Global Optimization of Bilinear Generalized Disjunctive Programs

Transcription

1 Global Optmzaton o Blnear Generalzed Dsunctve Programs Juan Pablo Ruz Ignaco E. Grossmann Department o Chemcal Engneerng Center or Advanced Process Decson-mang Unversty Pttsburgh, PA

2 Non-Convex Dscrete/Contnuous Optmzaton Models Mxed Integer Program (MIP) - Most common non-lnear dscrete/contnuous optmzaton model. - Purely equaton-based. - I some unctons n MIP are non convex non-convex MINLP. Generalzed Dsunctve Program (GDP) - Developed by Raman & Grossmann (1994) - Combnaton o algebrac equatons, dsunctons and logc propostons. - Natural representaton o engneerng problems. - I some unctons n GDP are non convex non-convex GDP. - I all non lnear unctons are gven by blnear terms Blnear GDP. Goal: Global Optmzaton o Blnear GDP wth mproved relaxatons 2

3 Blnear Generalzed Dsunctve Programs Mn D Z = ( x) + s.t. g( x) 0 c K Y r ( x) 0 c = γ Obectve Functon Global Constrants Dsunctons K Blneartes x R Ω(Y)= True n,c R,Y {True,False} Logc Propostons D, K Process engneerng models. (e.g. HENS, Water Treatment Networs, Process Networs n general) Blneartes may lead to multple local mnma Global Optmzaton technques are requred Global optmzaton method by Lee & Grossmann (2001) Is ther relaxaton the tghtest? Relaxaton o Blnear terms usng McCormc envelopes leads to a LGDP Improved relaxatons or Lnear GDP has recently been obtaned (Sawaya & Grossmann, 2007) 3

4 LGDP to Dsunctve Programmng Reormulaton Sawaya N.W. and Grossmann I.E. (2006) T Mn Z = c + d x K st.. Bx b J J A x a K Ω ( Y) = True L U x x x Y { True, False} J, K c Y c = γ Y K 1 R K 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 Hλ h c λ λ = γ 1 R K LGDP Integralty λ guaranteed DP 4

5 Equvalent Dsunctve Programs Regular Form (RF): orm represented by ntersecton o unons o polyhedra There exsts many orms between CNF and DNF that are equvalent 5

6 Illustratve Example: Basc Steps Then F can be brought to DNF through 2 basc steps. 6

7 A Herarchy o Relaxatons or DP 7

8 A Herarchy o Relaxatons or GDP Proposton 4 (Sawaya & Grossmann, 2006) For = 1,2,... T + K 1 let regular orms o the dsunctve set: FGDP be a sequence o n+ J + K K : (,, ) : ˆ mn mn F = z = x λ c R b z b ˆ 0 ( A% ) z a% ) ( A z a ), such that T K% J ˆ n K m Jn ) F GDP 0 corresponds to the dsunctve orm: 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 ) or = 1,2,... T + K 1, F GDP s obtaned rom F by a basc step. GDP 1 Then, h rel F h rel F L h rel F = clconv F = clconv F.(true convex hull) GDP GDP GDP GDP t 0 1 T + K 1 T + K 1 8

9 Blnear GDP Relaxaton Illustratve Example: Optmal reactor selecton Feasble regon proected onto the FX space F Demand F: Flow X: Converson GDP Formulaton Max Z = θfx γf CP Obectve Functon Demand constrant 8 I II Converson s.t. FX d Y11 F = α1x + β1 X1 X X CP = Cp1 1 Y21 F = α2x + β2 X 2 X X CP = Cp2 2 Reactor Curves Y11 Y21 = True Feasble Regon CP,X,F R F F F Y 11,,Y 21 {True, False} X 9

10 Blnear GDP Relaxaton (Lee & Grossmann 2001) F Blnear Terms 8 II Convex Envelopes Max Z = θp θfx γf CP CP s.t. FX P d P F.X + F.X - F.X P F.X + F.X F.X P F.X + F.X F.X P F.X + F.X F.X Relaxaton (No Basc Steps) I Convex Hull Relaxaton X Y11 F = α1x + β1 X1 X X CP = Cp1 1 Y11 Y21 = True CP,X,F R F F F Y 11,,Y 21 {True, False} Y21 F = α2x + β2 X 2 X X CP = Cp2 2 10

11 Blnear GDP Relaxaton Proposed Relaxaton F Blnear Terms 8 Relaxaton (Basc Steps) I II Convex Envelopes Convex Basc Hull Steps Relaxaton X Max Z = θp θfx γf CP CP s.t. FX P d Y11 P F.X + F.X - F.X Y21 P P d F.X + F.X F.X P d P F. X + F P. X F.X F +. XF.X F P.X F. X + F. X F P F.X P F. X + F. X F + F. X.X F.X P F. X + F. X F P F. X + F. X F. X P F. X + F. X F Y11 Y21 P F. X + F =. X α F. X 1X + β 1 FP = F α. + F. X F 2X + β2 F = α + 1X β 1 F = α2x + β2 X1 X X1 X 2 X X 2 X1 X CP X= 1 Cp X 2 X X 1 CP = Cp 2 2 CP = Cp1 CP = Cp2 Y11 Y21 = True CP,X,F R F F F Y 11,,Y 21 {True, False}. X. X. X. X 11

12 F 8 Blnear GDP Relaxaton Comparson Relaxaton (Lee & Grossmann, 2001) Actual Feasble Regon Proposed Relaxaton X The applcaton o basc steps pror to the dscrete relaxaton leads to a tghter relaxed easble regon 12

13 Rules to apply basc steps. When they mae a derence. Motvatng example: (cl conv S 2 ) (cl conv S 1 ) S 1 = P 1 P 2 P 1 S 2 = P 3 P 4 Extreme ponts P 3 P 4 P 2 cl conv ( S 2 S 1 ) No basc steps are necessary! In general the hypothess o the theorem s not easy to very To gan nsght, the structure o the program should be exploted 13

14 Rules to apply basc steps. When they mae a derence. Proposton : Let S 1 and S 2 be two dsunctons n whch the varables restrcted n S are not restrcted n the S (=1,2 ; =1,2 and ) Then the system satses the hypothess o theorem 4.2. Example: cl conv ( S 2 S 1 ) (cl conv S 2 ) (cl conv S 1 ) x 1 x 1 S 1 S 2 x 2 x 2 14

15 Summary o practcal rules to apply basc steps 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 expected I blneartes are outsde the dsunctons apply basc steps by ntroducng them n the dsunctons prevous to the relaxaton. I blneartes are nsde the dsunctons a less tghtenng eect s expected. A less ncrease n the sze o the ormulaton s expected when basc steps are appled between mproper dsunctons and proper dsunctons. 15

16 Methodology Step 1: GDP reormulaton (Apply basc steps ollowng 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 16

17 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. S = M = MU SU S3 M3 G/H/I S6 S ζ = 1 SU S1 N o cont. vars. : 114 N o dsc. vars. : 9 N o blnear terms: 36 Optmal structure M1 A S4 D h = ζ h YP h = β ' F =,, OPU CP = OPU F S, ' IPU, SU PU 0 ζ 1, S2 M2 D S5 M4 0,,, 0 CP S3 Z* = YP h { true, alse} h D PU 17

18 Case Study II: Poolng networ desgn Process superstructure Stream Pool Product S1 S2 S3 S4 S5 S1 S2 S5 P1 P2 P3 P4 N o cont. vars. : 76 N o dsc. vars. : 9 N o blnear terms: 24 Optmal structure Stream Pool Product P1 P3 Z* = w Generalzed Dsunctve Program Mn Z = CP + CST + c w d s.t. YP lo w I w W w = w, w W K I = ζ w, w W, K I ζ = 1 K CP = γ I w W J w W w w = J I J I w W K J w W K w W S w = λ w w' w' W Z = 0 w w J J w' W lo YST J w W CST = α w 0 ζ 1;0, w w w J K I, J, w W ' = 0 K, w W YST w = 0 CST = 0 w w up w w 0 CST, CP ; YST, YP { true, alse} YP = 0, I, w W = 0, K, w W CP = 0 I J 18

19 Perormance 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 % 19

20 Conclusons and Remars Tghter reormulaton o blnear GDPs. Proposed general rules to mplement basc steps. Proposed methodology to solve Blnear GDPs. Applcaton n two cases showed mproved perormance. 20