Generalized disjunctive programming: A framework for formulation and alternative algorithms for MINLP optimization

Transcription

1 Generalzed dsunctve programmng: A framewor for formulaton and alternatve algorthms for MINLP optmzaton Ignaco E. Grossmann Center for Advanced Process Decson-mang Carnege Mellon Unversty Pttsburgh PA USA IMA Hot Topcs Worshop: Mxed-Integer Nonlnear Optmzaton: Algorthmc Advances and Applcatons November

2 Motvaton Dscrete/Contnuous Optmzaton Nonlnear models 0-1 and contnuous decsons Optmzaton Models Mxed-Integer Lnear Programmng (MILP) Mxed-Integer Nonlnear Programmng (MINLP) Alternatve approach: Logc-based: Generalzed Dsunctve Programmng (GDP) Challenges How to develop best model? How to mprove relaxaton? How to solve nonconvex GDP problems to global optmalty? 2

3 Outlne 1. Overvew of maor relaxatons for nonlnear GDP and algorthms 2. Lnear GDP: herarchy of relaxatons 3. Global Optmzaton of nonconvex GDP Ph.D. Students Ramesh Raman Metn Turay Sangbum Lee Nc Sawaya Juan Ruz 3

4 4 MINLP f(xy) and g(xy) - assumed to be convex and bounded over X. f(xy) and g(xy) commonly lnear n y } {01} { } { 0 ) (.. ) ( mn a Ay y y b Bx x x x R x x X y X x y x g t s y x f Z m U L n = = = Mxed-Integer Nonlnear Programmng Obectve Functon Inequalty Constrants

5 Mxed-nteger Nonlnear Programmng Algorthms Branch and Bound (BB) Ravndran and Gupta (1985) Stubbs Mehrotra (1999) Leyffer (2001) Generalzed Benders Decomposton (GBD) Geoffron (1972) Outer-Approxmaton (OA) Duran and Grossmann (1986) Fletcher and Leyffer (1994) LP/NLP based Branch and Bound Quesada Grossmann (1994) Extended Cuttng Plane(ECP) Westerlund and Pettersson (1992) Codes: SBB GAMS smple B&B MINLP-BB (AMPL)Fletcher and Leyffer (1999) Bonmn (COIN-OR) Bonam et al (2006) FlMINT Lnderoth and Leyffer (2006) DICOPT (GAMS) Vswanathan and Grossman (1990) AOA (AIMSS) α ECP Westerlund and Peterssson (1996) MINOPT Schweger and Floudas (1998) BARON Sahnds et al. (1998) 5

6 Generalzed Dsunctve Programmng Motvaton 1. Facltate modelng of dscrete/contnuous optmzaton problems through use algebrac constrants and symbolc expressons 2. Reduce combnatoral search effort 3. Improve handlng nonlneartes 6

7 Generalzed Dsunctve Programmng (GDP) Raman and Grossmann (1994) (Extenson Balas 1979) mn Z = s.t. r c + ( x ) f 0 ( x ) Obectve Functon Common Constrants OR operator J g ( x ) 0 c γ = Ω = true x ( ) R n c R { true false } 1 K Dsuncton Constrants Fxed Charges Logc Propostons Contnuous Varables Boolean Varables 7

8 Process Networ wth fxed charges GDP model T Mn Z = c1+ c2 + c3 + d x st.. x = x + x x = x + x x = p x x = x = c1 = γ 1 c1 = x = p x x = x = c2 = γ 2 c2 = x = p x x = x = c3 = γ 3 c3 = x x U { True False} c c c R 1 8

9 Generalzed Dsunctve Programmng (GDP) Raman and Grossmann (1994) mn Z = s.t. r c + ( x ) f 0 ( x ) Obectve Functon Common Constrants OR operator J g ( x ) 0 c γ = Ω = true x ( ) R n c R { true false } 1 K Dsuncton Constrants Fxed Charges Logc Propostons Contnuous Varables Boolean Varables Relaxaton? 9

10 Bg-M MINLP (BM) MINLP reformulaton of GDP mn Z = γ λ + f ( x) st.. r( x) 0 g ( x) M (1 λ ) J K K J J λ = 1 K Aλ a x 0 λ {01} Bg-M Parameter Logc constrants NLP Relaxaton 0 1 λ 10

11 Consder Dsuncton K Convex Hull Formulaton J g c ( x) 0 = γ Theorem: Convex Hull of Dsuncton (Lee Grossmann 2000) Dsaggregated varables ν {( x c) x = 0 v J λ g λ v J ( v = 1 0 < λ / λ ) 0 J λ - weghts for lnear combnaton λ U c = λ γ J J 1 } => Convex Constrants - Generalzaton of Balas (1979) Stubbs and Mehrotra (1999) 11

12 Remars 1. h ( v λ) = λ g ( v / λ) If g(x) s a bounded convex functon h( v λ) s a bounded convex functon Hrart-Urruty and Lemaréchal (1993) h( ν 0) = 0 for bounded g(x) λ g ( ν / λ ) 0 0 ν U λ 2. Replace where by: ((1 ε ) λ + ε)( g ( ν / ((1 ε) λ + ε))) εg (0)(1 λ ) 0 a. Exact approxmaton of the orgnal constrants as ε 0. b. The constrants are exact at λ = 0 and at λ = 1 regardless of value of ε. f λ = 0 ( ε)( g (0)) εg (0) = 0 0 f λ = 1 ((1)( g ( ν / (1)) εg (0)(0) = (1) g ( ν / (1)) 0 c. The LHS of the new constrants are convex. Furman Sawaya & Grossmann (2007) 12

13 Convex Relaxaton Problem (CRP) CRP: mn Z = s. t. γ K J λ + f r( x) 0 ( x) 0 ν λ g x ν λ U ( ν x = J λ J / λ ) 0 0 λ K J = 1 K ν 0 J Aλ a 1 J K K K Convex Hull Formulaton Logc constrants Property: The NLP (CRP) yelds a lower bound to optmum of (GDP). Note: Hull relaxaton as ntersecton of convex hull for each dsuncton 13

14 Strength Lower Bounds Theorem: The relaxaton of (CRP) yelds a lower bound that s greater than or equal to the lower bound that s obtaned from the relaxaton of problem (BM): RBM: mn Z = γ λ + f( x) st.. r( x) 0 g ( x) M (1 λ ) J K K J J λ = 1 K Aλ a x 0 0 λ 1 14

15 15 MINLP Reformulaton ) / ( ) (.. ) ( mn K J x a A K J g K K J U K x x r t s x f Z J J K J = = = + = λ λ λ λ λ λ λ γ ν ν ν ν Specfy n CRP as 0-1 varables λ

16 Methods Generalzed Dsunctve Programmng GDP Logc based methods Reformulaton MINLP Outer-Approxmaton Generalzed Benders Extended Cuttng Plane Branch and bound (Lee & Grossmann 2000) Decomposton Outer-Approxmaton Generalzed Benders (Turay & Grossmann 1997) Convex-hull Bg-M Cuttng plane (Sawaya & Grossmann 2004) 16

17 A Branch and Bound Algorthm for GDP Tree Search NLP subproblem at each node Solve CRP of GDP lower bound CRP Branchng Rule Set the largest λ as 1 Dchotomy rule Logc nference CNF unt resoluton (Raman & Grosmann 1993) Depth frst search When all the terms are fxed upper bound Repeat Branchng untl Z L > Z U. CRP + fx a term n dsuncton CRP + convex hull of remanng terms 17

18 GDP Example Fnd x 0 (x S 1 )(x S 2 )(x S 3 ) to mnmze Z = (x 1-3) 2 + (x 2-2) 2 + c Obectve Functon = contnuous functon + fxed charge (dscontnuous). x 2 S 3 Contour of f (x) Local solutons Global Optmum ( ) Z* = S 1 S 2 (00) x 1 18

19 Example : convex hull x 2 S 3 Convex hull = conv(us ) S 2 S 1 x 1 19

20 Example: CRP soluton Convex hull = conv(us ) x 2 S 3 Weght λ 1 = λ 2 = λ 3 = Convex combnaton of z x* S 2 z = v /λ S 1 Local soluton pont (00) Convex hull optmum Z L = x L = ( ) Infeasble to GDP x 1 20

21 Example : branch and bound Frst Node: S 2 Optmal soluton: Z U = x 2 S 3 Optmal Soluton ( ) Z* = (00) S 1 S 2 x 1 21

22 Example : branch and bound Second Node: conv(s 1 U S 3 ) Optmal soluton: Z L = x 2 S 3 Lower Bound Z L = Upper Bound Z U = S 1 S 2 (00) x 1 22

23 Example: Search Tree Branchng Rule: λ - the weght of dsaggregated varable Fx as true: fx λ as 1. Root Node Convex hull of all S Z = λ = [ ] 2 Frst Node Fx λ 2 = 1 Z = [x 1 x 2 ] = [ ] λ = [010] Z L = Branch on 2 2 Second Node Convex hull of S 1 and S 3 Z = λ = [ ] Z U = Bactrac Z L = > Z U Stop 23

24 Process Networ wth Fxed Charges Türay and Grossmann (1997) Superstructure of the process A x 1 x 2 x4 1 x 12 x 13 E x 3 x 5 x 11 B x x 19 x x 20 x 22 x 23 x 24 F x 6 x 15 x 25 x 18 x 16 x D C x 8 3 x 9 x 10 : Unt x 7 Specfcatons 24

25 Optmal soluton Mnmum Cost: $ 68.01M/year x 14 x 19 6 x 20 A x 1 Raw Materal x4 2 Reactor x 12 x 5 x 11 B 4 x 13 x 25 E x 17 x 6 x 10 Reactor 8 x 23 x 24 F x 18 Products D x 8 : Unt x 7 25

26 Proposed BB Method Proposed BB Z L = λ = [ ] Z L = λ = [ ] Fx λ 3 = 1 Fx λ 3 = 0 Stop Z L = > Z U Z U = Z U = = Z* λ = [ ] λ = [ ] Feasble Soluton Optmal Soluton 0 Fx λ 2 = 1 Fx λ 2 = 0 4 λ = [ ] 8 = 0 Z L = Bg-M Std. BB 0 4 = 0 4 = * 1 = 0 1 = 1 6 = 0 6 = = 1 8 = 0 8 = 1 2 = 0 2 = 1 1 = 1 3 = 0 3 = 1 5 nodes vs. 17 nodes of Standard BB (lower bound = 15.08) 26

27 Logc-based Outer Approxmaton Man pont: avods solvng MINLP n full space NLP Subproblem: (reduced) mn Z = c SD + f ( x) st.. g( x) 0 h ( x) 0 for = true ˆ D SD ˆ c = γ B x = 0 for = false D ˆ SD c = 0 x R n c R m (NLPD) Turay Grossmann (1997) Redundant constrants are elmnated wth false values Master Problem: s. t. Mn Z = c +α l l T l α f ( x ) + f ( x ) ( x x ) l l T l g( x ) + g( x ) ( x x ) 0 l = 1... L (MGDP) Master problem solved wth dsunctve branch and bound or wth MILP reformulaton D l l T l h ( x ) + h ( x ) ( x x ) 0 l L c = γ SD Ω () = True α R x R n c R m {true false} m Proceed as OA. Requres ntalzaton several NLPs to cover all dsunctons 27

28 LogMIP Aldo Vecchett INGAR Part of GAMS Modelng System -Dsunctons specfed wth IF Then ELSE statements DISJUNCTION D1(IKJ); D1(IKJ) wth (L(IKJ)) IS IF (IKJ) THEN NOCLASH1(IKJ); ELSE NOCLASH2(IKJ); ENDIF; -Logc can be specfed n symbolc form ( OR AND NOT ) or specal operators (ATMOST ATLEAST EXACTL) -Lnear case: MILP reformulaton bg-m convex hull -Nonlnear: Logc-based OA 28

29 Lnear Generalzed Dsunctve Programmng LGDP Model Raman R. and Grossmann I.E. (1994) (Extenson Balas (1979)) (LGDP) T Mn Z = c + d x K st.. Bx b J J A x a K Ω ( ) = True L U x x x { True False} J K c c = γ K 1 R K Logcal OR operator Can we obtan stronger relaxatons? Carnege Mellon Obectve functon Common constrants Dsunctve constrants Logc constrants Contnuous varables Boolean varables 29

30 Carnege Mellon Dsunctve Programmng Dsuncton: A set of constrants connected to one another through the logcal OR operator Conuncton: A set of constrants connected to one another through the logcal AND operator Constrant set of a DP can be expressed n two equvalent extreme forms - Dsunctve Normal Form (DNF). A dsuncton whose terms do not contan further dsunctons { } R n : ( ) F = x A x a Q - Conunctve Normal Form (CNF). A conuncton whose terms do not contan further conunctons { : ( ) 1... } R F = x Ax a d x d = t n h h 0 hq 30

31 T Mn Z = c + d x K st.. Bx b A x a J c = γ K K J Ω ( ) = True L U x x x { True False} J K c Lnear Generalzed Dsunctve Programmng LGDP Model 1 R K Boolean varables (LGDP) Obectve functon Common constrants Dsunctve constrants Logc constrants Carnege Mellon How to deal wth Boolean and logc constrants n Dsunctve Programmng? 31

32 Reformulatng LGDP nto Dsunctve Programmng Formulaton Sawaya N.W. and Grossmann I.E. (2008) T Mn Z = c + d x K st.. Bx b A x a J c = γ K K J Ω ( ) = True L U x x x { True False} J K c 1 R K T Mn Z = c + d x K st.. Bx b λ = 1 A x a K J c = γ λ = 1 K J Hλ h L U x x x 0 λ 1 J K c 1 R K Carnege Mellon LGDP LDP => Integralty λ guaranteed Proposton. LGDP and LDP have equvalent solutons. 32

33 Equvalent Forms n DP Through Basc Steps There are many forms between CNF and DNF that are equvalent Regular Form (RF): form represented by ntersecton of unons of polyhedra Thus the RF s: where for F = tt t T S t S = P P a polyhedron Q. t t Q t Proposton 1 (Theorem 2.1 n Balas (1979)). Let F be a dsunctve set n RF. Then F can be brought to DNF by T 1 whch preserve regularty: recursve applcatons of the followng basc steps For some rs Tr s brng Sr Ss to DNF by replacng t wth: S = ( P P). rs Q t tq r s Carnege Mellon 33

34 Carnege Mellon Illustratve Example: Basc Steps F = S S S S1 = ( P11 P21) S2 = ( P12 P22) S3 = ( P13 P23) Then F can be brought to DNF through 2 basc steps. Apply Basc Step to: S S = ( P P ) ( P P ) S = ( P P ) ( P P ) ( P P ) ( P P ) S We can then rewrte F = S S S Apply Basc Step to: as F = S12 S3 S S = (( P P ) ( P P ) ( P P ) ( P P )) ( P P ) ( P11 P12 P13) ( P11 P22 P13) ( P21 P12 P13) ( P21 P22 P13) = ( P11 P12 P23 ) ( P11 P22 P23 ) ( P21 P12 P23) ( P21 P22 P23 ) We can then rewrte F = S S as F = S whch s ts equvalent DNF 34

35 Equvalent Forms for GDP T Mn Z = c + d x K st.. Bx b J J A x a K Ω ( ) = True L U x x x { True False} J K c c = γ K 1 R K LGDP T Mn Z = c + d x K st.. Bx b J J = 1 A x a K L U x x x = 1 K 0 λ 1 J K c H λ h c λ λ = γ 1 R K LDP n+ J + K K F = z: = ( x λ c) R : b z b0 ( A z a ) T K J LDP Carnege Mellon All possble equvalent forms for GDP obtaned through any number of basc steps are represented by: n+ J + K K : ( ) : ˆ mn mn F = z = x λ c R b z b ˆ 0 ( A z a ) ( A z a ) T K J ˆ nk mjn 35

36 Proposton 2 (Theorem 3.3 combned wth Corollary 3.5 n Balas (1979)). Let n { R 0 } F = P P = x : A x a Q Q 0 Convertng LDP to MIP reformulatons where Q s an arbtrary set and each ( A a ) s an m ( 1) n+ matrx such that every P s a bounded non-empty polyhedron. Furthermore let ζ ( Q) be the set of all those n x R such that there exst vectors n+ 1 ( ) v y R Q satsfyng x v = 0 Q A v a y 0 Q 0 y 0 Q Q y = 1 Q ν dsaggregated varables => Convex Hull Then cl conv F = ζ ( Q). Proposton 3 (Corollary 3.7 n Balas (1979). Let ζ ( Q): { x ζ( Q): y {01} Q} I Then ζ I ( Q) = F. =. => MIP representaton 36 Carnege Mellon

37 Famly of MIP Reformulatons For GDP n+ J + K K : ( ) : ˆ mn mn F = z = x λ c R b z b ˆ 0 ( A z a ) ( A z a ) T K J ˆ nk mjn LDP General template for any MILP reformulaton T Mn Z = γ y + d x K J st.. bx b0 IB hy h0 IH x x x I L U X mn 2 S H mj y = uˆ ( ) L K I nn mn x = vˆ nn bvˆ J mj mn n n 0 mn 0 2n 2n 2n n 2n 2n b yˆ I mj nn mn B n mn uˆ = yˆ K mj nn mn S n huˆ h yˆ I mj nn mn H n mn uˆ = yˆ ( ) M mj nn mn mn n mn A vˆ a yˆ ( ) M mj nn L mn U x yˆ vˆ x yˆ mj nn 0 uˆ yˆ ( ) L mj nn mj n mq J n mn mn 3 n yˆ = 1 nn yˆ = y n N J K mn mn n mn mn n y mn mn = 1 K yˆ 0 mj nn mn y {01} J K n n MIP Carnege Mellon 37

38 Partcular case: Convex Hull Reformulaton of LGDP Raman and Grossmann I.E. (1994) T Mn Z = γ y + d x K J st.. Bx b x= v K A v a y J K L U x y v x y J K J J y = 1 K {01} Hy h y J K (CH) Dsaggregated varables Whle ths MILP formulaton has stronger relaxaton than bg-m t s not strongest!! 38 Carnege Mellon

39 A Herarchy of Relaxatons for GDP Proposton 4. For T + K 1 let F GDP be a sequence of regular forms of the dsunctve set: n+ J + K K : ( ) : ˆ mn mn F = z = x λ c R b z b ˆ 0 ( A z a ) ( A z a ) T K J ˆ nk mjn such that ) F GDP 0 corresponds to the dsunctve form: n+ J + K K F = z: = ( x λ c) R : b z b0 ( A z a ) ; T K J ) F GDP T + K 1 : = F s n DNF; t ) for = 1 t F GDP s obtaned from F by a basc step. GDP 1 Then h rel F h rel F h rel F = clconv F = clconv F.(true convex hull) GDP GDP GDP GDP t 0 1 T + K 1 T + K 1 Carnege Mellon 39

40 Illustratve Example: Herarchy of Relaxatons x x x x x1 = 0 x1 = 1 0 x 1 0 x Convex Hull of dsuncton x 2 LP Relaxaton Applcaton of 2 Basc Steps x1 x x1 x x1 x2 1 0 x1 x x1 = 0 x1 = 1 0 x 1 0 x x 1 Convex Hull of dsuncton Tghter Relaxaton! Carnege Mellon 40

41 Problem statement: Hf (1998) Numercal Example: Strp-pacng problem Gven a set of small rectangles wth wdth H and length L. Large rectangular strp of fxed wdth W and unnown length L. Obectve s to ft small rectangles onto strp wthout overlap and rotaton whle mnmzng length L of the strp. y (00) (x y ) W L =? x Set of small rectangles 41 Carnege Mellon

42 GDP/DP Model for Strp-pacng problem Mn lt st.. lt x + L N < x+ L x x+ L x y H y y H y x UB L N N H y W N lt x y R { True False} N < Obectve functon Mnmze length Dsunctve constrants No overlap between rectangles Bounds on varables Mn lt st.. lt x + L Carnege Mellon λ = 1 λ = 1 λ = 1 λ = 1 N < x+ L x x+ L x y H y y H y λ + λ + λ + λ = 1 N < x N UB L N H y W R N lt x y λ λ λ λ N < 42

43 25 Rectangle Problem Optmal soluton= 31 Orgnal CH varables 4940 cont vars 7526 constrants LP relaxaton = 9 => Strengthened varables 5783 cont vars 8232 constrants LP relaxaton = 27! 31 Rectangle Problem Optmal soluton= 38 Orgnal CH varables 9716 cont vars constrants LP relaxaton = => Strengthened varables cont vars constrants LP relaxaton = 33! 43 Carnege Mellon

44 Cuttng Planes for Lnear Generalzed Dsunctve Programmng GDP Model: Mn Z = + h T x c K Sawaya Grossmann (2004) Obectve Functon s.t. Bx b Common Constrants OR Operator J A c x a = γ K Dsunctve Constrants Ω() = True x R n {True False} c R J K Logc Constrants Boolean Varables 44

45 Motvaton for Cuttng Plane Method Trade-off: Bg-M fewer vars/weaer relaxaton vs Convex-Hull tghter relaxaton/more vars x 2 x R BM Strengthened Bg-M Relaxed Feasble Regon x SEP Bg-M Relaxed Feasble Regon Cuttng Plane (x - x SEP ) T (x SEP -x R BM ) 0 Convex Hull Relaxed Proected Feasble Regon x 1 45

46 Global Optmzaton Algorthms Most algorthms are based on spatal branch and bound method (Horst & Tuy 1996) Nonconvex NLP/MINLP αbb (Adman Androulas & Floudas 1997; 2000) BARON (Branch and Reduce) (Ryoo & Sahnds 1995 Tawarmalan and Sahnds (2002)) OA for nonconvex MINLP (Kesavan et al. 2004) Branch and Contract (Zamora & Grossmann 1999) Nonconvex GDP Two-level Branch and Bound (Lee & Grossmann 2001) 46

47 Spatal Branch and Bound to obtan the Global Optmum Guaranteed to converge to global optmum gven a certan tolerance between lower and upper bounds 47

48 Global optmum search Branch and bound tree Multple mnma LB < UB Obectve LB LB Lower bound UB = Upper bound LB > UB LB < UB LB 48

49 Nonconvex GDP mn Z = s.t. r c + ( x ) f 0 ( x ) Obectve Functon Common Constrants OR operator J g ( x ) 0 c γ = Ω = true x ( ) R n c R { true false } 1 K Dsunctons Logc Propostons f g and r: nonconvex 49

50 Convex Underestmator GDP (R) Introducng convex underestmators J mn f s.t. x r Z g ( x ) 0 c γ = Ω = true = ( ) R { true false } and c + r ( n g c x ) : f R 0 1 convex ( x ) K Convex underestmators Blnear: Lnear McCormc (1976) Al-Khayyal (1992) Lnear fractonal: Convex nonlnear Quesada and Grossmann (1995) Concave separable: Lnear secant Problem (R) yelds a vald lower bound to Problem (GDP) 50

51 Convex envelopes Concave functon Secant g(x) f(x) a b x [ f( b) f( a)] g( x) = f ( a) + ( x a) b a 51

52 Blnear w = xy L U L U x x x y y y McCormc convex envelopes L L L L w x y+ y x x y U U U U w x y + y x x y L U L U w x y + y x x y U L U L w x y + y x x y For other convex envelopes/underestmators see: Tawarmalan M. and N. V. Sahnds Convexfcaton and Global Optmzaton n Contnuous and Mxed-Integer Nonlnear Programmng: Theory Algorthms Software and Applcatons Vol. 65 Nonconvex Optmzaton And Its Applcatons seres Kluwer Academc Publshers Dordrecht

53 Basc Ideas Global Optmzaton GDP 1. Branch and bound enumeraton on dsunctons of convex GDP (R) 2. When feasble dscrete soluton found swtch to spatal branch and bound (NLP subproblem) Dsunctve B&B Feasble dscrete Spatal B&B 53

54 A B C Synthess Multproduct Batch Plant (Brewar & Grossmann 1990) Mxng Reacton Crystallzaton Dryng Tass S S Equpment S A B C Unt 1 Unt 2 Unt 3 Unt 4 Unt 5 Cast Iron Stanless Steel Cast Iron w/ Agtator w/ Agtator Jaceted Stanless Steel Jaceted w/ Agtator Tray Dryer More than 100 alternatves: each requres nonlnear optmzaton 54

55 Synthess Multproduct Batch Plant Nonconvex GDP Model s. t. pt mn COST = N V t T = tt pty n B B S t t Q = 1 = 1... N = 1... N ; M EQ C P = 1... N + CS P ; t = 1... T = 1... M P Obectve functon Szng Process tme Demand N p nt = 1 L H Horzon tme J t t T V V t T pty = pt t t pty = 0 ' t ' t T Dsuncton for Tas Assgnments Nonconvex functons 55

56 56 } { ; 0 ) ( ) ( ) ( ) ( ) ( false true W C EX pty N pt T B n V V C W W W W W W W W EX EX EX EX EX l c t t t EQ L T t J CS VST B B S VST CS VST NEQ B S VST NEQ B S VST B B S J T pt N V C EX pt T N C pt T N C pt T N C pt T N C V V V V C EX L EQ L EQ L EQ L EQ L EQ U L = = = + = = = = = = = = = + = ' ' ' 0.5 ' ' ' φ φ α γ Dsuncton for Equpment Dsuncton for Storage Tan Logc Propostons GDP model (contnued)

57 Proposed Algorthm for Nonconvex GDP Step 0 Nonconvex MINLP OA (Vswanathan and Grossmann 1990) Z U Step 1 Bound Contracton (Zamora and Grossmann 1999) New Bound Step 2 BB wth s Update Z L Stop when Z L Z U (Lee and Grossmann 2000) When soluton s Integral Add Integer Cut Step 3 Spatal BB Update Z U Fxed s (Quesada and Grossmann 1995) 57

58 Upper Bound Soluton Cost = $ (by GAMS/DICOPT++) Use 4 Stages (6 unts) wthout Storage Tan A B C Mxng Reacton Crystallzaton Dryng = 1 = 2 = 4 = 5 V 1 = 4842 L V 2 = 2881 L V 4 = 2469 L V 5 = 8071 L A B C A 243 batches 4.5hrs B 260 batches 6hrs C 372 batches 9hrs 6000 hrs hrs 1562 hrs 3345 hrs

59 Optmal Soluton: Multproduct Batch Plant Global optmal cost = $ (5% mprovement) 3 Stages + 1 storage tan (5 unts) (43 nodes 48 sec) A B C Mxng Reacton Storage Tan S Crystallzaton Dryng = 2 = 3 = 5 V 2 = 4309 L VST 2 = 4800 L V 3 = 3600 L V 5 = L A 250 batches 5hrs B 293 batches 3hrs C 418 batches 5.5hrs A B C 6000 hrs Storage 1503 hrs 2202 hrs 2295 hrs batches 9hrs batches 12 hrs batches 9hrs 59

60 Global Optmzaton of Blnear Generalzed Dsunctve Programs Juan Ruz Mn D Z = f ( x) + s.t. g( x) 0 c K r ( x) 0 c = γ Obectve Functon Global Constrants Dsunctons K Blneartes x R Ω()= True n c R {TrueFalse} Logc Propostons D K Blneartes may lead to multple local mnma Global Optmzaton technques are requred Relaxaton of Blnear terms usng McCormc envelopes leads to a LGDP Improved relaxatons for Lnear GDP has recently been obtaned (Sawaya & Grossmann 2007) 60 Carnege Mellon

61 Gudelnes for applyng basc steps n Blnear GDP Replace blnear terms n GDP by McCormc convex envelopes (LGDP) Apply basc steps between those dsunctons wth at least one varable n common. The more varables n common two dsunctons have the more the tghtenng can be expected If blneartes are outsde the dsunctons apply basc steps by ntroducng them n the dsunctons prevous to the relaxaton. If blneartes are nsde the dsunctons a smaller tghtenng effect s expected. A smaller ncrease n the sze of the formulaton s expected when basc steps are appled between mproper dsunctons and proper dsunctons. Carnege Mellon 61

62 Methodology Step 1: GDP reformulaton (Apply basc steps followng the rules presented) Step 2: Bound Contracton (Zamora & Grossmann 1999) Step 3: Branch and Bound Procedure (Lee & Grossmann 2001) Spatal B&B Contractng Bounds Intersectng dsunctons 62 Carnege Mellon

63 Case Study I: Water treatment networ desgn Process superstructure Generalzed Dsunctve Program S2 S1 M1 M2 A/B/C D/E/F S4 S5 M4 Mn Z = CP PU s.t. f S = f M = f f MU SU S3 M3 G/H/I S6 S ζ = 1 SU S1 N of cont. vars. : 114 N of dsc. vars. : 9 N of blnear terms: 36 Optmal structure M1 A S4 f f D h = ζ f h P h = β f' F = f OPU CP = OPU F S ' IPU SU PU 0 ζ 1 S2 M2 D S5 M4 0 f f 0 CP S3 Z* = P h { true false} h D PU 63 Carnege Mellon

64 Case Study II: Poolng networ desgn Carnege Mellon Process superstructure Stream Pool Product S1 S2 S3 S4 S5 S1 S2 S5 P1 P2 P3 P4 N of cont. vars. : 76 N of dsc. vars. : 9 N of blnear terms: 24 Optmal structure Stream Pool Product P1 P3 Z* = f w Generalzed Dsunctve Program Mn Z = CP + CST + c fw d s.t. P lo f fw I ww f w = fw ww f K I f = ζ fw ww K I ζ = 1 K CP = γ I ww J ww f f w w = J I J I ww K J ww K ww S f w = λ f f w w' w' W Z = 0 w w J J w' W f lo ST J ww CST = α f w 0 ζ 1;0 f f f f w w f w J K I J ww ' = 0 K ww ST fw = 0 CST = 0 w w f up w w 0 CST CP ; ST P { true false} P = 0 I w W = 0 K w W CP = 0 I J 64

65 Performance Global Optmzaton Technque usng Lee & Grossmann relaxaton Global Optmzaton Technque usng proposed relaxaton Relatve Improvement Example 1 Intal Lower Bound % Bound contracton 99.7% Nodes % Global Optmzaton Technque usng Lee & Grossmann relaxaton Global Optmzaton Technque usng proposed relaxaton Relatve Improvement Example 2 Intal Lower Bound % Bound contracton 8% Nodes % 65 Carnege Mellon

66 Conclusons GDP modelng framewor - Provdes a logc-based framewor for dscrete-contnuous optmzaton - bg-m and convex hull alternatve formulatons dfferent relaxatons - Soluton methods: reformulaton branch and bound decomposton Unfed Lnear GDP wth Dsunctve Programmng - Developed DP equvalent formulaton for GDP - Developed a famly of MIP reformulatons for GDP - Developed a herarchy of relaxatons for GDP - Numercal results have shown great mprovement n lower bound for strp pacng problem Nonconvex GDPs - Spatal branch and bound methods can be developed - Tghter lower bounds can be obtaned n blnear problems by applyng basc steps 66 Carnege Mellon

67 Open Cybernfrastructure for Mxed-nteger Nonlnear Programmng: Collaboraton and Deployment va Vrtual Envronments CMU: Grossmann Begler Belott Cornueols Margot Ruz Sahnds IBM: Lee Wächter General Goals (a) Create a lbrary of optmzaton problems n dfferent applcaton areas n whch one or several alternatve models are presented wth ther dervaton. In addton each model has one or several nstances that can serve to test varous algorthms. (b) Provde a mechansm for researchers and users to contrbute towards the creaton of the lbrary of optmzaton problems. (c) Provde a forum of dscusson for algorthm developers and applcaton users where alternatve formulatons can be dscussed as well as performance and comparson of algorthms. (d) Provde nformaton on MINLP tutorals and bblography to dssemnate ths nformaton. Maor emphass Collect optmzaton problems n whch alternatve model formulatons are documented wth correspondng computatonal results (engneerng fnance operatons management bology) 67 Carnege Mellon