Strengthening of Lower Bounds in the Global Optimization of Bilinear and Concave Generalized Disjunctive Programs

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1 Strengthenng of Lower Bounds n the Gbal Optmzaton of Blnear and Concave Generalzed Dsunctve Programs Juan Pab Ruz, Ignaco Grossmann* Department of Chemcal Engneerng, Carnege Meln Unversty Pttsburgh, PA, USA 53 March 009 Abstract Ths paper s concerned wth gbal optmzaton of Blnear and Concave Generalzed Dsunctve Programs. A maor obectve s to propose a procedure to fnd relaatons that yeld strong wer bounds. We frst present a general framewor for obtanng a herarchy of lnear relaatons for nonconve Generalzed Dsunctve Programs (GDP). Ths framewor combnes lnear relaaton strateges proposed n the lterature for nonconve MINLPs wth the results of the wor by Sawaya & Grossmann (008) for Lnear GDPs. We further ept the theory behnd Dsunctve Programmng to gude more effcently the generaton of relaatons by consderng the partcular structure of the problems. Fnally, we show through a set of numercal eamples how these new relaatons can strenghten substantally the wer bounds for the gbal optmum, often leadng to a sgnfcant reducton of the number of nodes when used wthn a spatal branch and bound framewor. *Author to whom correspondence should be addressed

2 - Introducton Generalzed Dsunctve Programmng (GDP), deveped by Raman and Grossmann (994), has been proposed as a framewor that facltates the modelng of dscretecontnuous optmzaton problems by alwng the use of algebrac and gcal equatons through dsunctons and gc propostons that are epressed n terms of Boolean and contnuous varables. In order to tae advantage of estng solvers (Westerlund & Pettersson,995; Vswanathan & Grossmann, 990; Sahnds, 996; Leyffer, 00; Kesavan et al, 004; Bonam et. al, 008), GDPs are often reformulated as MILP/MINLP problems by usng ether the Bg-M (BM) (Nemhauser & Wolsey, 988), or the Conve Hull (CH) (Lee & Grossmann, 000) reformulaton (See Append ). It s mportant to note that GDP problems can always be reformulated as an MINLP. However, these reformulatons are not unque and may have assocated relaatons that are not very tght, consequently havng an adverse effect on the effcency of the algorthm that s used. In general, the tghter the relaaton of the reformulaton and the fewer the number of varables and constrants, the smaller the computatonal effort s. In the partcular case of nonconve GDP problems the drect applcaton of tradtonal algorthms to solve the reformulated MINLPs such as Generalzed Benders Decomposton (GBD) (Benders, 96 and Geoffron, 97) or Outer Appromaton (OA) (Duran & Grossmann, 986), may fal to fnd the gbal optmum snce the soluton of the NLP subproblem may correspond to a cal optmum and the cuts n the master problem may not be vald. Therefore, specalzed algorthms should be used n order to fnd the gbal optmum (Horst & Tuy, 996, Tawarmalan & Sahnds, 00 and Fudas, 000). Nonconve GDP problems wth blnear constrants are of partcular nterest snce these arse n many applcatons, for nstance, n the desgn of poolng problems (Meyer & Fudas, 006), n the synthess of ntegrated water treatment networs (Karpah & Grossmann, 006), or generally, n the synthess of process networs wth multcomponent fws (Quesada & Grossmann, 995). In addton, nonconve GDP problems wth concave constrants frequently arse when nonlnear nvestment cost functons are consdered (Turay & Grossmann, 996). To tacle ths problem, Lee and Grossmann (003) proposed a gbal optmzaton method that frst relaes the blnear terms by usng the conve envepes of McCormc (976) and the

3 concave terms by usng lnear underestmators. The conve hull (Balas, 985) s then appled to each dsuncton. Ths formulaton s then used wthn a spatal branch and bound technque n whch the branchng s frst performed on the Boolean varables folwed by the contnuous varables. Whle the method proved to be effectve n solvng several problems, a maor queston s whether one mght be able to obtan stronger wer bounds to enhance the effcency for gbally optmzng GDP problems. Sawaya and Grossmann (008) have recently establshed new connectons between Lnear GDP and the Dsunctve Programmng theory by Balas (979). As a result, a famly of tghter reformulatons has been dentfed. These are obtaned by performng a sequence of basc steps on the orgnal dsunctve set (.e. each basc step s characterzed by generatng a new set of dsunctons by ntersectng the former), brngng t to a form cser to the Dsunctve Normal Form (DNF), and hence tghtenng ts dscrete relaaton (Balas, 985). It s mportant to note that each ntersecton usually creates new varables and constrants. Therefore, t s mportant to recognze when t may be useful to mae these ntersectons. Some general rules are descrbed n ths wor. In ths wor we buld on the wor by Sawaya and Grossmann (008) eptng the newly dscovered herarchy of relaatons n order to solve more effcently nonconve GDP problems, partcularly, wth blneartes and concave functons n ther constrants, namely Blnear GDP and Concave GDP. Ths paper s organzed as folws. In secton we present the general structure and partcular propertes of the problems for whch we am at fndng better relaatons (.e. Blnear GDP and Concave GDP). In secton 3 and 4, a general theoretcal framewor s proposed for obtanng tghter lnear relaatons effcently for nonconve GDPs. The mplementaton of ths framewor s then llustrated n secton 5 by fndng a relaaton for two small eamples, one of them formulated as a Blnear GDP and the other as a Concave GDP. Secton 6 outlnes the mplementaton of the tghter reformulaton wthn a spatal branch and bound procedure whose performance s compared wth current methodoges (.e. Lee & Grossmann, 003) n secton 7. 3

4 - Nonconve Generalzed Dsunctve Programs The general structure of a nonconve GDP can be represented as folws (Raman & Grossmann, 994, Turay & Grossmann, 996, Lee & Grossmann, 000): Mn Z = f ( ) + s.t. g l ( ) 0 c K l L D r ( ) 0 c Y J = γ ( ) K ( GDP NC ) Ω ( Y ) = True R n, c R, Y { True, False} where f : R R n s a functon of the contnuous varables n the obectve functon, g l : R R n, l L, bengs to the set of gbal constrants, the dsunctons K, are composed of a number of terms D, that are connected by the OR operator. In each term there s a Boolean varable Y, a set of nequaltes r ( ) 0, r n : R R, J and a cost varable c. If Y s true, then r ( ) 0 and c = γ () are enforced; otherwse they are gnored. Also, Ω(Y) = True are gc propostons for the Boolean varables. As ndcated n Sawaya & Grossmann (008), we assume that the gc constrants Y are contaned n Ω(Y) = True. In a nonconve GDP, f, r, γ and/or g l J are nonconve functons. Blnear GDPs (BGDP) are the frst class of nonconve GDP problems that we address n ths paper. A BGDP s a nonconve GDP where the functons n the constrants only contan blnear and lnear terms. In general we can represent a BGDP as: 4

5 Mn Z = d T + K c s.t. T Q l l l + a b l L D T Q + a c Y b = γ J K (GDP B ) Ω ( Y ) = True R n, c R, Y { True, False} where some of the matrces Q l, Q are ndefnte REMARK: Note that f all matrces Q l, Q are postve semdefnte then the problem s conve and no gbal optmzaton methods are requred to fnd the optmal soluton. In order to solve (GDP B ) wth a spatal branch and bound method a conve GDP relaaton s requred. A vald Lnear GDP relaaton (See Proposton ) can be obtaned by fndng sutable under and over estmatng functons of the nonconve constrants. Although ths set of estmators s not unque, we propose to use the conve envepes proposed by McCormc (976) for blnear terms (See also Al-Khayyal & Fal, 983). Defnng X = T we can fnd a relaaton for each term X = as: X X X X =,.n, < <n+ Ths leads us to the folwng Lnear GDP, 5

6 Mn s.t. Z Q = d + c L T K l l l X + a b l L D Q X + a c Y b = γ J K (GDP RB ) X X X X Ω ( Y ) = True =,.n, < <n+ n X R n, R n, c R, Y { True, False} where represents the scalar product of matrces. Tradtonally, (GDP RB ) has been used to predct wer bounds n the spatal branch and bound method (Lee & Grossmann, 003). In ths wor we wll show that by the applcaton of a systematc procedure descrbed n sectons 3 and 4, we can mprove the strength of the contnuous relaaton of (GDP RB ), leadng to stronger wer bound predctons for (GDP B ). The second class of nonconve GDP problems we are nterested n are Concave GDPs. These problems often arse when economes of scale are consdered n the economc evaluaton of potental desgns (Turay & Grossmann, 996). A Concave GDP s a nonconve GDP where the functons n the constrants are concave and lnear. The general representaton of these problems s as folws: Mn Z = f ( ) + s.t. g l ( ) 0 c K l L D r ( ) 0 c Y J = γ ( ) K ( GDP Co ) 6

7 Ω ( Y ) = True n R, c R, Y { True, False} where r, γ and/or g l are concave or lnear. REMARK: Note that the effect of the economes of scale are often defned n the equalty c = γ (). Frequently, the functon γ () s concave unvarate. A vald Lnear GDP relaaton of the concave GDP (GDP RCo ) can be obtaned by fndng sutable conve under and over lnear estmators for the concave functons. As n the case of BGDP, ths set of estmators s not unque. However, n ths wor we propose mang use of the framewor proposed by Tawarmalan and Sahnds (Theorem.5,.6 and Corollary.8) to obtan vald under estmators and polyhedral outer appromatons to obtan vald over estmators. For the partcular case n whch γ () s a concave unvarate functon defned n the doman [, ], a vald lnear under estmator s gven by the secant γ ( ) γ ( ) γ *= ( ) + γ ( ) As n the case of (GDP RB ) we wll show that the framewor we propose n secton 3 and 4 can also strengthen the relaaton gven by GDP RCo REMARK: It s mportant to note that sometmes concave functons and blnear terms arse n the same GDP formulaton (See Append 4 Eample 3 and Eample 6). The method we propose can be readly etended to solve ths class of problems. 3 - A herarchy of lnear relaatons for nonconve GDP In ths secton we present a general framewor to obtan a herarchy of lnear relaatons for the nonconve GDP problem (GDP NC ) that can serve as a bass to predct tght wer bounds to the gbal optmum. Ths framewor wll be presented for general nonconve 7

8 GDP problems, although t wll later be only appled to the blnear and concave case (.e. GDP B and GDP Co ) Wthout ss of generalty, we can consder n (GDP NC ) f to be a lnear functon of. Hence, we can represent t as d T. Let us defne the folwng sets: L c := { l L g l () s conve} L nc := { l L g l () s nonconve} J c := { J, D, K r () s conve} J nc := { J, D, K r () s nonconve} D c := { D, K γ () s conve} D nc := { D, K γ () s nonconve } In the frst step of ths approach we replace each nonconve functon wth a vald lnear under/over-estmator as was done wth (GDP RB ) and (GDP RCo ). Generalzng, g l l l ( ) 0 s replaced by A b, l L nc, r g g ( ) 0 s replaced by A b, J nc, D, K, r r c = γ (), s replaced by Aγ (, c ) b, D γ nc, K Note that the dmensons of the matrces, l l A, g A, r A γ, and the rght-hand sde vectors b r l, b g, b γ, depend on the partcular under/over estmators that are chosen as was dscussed n secton. Smlarly, we replace each conve nequalty wth a vald lnear outer-appromaton. Namely, g l l l ( ) 0 s replaced by A b, l L c, g g ( ) 0 s replaced by A b, J c, D, K, r r r 8

9 c = γ (), s replaced by Aγ (, c ) b, D γ c, K Note that the dmensons of the matrces l l A, g A, r A γ and the vectors b r,, depend l b g b γ on the polyhedral outer-appromaton technque that s chosen (Tawarmalan & Sahnds, 00, Gruber & Kenderov, 98). Replacng the nonconve and concave functons n (GDP NC ) by the correspondng under/over-estmators and outer-appromatons, leads to the folwng Lnear GDP: Mn s.t. D A b l L l g A l g Y r br Aγ (, c ) Ω ( Y ) = True Z L = d R n, c T + R, Y K J b c γ { True, False} K ( GDP RLP ) Note that (GDP RB ) and (GDP RCo ) are partcular cases of GDP RLP. In the folwng proposton (PROPOSITION ) we prove that GDP RLP s ndeed a vald Lnear GDP relaaton for (GDP NC ) n+ K K PROPOSITION : Let us defne the set S NC as the set of ponts (,c,y) R B such that (,c,y) bengs to the feasble regon defned n GDP NC. Smlarly, let us defne D the set S RLP as the set of ponts (,c,y) R D + n K K B such that (,c,y) bengs to the feasble regon defned n GDP RLP. Then S NC S RLP PROOF: Defnng NC g = {(,c,y) R NC r = {(,c,y) R NC γ = {(,c,y) R D + n K K B g l ( ) 0 l L } D + n K K B r 0 D, K, J } D + n K K B c = γ ( ) D, K } 9

10 C g = {(,c,y) R C r = {(,c,y) R C γ = {(,c,y) R G = {(,c,y) R D + n K K B l l A b l L } g g D + n K K B Ar br D K, J D + n K K B D + n K K B The folwng relatons clearly hold, S NC = NC NC, } A, c ) b D, K ( } γ γ Ω(Y)= true, NC g { r } { γ K D J K D } } G S RLP = C C } { C g { r γ K D J K D } G Snce NC g C g, NC r C r and NCγ Cγ by defnton of the lnear over/under-estmators or polyhedral outer-appromatons (Tawarmalan & Sahnds 00) t folws that, S C C = NC { NC r } { NCγ } G NC g K D J NC g r } { K D J K D K D { NCγ } G C { C r } { NCγ } G g { C r } { Cγ K D J K D } G = S RLP g K D J K D Hence, S NC S RLP. COROLLARY : The optmal soluton of (GDP RLP ) yelds a vald wer bound Z L to the gbal optmum of problem (GDP NC ) Snce the obectve functon for (GDP RLP ) and (GDP NC ) are the same, the proof folws trvally from PROPOSITON. In order to predct strong wer bounds for the gbal optmum of (GDP NC ), we consder the herarchy of relaatons for (GDP RLP ) from the wor of Sawaya & Grossmann (008). These authors proved that any Lnear Generalzed Dsunctve Program (LGDP) that nvolves Boolean and contnuous varables can be equvalently formulated as a Dsunctve Program (DP), that only nvolves contnuous varables (See 0

11 Append ). Ths means that we are able to ept the wealth of theory behnd DP from Balas (979,985) n order to solve LGDP. One of the propertes of dsunctve sets s that they can be epressed n many dfferent equvalent forms. Among these forms, two etreme ones are the Conunctve Normal Form (CNF), whch s epressed as the ntersecton of elementary sets (.e. sets that are the unon of half spaces), and the Dsunctve Normal Form (DNF), whch s epressed as the unon of polyhedra. One mportant result n Dsunctve Programmng Theory, as presented n the wor of Balas (985), s that we can systematcally generate a set of equvalent DP formulatons gong from the CNF to the DNF by performng an operaton called basc step that preserves regularty. A basc step s defned by the folwng theorem. THEOREM.. (Balas, 985) Let F be the dsunctve set n RF gven by F = S where S = P, P a polyhedron, Q. Then F can be brought to DNF by T - Q applcatons of the folwng basc steps, whch preserve regularty: For some, l T, l, brng S Sl to DNF by replacng t wth S = ( P P ) Note that from the above theorem, a basc step nvolves ntersectng a gven par of dsunctons S and S l. REMARK: A partcular case arses when for a gven T, l Q Ql T S s gven by a sngle polyhedron. In ths case we denote S, as an mproper dsuncton (otherwse, we denote t as a proper dsuncton). For eample, the set of gbal constrants A b l L that l g l g appear n (GDP RLP ) are mproper dsunctons. From the practcal pont of vew, one mportant property of mproper dsunctons s the fact that when a basc step s appled between one of them and a proper dsuncton (.e. ntersectng them), the number of polyhedra n the resultng dsunctve set s not ncreased. Ths wll become mportant later when the mplementaton of basc steps are dscussed (See secton 4). Although the formulatons obtaned after the applcaton of basc steps on the dsunctve sets are equvalent, ther contnuous relaatons are not. We denote the contnuous relaaton of a dsunctve set F = S n regular form where each S s a unon of T

12 polyhedra, as the hull-relaaton of F (or h-rel F). Where h rel F : = cl conv S and T cl conv S denotes the csure of the conve hull of S. That s, f S = P, n P = { R, A b }, then cl conv S s gven by, = v Q, λ 0, λ =, A v b λ Q. Note that the conve hull of F s n Q general dfferent from ts hull-relaaton. As descrbed n Theorem 4.3., the applcaton of a basc step on a dsunctve set leads to a new dsunctve set whose relaaton s at least as tght, f not tghter, as the former. Q THEOREM 4.3 (Balas, 985) For =0,,.,t let F = S be a sequence of regular forms T of a dsunctve set, such that: ) F 0 s n CNF, wth P 0 = S, ) F t s n DNF, T 0 ) for =,.,t, F s obtaned from F - by a basc step Then h rel F... 0 h rel F h rel Ft. Consder now the lnear relaaton of (GDP NC ), namely (GDP RLP0 ), whch s equvalent to (GDP RLP ). We ntroduce the subscrpt to ndcate the number of basc steps that has been appled to the ntal Lnear GDP relaaton (GDP RLP0 ) that s obtaned by the under/over-estmaton of the nonconve terms from the (GDP NC ). Clearly, (GDP RLP0 ) represents a dsunctve set that s between the CNF and the DNF. We denote (GDP RLPt ) as the DNF form of the Lnear GDP relaaton. By THEOREM. (Balas, 985) we can wrte: GDP ~... GDP... ~ GDP RLP0 ~ GDPRLP RLP RLPt where ~ means equvalent dsunctve sets and where GDPRLP can be obtaned from GDP RLP by the applcaton of a basc step. From PROPOSITION we now that GDPRLP 0 GDPNC then GDP RLP0 ~ GDPRLPt GDPNC. Hence, the herarchy of contnuous lnear relaatons for the nonconve GDP NC can be descrbed as folws: h rel GDP h rel GDP.. h rel GDP.. h rel GDP RLP0 RLP We can then establsh the folwng proposton. RLP RLPt GDP NC

13 PROPOSITION : The wer bounds for the gbal optmum obey the folwng relatonshp: Z RLP0 Z RLP Z RLP Z RLPt Z NC where Z RLP0, Z RLP, Z RLP, Z RLPt are the optmal solutons of the hull-relaatons of problems GDP RLP0, GDP RLP,. GDP RLP, GDP RLPt respectvely and Z NC s the optmal soluton of GDP NC. PROOF: Snce Z RLP0, Z RLP, Z RLP, Z RLPt, Z NC, are defned by the same obectve functon and h rel GDP.. RLP0.. h rel GDPRLP h rel GDPRLPt GDPNC, then the proof folws trvally. Note that h rel GDP RLP 0 s equvalent to the relaaton proposed by Lee and Grossmann (003) when the under/over-estmaton functons used on GDP NC are lnear. REMARK: It s mportant to note that every tme a basc step s appled, there mght be an ncrease n the sze of the problem. Hence, some rules to gude the mplementaton of ths operaton are necessary. These rules should consder two aspects. Frstly, the effect on the tghtenng of the relaaton gven by a basc step, and secondly, the effect on the ncrease n the sze of the formulaton. 4- Rules to mplement the basc steps on GDP In order to mae good use of the herarchy of relaatons descrbed n the prevous secton, one mportant aspect s to understand whch basc steps wll lead to an mprovement n the tghtness of the relaaton, and hence n a potental ncrease n the wer bound of the gbal optmum. In other words, we need to be able to dfferentate among the basc steps that wll lead to a strct ncluson wth those that wll eep the relaaton unchanged. Wth ths obectve n mnd, Balas(985) proposed the folwng theorem whch can be readly etended to the case of lnear GDP. THEOREM 4.5. (Balas, 985) For =,, let where each P, S = P Q Q, =,, s a polyhedron. Then 3

14 clconv( S S ) = ( clconv S) ( clconv S ) f and only f every etreme pont (etreme drecton) of clconv S ) ( clconv ) s an etreme pont (etreme drecton) of P P for some (,) Q Q. ( S S = P P 3 S = P P 4 clconv Fg. Illustratve eample of Theorem 4.5. Fg. llustrates the basc dea behnd ths theorem. Clearly, ( S S S ) = ( clconv S ) ( clconv ). Ths s n agreement wth Theorem 4.5 consderng that the set of etreme ponts of clconv S ) ( clconv ) gven by {v, v, ( S v5, v6} s contaned n the set of etreme ponts of P } P } gven by {v, v, v3, v4, v5, v6} { P3 { P3 Although to the best of our nowledge t s not easy to fnd n general a systematc and computatonally effcent way to chec the hypothess of Theorem 4.5, we can stll mae use of t by eptng the structure of partcular problems. Some of the cases that are commonly found n GDP formulatons are descrbed bew. Case : PROPOSITION 3 : Let S = P where P, Q ={,} are polyhedra defned n the Q space, n R, H s a half space defned by + β 0 α and H* s a facet of H (.e. α + β = 0). If P s a pont n R n such that P H *, then clconv( H S) = clconv( H ) clconv( S). PROOF: The system trvally satsfes the hypothess of Theorem

15 Eample: A partcular case s found n process systems when the unt operaton decsons are represented as folws: Y B Y b = 0 T α 0 = cp = s cp 0 Fg : Illustraton of Proposton Clearly the hull relaaton of the above dsuncton s the same as the hull relaaton of the folwng system (See Fg. ), T α Ths partcular case was already noted by Vecchett and Grossmann (003) Case : 0 Y B Y b = 0 = cp = s cp 0 PROPOSITION 4: Let S P Q = and S = P be two dsunctve sets defned n Rn. If Q the set of varables constraned n S are not constraned n S, and the set of varables constraned n S are not constraned n S, then clconv S S ) = clconv( S ) clconv( ) ( S. PROOF: The system trvally satsfes the hypothess of Theorem 4.5. Eample: S = [ a b], S = c d] [ e ] [ f 5

16 As can be seen from Fg 3, clconv S S ) = clconv( S ) clconv( ) ( S Fg 3 Illustraton of Proposton 3 Another mportant aspect to consder s the effect that a partcular basc step has n the ncreasng of the sze of the formulaton. In ths respect we can dfferentate two types of basc steps. Frstly, the ones that are mplemented between two proper dsunctons, and second, the ones that are mplemented between a proper and an mproper dsuncton. In ths wor we propose to use the latter approach. Note that n ths case, parallel basc steps (.e. smultaneous ntersecton of each mproper dsuncton wth all proper dsunctons) wll not lead to an ncrease n the number of polyhedra n the dsunctve set, eepng the sze of the formulaton smaller. 5- Illustratve Eamples In ths secton our am s to llustrate how mproved wer bounds can be obtaned wth the proposed framewor n two smple eamples. One of them s descrbed by a BGDP, and the other descrbed by a Concave GDP. 5-- Eample : Smple Blnear GDP problem In ths secton we llustrate how to obtan a strong relaaton n a smple BGDP. Fg. 4 shows a small serstructure consstng of two reactors, each characterzed by a fwconverson curve, a converson range for whch t can be desgned, and ts correspondng cost as can be seen n Table. The problem conssts n choosng the reactor and 6

17 converson that mamze the proft from sales of the product consderng that there s a lmt on the demand. Table : Data for the reactors Reactor Curve* Range Cost α β X X Cp I II The characterstc curve s defned as F = αx + β n the range of conversons [X,X ] where F and X are the fw of raw materal and converson respectvely. Fg 4. Two reactor networ The blnear GDP model, whch mamzes the proft, can be stated as folws: Ma Z = θfx γf CP s.t. FX d Y F = α X + β X X X CP = Cp Y Y = True Y F = α X + β (GDP NC ) X X X CP = Cp X,F,CP R, F F F, Y, Y {True, False} The assocated Lnear GDP relaaton s obtaned by replacng the blnear term, FX, usng the McCormc conve envepes: Ma Z = θp γf CP s.t. P d P FX + F X F X ; P FX + F X F X P FX + F X F X ; P FX + F X F X Y F = αx + β X X X CP = Cp Y Y = True Y F = α X + β (GDP RLPC ) X X X CP = Cp X,F, CP R F F F, Y, Y {True, False} 7

18 Intersectng the mproper dsunctons gven by the nequaltes of the relaed blnear term wth the only proper dsuncton (.e. by applyng fve basc steps), we obtan the folwng GDP formulaton, Ma Z = θp γf CP s.t. Y P d P FX + F X F X P FX + F X F X P FX + F X F X P FX + F X F X F = αx + β X X X CP = Cp Y Y = True Y P d P FX + F X F X P FX + F X F X P FX + F X F X P FX + F X F X F = αx + β X X X CP = Cp (GDP RLP ) X,F,CP R, F F F, Y, Y {True, False} Fg. 5 shows the actual feasble regon of (GDP NC ) and the proecton onto the F-X space of the hull relaatons of (GDP RLP0 ) and (GDP RLP ). Notce that n ths case the choce of reactor II s nfeasble. (a) GDP NC (b) GDP RLP0 (c) GDP RLP Fg. 5 a) Proected feasble regon of GDP NC, b) Proected feasble regon of relaed GDP RLP0 c) Proected feasble regon of relaed GDP RLP From Fg 5 t s clear that the relaed feasble regon of (GDP RLP ) s contaned n (GDP RLP0 ). Ths mples that f we solve the relaed problem (GDP RLP ) we obtan an per bound of the obectve Z of. that s cser to the gbal optmal soluton of.0. In contrast GDP RLP0 predcts a weaer per bound of.8. 8

19 5-- Eample : Concave GDP problems The folwng s a smlar smple eample to llustrate the proposed framewor n Concave GDP problems. The problem conssts n selectng one of the two reactors and ts sze wth the obectve of mnmzng the ss whle satsfyng two demands, each specfed wthn a gven range (.e. Range Demand : [Dc, Dc ], Range Demand : [Dc, Dc ] ) and a sellng prce (.e. Prce Demand : p, Prce Demand : p ). Note that ths problem can also be stated as a mamzaton of the proft. Table : Data for the reactors Reactor Curve* Range Cost α β Fb Fb γ δ I II * The characterstc curve s defned as Fb = αfa + β n the range of fws [Fb LO,Fb UP ] where Fb and Fa are the fw of fnal product and raw materal, respectvely. The processng cost s gven by Cp = γ Fb δ Fg 6 Serstructure for two reactor system The concave GDP model, whch mnmzes the ss, can be stated as folws: Mn Z = Cp p Fc p Fc s.t. Fa = Fc + Fc Y Fb = α Fa + β 0.7 Cp = γ + Fb δ LO UP Fb Fb Fb Y Fb = α Fa + β (GDP NC ) 0.7 Cp = γ + Fb δ LO UP Fb Fb Fb Y Y LO Fc LO Fc UP Dc Dc Dc Dc LO Fb UP Fb Fb UP 9

20 The assocated relaed GDP program s obtaned by replacng the concave terms usng the secant as an under estmator. Note that n ths partcular case no over estmators are necessary snce the soluton of the relaaton wll always be actve n the nequalty for the underestmator (.e. secant n secton ). (For the general case, as noted n secton 3, sportng hyperplanes can be used as over-estmators): Mn Z = Cp p Fc p Fc s.t. Fb = Fc + Fc (GDP RLP0 ) ( Fb ) Fb* Fb 0.7 Y Fb = α Fa + β Cp = γfb* + δ 0.7 ( Fb ) ( Fb Fb Fb Fb Fb Fb ) + ( Fb ) 0.7 ( Fb ) Fb* Fb 0.7 Y Fb = α Fa + β Cp = γ Fb * + δ 0.7 ( Fb ) ( Fb Fb Fb Fb Fb Fb ) + ( Fb ) 0.7 Y Y LO UP Fc Dc LO Fc Dc Dc Dc LO UP Fb Fb Fb UP Intersectng the mproper dsunctons that are gven by the bounds on the demand and the relatonshp between producton and demand, wth the only proper dsuncton present (.e. by applyng seven basc steps), we obtan the folwng GDP formulaton, Mn Z = Cp p Fc p Fc s.t. (GDP RLP ) Y Y Fb = αfa + β Fb = αfa + β Cp = γ Fb + * δ Cp = γ Fb * + δ ( Fb ) Fb* Fb Y Y ( Fb ) ( Fb Fb Fb Fb Fb Fb Dc Dc Fb = Fc + Fc LO LO Fc Fc Dc Dc UP UP ) + ( Fb ) 0.7 ( Fb ) Fb* Fb ( Fb ) ( Fb Fb Fb Fb Fb Fb Dc Dc Fb = Fc + Fc LO LO Fc Fc Dc Dc UP UP ) + ( Fb ) 0.7 Fg. 7 shows the actual feasble regon of (GDP NC ) and the proecton onto the Cp-Fb space of the hull relaaton of (GDP RLP0 ) and (GDP RLP ). As can be seen the 0

21 relaaton of (GDP RLP ) predcts a stronger wer bound at 5.33 versus the weaer wer bound of (GDP RLP0 ), a) GDP NC b) GDP RLP0 c) GDP RLP Fg 7. a) Proected feasble regon of (GDP NC ), b) Proected feasble regon of relaed (GDP RLP0 ) c) Proected feasble regon of relaed (GDP RLP ) 6- Gbal Optmzaton algorthm wth mproved relaatons The gbal optmzaton methodogy of the GDP that we propose folws the well nown spatal branch and bound method (Horst,996) s obtaned n the net secton. I. GDP REFORMULATION: The frst step n the procedure conssts of mang use of the framewor proposed n secton 3 and 4 to obtan a tght GDP formulaton. In summary: a) Rela the nonconve terms usng sutable lnear conve under/over-estmators. Ths wll lead to the Lnear GDP (GDP RLP0 ). b) Apply basc steps accordng to the rules descrbed n secton 3 and 4 (.e. Parallel basc steps between mproper dsunctons wth proper dsunctons folwng Proposton 3.4 ).

22 II. UPPER BOUND AND BOUND TIGHTENING: After a reformulaton s obtaned, the procedure contnues by fndng an optmal or suboptmal soluton of the problem to obtan an per bound. Ths s accomplshed by solvng the nonconve GDP reformulated as a MINLP (ether as bg-m or conve hull formulaton) wth a cal optmzer such as DICOPT++/GAMS (Vswanathan & Grossmann,990). By usng the result obtaned n the prevous step, a bound contracton of each contnuous varable s performed (Zamora & Grossmann, 999). Ths s done by solvng mn/ma subproblems n whch the obectve functon s the value of the contnuous varable to be contracted subect to the condton that the obectve of the orgnal problem s less than the per bound. III. SPATIAL BRANCH AND BOUND: After the relaed feasble regon s contracted, a spatal branch and bound search procedure s performed. Ths technque conssts of splttng the feasble regon recursvely nto subproblems that are elmnated when t s establshed that ther descendents cannot contan a better soluton than the one that has been obtaned so far. The splttng s based on a branchng rule, and the decson about when to elmnate the subproblems s performed by comparng the wer bound LB (.e. the soluton of the subproblem) wth the per bound UB (.e. the feasble soluton wth the west obectve functon value obtaned so far). The latter can be obtaned by solvng an NLP wth all the dscrete varables fed n the correspondng subproblem); f UB-LB < ε, where ε s a gven tolerance, then the node (.e. subproblem) s elmnated. From the above outlne of the algorthm, there are two features that characterze the partcular branch and bound technque: the branchng rule and the way to choose the net subproblem to splt. In the mplementaton of ths wor we have chosen to frst branch on the dscrete varable whch most volates the ntegralty condton n the relaed LP (.e. choosng the dscrete varable csest to /), and then on the contnuous varables by choosng the one that most volates the feasble regon n the orgnal problem (.e. the volaton to the feasble regon s computed by tang the dfference between the nonconve term and the assocated relaed varable). To generate the subproblems when branchng on the contnuous varables, we splt ther doman by usng the bsecton method. To choose the node to branch net, we folwed the Best Frst

23 heurstc that conssts n tang the subproblem wth west LB. The search ends when no more nodes reman n the queue. REMARK : The framewor presented does not restrct the use of dfferent set of basc steps than the one proposed n ths algorthm (e.g. between proper dsunctons) REMARK : There s an nherent gap between the best relaaton attanable by the applcaton of basc steps to DNF form and the gbal optmal soluton. The obectve value of ths relaaton can be obtaned from the MIP soluton of the Lnear GDP (GDP RLP ) (See Corollary.. n Balas (998)). REMARK 3: An alternatve approach to the above algorthm s to consder only partal use of the proposed relaaton. Ths consders the fact that the tghter relaatons of the dsunctve sets obtaned through the proposed framewor are often folwed by an ncrease n the sze of the reformulaton. Ths leads to a potentally hgher computatonal effort that mght not be compensated by the strength n the relaaton. In future wor we propose to use the proposed relaaton to calculate new bounds on the varables and then feed these new bounds to the Lee & Grossmann relaaton, whch s of wer dmenson. In Fg. 8 we show a schematc of the two orgnal approaches and the alternatve proposed. (a) (b) (c) Fg. 8 a) Lee & Grossmann Method, b) Orgnal Proposed Approach, c) Alternatve Approach 7- Numercal performance In ths secton we analyze the performance of the proposed algorthm through two sets of numercal eamples. The frst set conssts of 4 nstances of an analytcal Blnear GDP problem (.e. Eample 0) whch ams at showng two of the man strengths of the proposed approach, namely, ts capablty to predct stronger wer bounds of the gbal optmum at the root node and ts effect on the bound contracton procedure to produce 3

24 tghter wer and per bounds for the contnuous varables (See Append 3). Table 3 summarzes the characterstcs of the nstances. Table 3 Characterstcs and sze of eample problems Blnear Terms Dscrete Varables Contnuous Varables Instance Instance Instance Instance Table 4 Lower bounds of proposed framewor Gbal Optmum Lower Bound (Lee & Grossmann Relaaton) Lower Bound (Proposed Relaaton) Best Lower Bound Instance Instance Instance Instance As t s shown n Table 4 a stronger wer bound s predcted by our approach. Moreover, as t can be seen n Table 5, the bound contracton acheved (.e. 00%) n the contnuous varables leads the gbal optmzaton technque to fnd the solutons at the root node. Note that for the larger nstances, ths leads to reasonable computatonal tmes. Table 5 Performance of proposed methodogy wth spatal branch and bound. Gbal Optmzaton Technque usng Lee & Grossmann Relaaton Gbal Optmzaton Technque usng Proposed Relaaton Bound Bound Gbal Optmum Nodes contract. (% Avg) CPU Tme (sec) Nodes contract. (% Avg) CPU Tme (sec) Instance Instance -.5 > > Instance 3-5 > > Instance 4-50 > > The second set of numercal eamples conssts of 6 problems that frequently arse n Process Systems, whch nclude the llustratve problems n secton 5 (.e. Eample and Eample ) and four more (See Append 4). Eample 3 and Eample 6 deal wth the optmzaton of a Heat Echanger Networ wth dscontnuous nvestment costs for the echangers and can be represented by a nonconve GDP wth blnear and concave constrants (Turay & Grossmann, 996). Eample 4 deals wth the optmzaton of a Wastewater Treatment Networ whose assocated nonconve GDP formulaton s a 4

25 blnear GDP (Galan & Grossmann, 998). Fnally, Eample 5 s a Poolng Desgn problem that can be also represented as a blnear GDP (Lee & Grossmann, 003). Table 6 summarzes the characterstcs and sze of the eamples, and Table 7 shows the computatonal performance of the Lee and Grossmann (003) relaaton and the one proposed n ths wor. Table 6 Characterstcs and sze of eample problems Blnear Terms Concave Functons Dscrete Varables Contnuous Varables Eample 0 3 Eample 0 5 Eample Eample Eample Eample Table 7 Lower bounds of proposed framewor Gbal Optmum Lower Bound (Lee & Grossmann Relaaton) Lower Bound (Proposed Relaaton) Best Lower Bound Eample Eample Eample Eample Eample Eample All the eamples solved show an mprovement n the wer bound. For nstance, n Eample 3 t ncreased from 967 to 9495 whch s a drect ndcaton of the reducton of the relaed feasble regon. The column Best Lower Bound, as descrbed prevously, can be used as an ndcator of the performance of our set of rules to apply basc steps. Note that n the Eamples, and 4, the wer bound proposed n our methodogy s the best wer bound that can be obtaned by solvng the relaed DNF, whch s qute remarable. A further ndcaton of tghtenng s shown n Table 8 where numercal results of the branch and bound algorthm proposed n secton 6 are presented. As t can be seen, the number of nodes that the spatal branch and bound algorthm requres for fndng the gbal soluton s sgnfcantly reduced n Eamples, 3, 4 and 6. ecept n Eample 5. Ths may be attrbuted to the caton n whch the blnear terms arse n the formulaton (.e. blnear terms nsde the dsunctons) (See Append 5). 5

26 Table 8 Performance of proposed methodogy wth spatal branch and bound. Gbal Optmzaton Technque usng Lee & Grossmann Relaaton Gbal Optmzaton Technque usng Proposed Relaaton Bound Bound Gbal Optmum Nodes contract. (% Avg) CPU Tme (sec) Nodes contract. (% Avg) CPU Tme (sec) Eample Eample Eample Eample Eample Eample Table 9 shows the sze of the LP relaaton problem obtaned wth each methodogy. Note that even when the proposed methodogy leads to a sgnfcant ncrease n the sze of the formulaton, ths s not translated proportonally to the soluton tme of the resultng LP. Ths s largely due to the preprocessng step whch effectvely reduces the sze of the LP problem to be solved. Table 9 Sze of the LP relaaton for eample problems Sze of the LP Relaaton Lee & Grossmann Proposed Constrants Varables Constrants Varables Eample Eample Eample Eample Eample Eample Conclusons In ths paper we have proposed a novel approach for obtanng stronger relaatons, and hence stronger wer bounds for the gbal optmzaton of blnear and concave Generalzed Dsunctve Programs. Wth ths am we proposed a general framewor that combnes the theory presented n the lterature to obtan lnear relaatons for nonconve MINLPs (McCormc, 976, Al-Khayyal & Fal,983, Tawarmalan & Sahnds, 00, Fudas, 000) wth the results of the wor of Sawaya & Grossmann (008) to obtan tghter relaatons for the case of Lnear GDPs. We further epted the theory behnd Dsunctve Programmng to gude the dervaton of relaatons more effcently by 6

27 consderng the partcular structure of the problems. The performance of ths procedure was shown through a set of eamples, s of whch correspond to the Process Systems feld. All of the eamples showed mprovements n the wer bounds at the root node, leadng to a sgnfcant decrease n the enumerated nodes by the spatal branch and bound method. Ths s a drect ndcaton of tghtenng that was acheved. Moreover, the fact that the wer bounds obtaned by the proposed approach are cse to the best achevable (.e. bound obtaned as a result of solvng the DNF) s a further ndcaton of the good performance of the method. A maor queston that remans to be answered s how to mplement the strong relaatons proposed n ths wor wthn a spatal branch and bound framewor effcently when dealng wth large scale systems. As a future wor we am at taclng ths problem by consderng a hybrd approach that combnes constrant programmng technques and cut generaton strateges (Hooer, 007). Acnowledgements: The authors would le to acnowledge fnancal sport from the Natonal Scence Foundaton under grant NSFOCI

28 Append. Lee & Grossmann GDP reformulaton (Lee & Grossmann, 000) Consder the folwng GDP formulaton: Mn Z = f ( ) + s.t. g ( ) 0 c D r c Y ( ) 0 K = γ Ω (Y ) = True n R, c R, Y { True, False}, K, D The Bg-M (BM) reformulaton (see Nemhauser & Wolsey,988) s as folws: Mn Z = f ( ) + K D γ λ s.t. g ( ) 0 r ( D ) M ( λ ) λ = K Aλ a n R, λ {0,}, K, K, D (BM) D where the varable λ has a one-to-one correspondence wth the Boolean varable Y. Note that when λ = 0 and the parameter M s large enough, the assocated constrant becomes redundant; otherwse, t s enforced. Also, Aλ a s the reformulaton of the gc constrants n the dscrete space, whch can be easly accomplshed as dscussed n Raman and Grossmann (994). The conve hull reformulaton proposed by Lee & Grossmann (000) yelds, 8

29 Mn Z = f ( ) + K D γ λ s.t. g ( ) 0 = K v D λ r ( v / λ ) 0 0 v λuv D K, D (CH) K, D λ = K Aλ a n R, v R, λ {0,}, K, D where the functon λ r (ν /λ ) s conve f r (.) s conve. Furthermore, specal treatment s requred to mplement ths functon as descrbed n Sawaya and Grossmann (006) Append. Equvalence between Lnear Generalzed Dsunctve Programs and Dsunctve Programs (Sawaya & Grossmann, 008). Consder the folwng lnear generalzed dsunctve programmng problem, (Raman & Grossmann, 994) Mn Z = d T + c s.t. B b D A c Y a = γ K (LGDP) Ω (Y ) = True n R, c R, Y { True, False}, K, D Based on Proposton.. bew (for proof see Sawaya & Grossmann, 008) (LGDP) can be stated as an equvalent Dsunctve Program (Balas, 974) by replacng 9

30 Boolean varables Y, J, K nsde the dsunctons by equaltes λ =, D, K, where λ s a vector of contnuous varables whose doman s [0, ]. Furthermore, the gcal relatons Y D, K and Ω ( Y) = Trueare converted nto algebrac equatons, λ =, K, and Hλ h, respectvely. D Ths yelds the folwng model: Mn Z = d T + c s.t. B b D D λ = A a K c = γ λ = K (DP) Hλ h n R, c R, λ {0,}, K, D PROPOSITION. (Sawaya & Grossmann, 008) The lnear GDP model n (LGDP) s equvalent to the dsunctve program n (DP), n the sense that there ests a one-to-one D n+ K K correspondence between a feasble soluton (, c, Y ) R { True, False} to (LGDP) and a feasble soluton n+ K + D K (, c, λ ) R to (DP). Append 3 Snce the strength of the relaaton of the nonconve terms reles heavly on the bounds of each ndvdual varable, t s of man mportance to mplement an effcent bound tghtenng procedure as part as the gbal soluton method (Ryoo & Sahnds, 995, Tawarmalan & Sahnds, 00, Zamora & Grossmann,999). In ths secton we show how the relaaton procedure we proposed leads to a sgnfcant mprovement n the 30

31 bound contracton performance when t s used n combnaton wth the methodogy proposed by Zamora & Grossmann (999) PROPOSITION 5: Let us consder the hyperrectangle Ω = { : } where s a 0 vector representng each complcatng varable (.e. varables that appear n at least one nonconve term) and and ts wer and per bounds eplctely defned n the L& G L& G formulaton. Also, let us defne Ω = { : } and L& G Ω prop = { : prop prop } as the hyperrectangles obtaned after applyng the bound contracton procedure proposed by Zamora & Grossmann n combnaton wth L&G relaaton and the proposed relaaton respectvely. Then Ω L & G Ω prop Proof Snce the L&G relaaton s contaned n the proposed relaaton then the proof folws trvally. Let us consder the folwng eample (Eample 0): Mn y s.t. = 0. 5 y Y Y y y Y Y = true =,.I 0 0 y Fg 3 Feasble regon =0.5 y=0.5 Clearly the feasble regon s defned by the pont = 0.5, y = 0.5 for =,. I Bew we show the relaed feasble regon obtaned after the bound contracton procedure when the Lee and Grossmann relaaton and the Proposed Relaaton are used 3

32 Fg 4: Relaed Feasble Regon Fg 5: Relaed Feasble Regon usng Lee & Grossmann relaaton usng Proposed Relaaton for for boundng. ( 0.3,y 0.8 ) boundng. ( 0.5,y 0.5 ) REMARK: Although smple n ts representaton, the above eample s nherently dffcult to solve. For I>5, the solver GAMS/BARON (Sahnds, 996) fals to cse more than 0% GAP after 0000 sec. Append 4. Case studes EXAMPLE 3: Process Systems wth dscontnuous Investment Cost-Multple Sze Regons (Turay & Grossmann, 996). Ths problem conssts of fndng the heat ads of utltes, the ntermedate temperatures and the area of each echanger that mnmzes the nvestment and operaton cost. Note that the structure of the networ s fed and that we use the arthmetc mean temperature as the heat drvng force. Fg 9 HEN structure of Eample 3 3

33 Table 9 Data for Eample 3 Heat Echanger Area (m ) Investment Cost ($/yr), and 3 0 A A A A A A Heat Echanger Overall Heat Transfer Stream FCP(W/K) T n (K) T out (K) Cost($/W yr) Hot Cold Coolng Water Steam The problem can be formulated as folws: = Mn Z CP + FCP ) C s.t. FCP ( T h FCP ( T h FCP ( T c n, h n, h out, c T ) = A U T ) = A U 3 h ( T Tout, h ) Ccu + FCPc ( Tout, c T T ) = A U ( T 3 n, h ( T ( T T ) + ( T n, cw FCPh ( Tn h T ) = FCPc ( T Tn,, c s + T out, h T n, c ) + ( T T ) + ( T T ) s ) out, c T ) out, cw ) hu CP Y 0.6 = 750A A 0 CP Y 0.6 = 500A A 5 CP Y 3 = 600A 5 A =,,3 33

34 Y Y Y 3 = True =,,3 T T T T T T Y { True, False} =,,3 =,,3 T R =, A R =,,3 Notaton: Parameters FCP : Heat Capacty of stream, where {h,c} T n, : Inlet temperature condtons of the stream, where {h,c} T out, : Outlet temperature condtons of the stream, where {h,c} U : Overall Heat Transfer Coeffcent for echanger E, where {,,3} C hu : Cost of hot utlty C cw : Cost of coolng water Varables T, : Intermedate temperatures A : Area of heat echanger E, where {,,3} Ths problem s a nonconve GDP where the nonconvetes arse from the blnear terms nsde the gbal constrants and concave functons nsde the dsunctons. EXAMPLE 4: Wastewater treatment networ desgn Ths eample conssts of the selecton of the equpment necessary to process a gven amount of contamnated water at the west cost to satsfy predetermned lmts (Galan & Grossmann, 998). See Fgure 0 and Tables 0 and. 34

35 Fg. 0: System serstructure Table 0: Stream qualty Stream Fwrate(ton/h) Pollutant ppm F.0 A 00 B 300 F.5 A 300 B 700 F3 0.5 A 500 B 000 Table : Equpment data Treatment Unt Equpment h Removal Rato ( % ) Cost Functon α (0.49F ) π + γf A B α γ π A B C D E F G H I

36 Notaton: Indces Streams Components Unts h Equpment Unts Sets J Components SU Spltters MU Mers M Streams beng to mer PU Process Unts IUP Stream nput to process unt S Stream beng to OUP Stream output to PU Parameters δ Cost coeffcent Varables f Indvdual component of stream h YP Selecton of eq. h n f Indvdual component of mer CP Cost of treatment for ζ Splt fracton of spltter nto stream The problem can be formulated as a BGDP as folws: Mn Z = s.t. f S S = f M = f f ζ = PU CP SU MU SU f = ζ f S SU h YP h f = β f', OPU, ' IPU D F = f, OPU CP = F h 0 ζ, 0 f, f,, 0 CP, PU YP h { true, false} h D PU 36

37 Ths problem s a nonconve GDP where the nonconvetes arse from the blnear terms nsde the gbal constrants. EXAMPLE 5: Poolng problem Ths eample consders the optmal selecton of a stream sply, pools and fws to mnmze the total cost of a poolng networ whle satsfyng the product requrements. The serstructure of the system as well as the data are shown bew: Fg. : System serstructure Table Sply stream qualty and f cost w/ S S S3 S4 S5 A B α ($) Table 3 Cost of fws ($/g) and f cost of pools / P P P3 P4 S S S S S γ ($) Table 4 Product qualty, prce and demand w/ 3 A B d ($/g) S (g)

38 Notaton: Indces Stream Pools Products w Components Sets I Sply streams I J Pools K Products W Components w Parameters S Demand of Product α Fed cost for sply stream γ Fed cost for pool d Prce of product c Cost for fw from to λ w Qualty of w from Z w Qualty requrement of w n f, Upper/Lower bound of fws Varables f w Indvdual fw of w from to f w Indvdual fw w from to ζ Splt fracton from to YP Estence of pool YST Estence of sply stream CP Cost of pool CST Cost of sply stream f The problem can be formulated as a BGDP as folws: w Mn Z = s.t. f I w W J w W K f = ζ f w f w w I K CP = γ = CP + CST + c fw d J I J I w W K J w W K w W S f w = λ f f w w' W Z w' = 0 w w J J w' W f YST J w W CST = α f w f f w w J K I, J, w W ' = 0 K, w W YST fw = 0 CST = 0 YP fw I w W = fw, w W f I f fw, w W, K ζ = 0 ζ ;0 f, f w w f w w 0 CST, CP ; YST, YP { true, false} I YP = 0, I, w W = 0, K, w W CP = γ f w J 38

39 Ths problem s a nonconve GDP where the nonconvetes arse from the blnear terms nsde the dsunctons. EXAMPLE 6: Process Systems wth dscontnuous Investment Cost-Multple Sze Regons (Turay & Grossmann, 996) As n the eample 3, ths problem conssts of fndng the heat ads of utltes, the ntermedate temperatures and the area of each echanger that mnmzes the nvestment and operaton cost. Note that the structure of the networ s fed (See Fg. ), and that we use the arthmetc mean temperature as the heat drvng force. The data for ths problem s presented n Table 5 Fg Table 5: Problem Data for Eample 6 Heat Echanger Area (m ) Investment Cost ($/yr), and 3 0 A A A A A A

40 Heat Echanger Overall Heat Transfer Stream FCP(W/K) T n (K) T out (K) Cost($/W yr) Hot Hot Hot Cold Cold Cold Cold Coolng Water Steam Ths problem s a nonconve GDP where the nonconvetes come from blnear terms representng energy balances and concave terms representng cost functons. Append 5 PROPOSITION 6: When the blnear terms are present outsde the dsunctons, a tghtenng effect can be epected from the applcaton of basc steps. PROOF: The proof folws trvally from Theorem 4.3 (Balas, 985). Illustratve eample: Gven the folwng equvalent formulatons (GDP B ) and (GDP B ) 40

41 Mn Z a) Blnear term outsde the dsuncton = d T Y.3.4 Y.8 = 0.9 = 0 (GDP B ) Y Y = True 0 0 Y, = { True, False},, R b) Blnear term nsde the dsuncton Mn Z = d T Y Y = = (GDP B ) Y Y = True 0 0 Y, = { True, False},, R It s clear from Fg.6 that the applcaton of basc steps on (GDP B ) leads to a tghter contnuous relaaton. On the other hand, no mprovement n the relaaton s observed when the applcaton of basc steps s performed on (GDP B ) 4

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