. MIN 2x + 3x 2 +.5y + 2y 2 0.5y 3 MIN 2x + 3x 2 +.5y + 2y 2 0.5y 3 s.t. x 2 + y =.25 s.t. x 2 + y.25 x2.5 +.5y 2 = 3 x2.5 +.5y 2 3 x + y.6 P X y.25.333x 2 + y 2 3 X 2.5y 2 3 P CONV y y 2 + y 3 0 x + y.6 x, x 2, x 3 0.333x 2 + y 2 3 (y, y 2, y 3 ) {0, } y y 2 + y 3 0 X = pl (x ), =, 2 x, x 2 ε; X ε 2 ; X 2 ε.5 (y, y 2, y 3 ) {0, } where X = pl (x ) represet the pecewse learzatos wth some sutable grd pots. The trasformatos used are X = x 2, X 2 = x2.5. I ths problem, o upper bouds were calculated at all. O the other had, the soluto approach used cossted of subsequetly solvg the problem (P cov ) where the soluto of the prevous (P cov ) were added to the set of grd pots the pecewse lear approxmatos of the trasformato costrats. The soluto to the ffth covex subproblem troduced o ew grd pots (t was exactly at some old grd pots), whch mpled that the soluto was feasble the orgal problem ad, hece, optmal. The global optmal objectve s 7.67. Example 4. The last example s take from [34] ad cludes blear equalty ad equalty costrats. MIN 6x + 6x 2 9x 5 + 0x 6 5x 9 s.t. x + x 2 x 3 x 4 = 0; x 3 x 5 + x 7 = 0 x 4 + x 8 x 9 = 0; x 6 + x 7 + x 8 = 0 2.5x 5 + 2x 7 + x 3 x 0 0 2x 8.5x 9 + x 4 x 0 0 3x + x 2 x 3 x 0 x 4 x 0 = 0 x 0, =,..., 9; x 0 x (300, 300, 00, 200, 00, 300, 00, 200, 200, 3).
. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 7 The blear equalty costrat s relaxed to two equaltes pror to soluto. If o upper boudg procedure was used, the global optmum (wth a objectve value of 400) was foud after the soluto of covexfed subproblems. After the th terato o ew grd pots were added. If a NLP search s used as a upper boudg procedure after the frst covexfed subproblem the global optmum was, fact, obtaed as the upper boud. The local NLP soluto was added as a ew grd pot, after whch the lower boud solutos form the foudato of the ew grd pots the followg teratos. Havg a good upper boud decreased the umber of teratos from to 9. More small ad medum scale examples are solved [32]. It s worth otcg that ths method requres oly oe trasformato for each varable that s foud a ocovex costrat. Ths meas that eve f the umber of ocovex costrats s large, but the umber of the varables the costrats s small, the proposed method has good possbltes to solve the problem, for further detals, see [25]. 7. Dscusso I ths paper a ovel reformulato techque for NLP ad MINLP problems, whch cota sgomal expressos, was preseted. It was show that, gve ay ocovex optmzato problem wth a objectve fucto ad equalty costrats that ca be decomposed to a covex ad sgomal part, t s possble to costruct a correspodg covex uderestmatg MINLP problem. For ths purpose, three dfferet trasformato techques were derved ad llustrated. The expoetal trasformato (ET) ad the verse trasformato (IT) were applcable to postve terms ad the potetal trasformato (PT) was used o egatvely sged terms. The olear trasformato costrats were dscretzed order to obta a pecewse lear formulato. The qualty of the uderestmato produced by the ET ad IT was also vestgated. It was foud that ET usually produces a tghter uderestmate tha IT. Ths cocluso s also supported by the boudg the umercal examples. The reformulato could further be used for the costructo of a teratve global optmzato procedure. The tal dscretzato s subsequetly updated order to tghte the uderestmatg problem. The dscretzato could be updated several ways. These updatg rules correspod to dfferet rectagular partto schemes used brach ad boud methods for global optmzato. Fally, the method was used to successfully solve four example problems from the lterature. More work should be doe to make the soluto of the covex MINLP problems less expesve. At the momet, every MINLP problem s solved from scratch. It would be preferred to collect formato after each subproblem ad corporate ths the soluto scheme. For example, t would be possble to keep some mportat cuts (from the ECP method) derved prevous teratos ad use them subsequet oes. Aother mprovemet would be to use formato drectly from dfferet odes the brach ad boud three to reduce the combatoral space the followg subproblems,.e. to dscard such regos from the tal feasble rego, whch caot cota the optmal soluto. Appedx A Proof of Property 2. Theorem. The fucto a e{r x +r 2 x 2 + +r x} x s x s 2 2 x s Proof. The deomator ca be rewrtte as a e{r x +r 2 x 2 + +r x } x s xs 2 2 xs = a s covex o + f a, s 0 ad r. e {r x +r 2 x 2 + +r x } e {s l(x )+s 2 l(x 2 )+ +s l(x )} = ae {r x +r 2 x 2 + +r x } e {s l(x )+s 2 l(x 2 )+ +s l(x )} = ae {r x +r 2 x 2 + +r x s l(x ) s 2 l(x 2 ) s l(x )} The l-fucto s cocave o +. The t follows that the fucto r x + r 2 x 2 + + r x s l(x ) s 2 l(x 2 ) s l(x ) s covex o + sce s 0. Sce a 0 ad the exp-fucto s covex ad creasg o t follows from stadard covex aalyss [35] that the expresso Property 2 s a covex fucto o +.
8. Pör et al. / Dscrete Optmzato 5 (2008) 08 20 Fg. 4. ET uderestmato errors for a blear term the box [, 4] [3, 5]. Appedx B Fg. 5. IT uderestmato errors for a blear term the box [, 4] [3, 5]. Here we vestgate the qualty of the covex uderestmato geerated by ET ad IT. We study a specal case of a posyomal, amely the blear term x x 2 the box [a, b ] [a 2, b 2 ]. However, all results ca be geeralzed to the case of a geeral posyomal term. The error, ε, for the pecewse lear approxmato of the trasformato costrats X = l(x ) s gve by ε (x ) = l(x ) X = l(x ) (α x + β ); α = l(b ) l(a ) b a ; β = b l(a ) a l(b ) b a for =, 2. The error betwee the blear term ad the correspodg covexfed term at a arbtrarly pot (x, x 2 ) the grd box [a, b ] [a 2, b 2 ] s the gve by the expresso ET (x, x 2 ) = x x 2 e X +X 2 = x x 2 e α x +α 2 x 2 +β +β 2. By usg smlar argumets the error, ε, betwee the pecewse lear approxmato ad the trasformato costrat X = x s ε = X x = ᾱ x + β x ; ᾱ = a b ; β = a + b ad the error the grd box s gve by I T (x, x 2 ) = x x 2 = x x 2 X X 2 (ᾱ x + β )(ᾱ 2 x 2 + β 2 ). As a llustrato we assume that the box s [, 4] [3, 5]. The error for ET ad IT s calculated at 00 separate pots the gve box. The errors are gve Fgs. 4 ad 5. It s clear that the verse trasformato geerates looser uderestmato ths case. The maxmal error IT s about twce as large as ET. The exact maxmal error ca be obtaed by solvg the correspodg maxmzato problems the gve grd box. Ths results max ET = 2.5392 ad I T max = 5.0273. A drawback wth these trasformatos s clearly that they are exact oly at the extreme pots of the box. If we compare the qualty of
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