Hierarchy of Relaxations for Convex. and Extension to Nonconvex Problems

Transcription

1 Hierarchy of Relaations for Conve Nonlinear Generalized Disjunctive Programs and Etension to Nonconve Problems Ignacio Grossmann Sangbum Lee, Juan Pablo Ruiz, Nic Sawaya Center for Advanced Process Decision-maing Department of Chemical Engineering Carnegie Mellon University it Pittsburgh, PA 5, U.S.A NNU, rondheim December 5, 0

2 Basic question: How can we obtain strong relaations for MINLP/GDP problems? Outline. Overview of major relaations for nonlinear GDP problems big-m and hull relaation). Conve nonlinear GDP: hierarchy of relaations Concept of basic steps Equivalent NLP formulation. Application to global Optimization of nonconve GDP Bilinear, concave and linear fractional functions

3 Global Optimization of MINLP - Global optimization techniques find the global optimum m by sequentially entiall approimating the non-conve problem with a conve relaation Global Optimum ighter Relaation Lower Bound Conve Relaation - ighter formulations lead to more efficient algorithms Finding strong relaations is a ey element in. Global Optimization. Efficient solution of conve MINLP problems

4 MINLP Mied-Integer Nonlinear Programming 0 ),.. ), min y g t s y f Z Objective Function Inequality Constraints },, {, b B R X Y y X U L n }, {0,} { a Ay y y Y m f,y) and g,y) - assumed to be conve and bounded over X. f,y) and g,y) commonly linear in y 4

5 Mied-integer Nonlinear Programming Algorithms Branch and Bound BB) Ravindran and Gupta 985), Stubbs, Mehrotra 999), Leyffer 00) Generalized Benders Decomposition GBD) Geoffrion 97) Outer-Approimation OA) Duran and Grossmann 986), Fletcher and Leyffer 994) LP/NLP based Branch and Bound Quesada, Grossmann 994) Bonami et al. 008) Etended Cutting PlaneECP) Westerlund and Pettersson 99) Codes: SBB GAMS simple B&B MINLP-BB AMPL)Fletcher and Leyffer 999) Bonmin COIN-OR) OR) Bonami et al 006) FilMIN Linderoth and Leyffer 006) KNIRO Nocedal 009) DICOP GAMS) Viswanathan and Grossman 990) AOA AIMSS) ECP Westerlund and Peterssson 996) MINOP Schweiger and Floudas 998) BARON Sahinidis et al. 998) Couenne Belotti, Margot 008) 5

6 Continuous MINLP Relaation L Nonlinear Programming Problem 0 ),.. ), min y g s t y f Z L } {, b B R X Y y X U L n }, 0 { },, { a Ay y y Y b B R X Z L = Lower bound to optimal MINLP solution 6 Can we develop tighter lower bounds?

7 Generalized Disjunctive Programming GDP) Raman and Grossmann 994) Etension Balas, 979) Motivation: Facilitate modeling discrete/continuous problems OR operator j J min Z c s.t. r ) f ) 0 Y j g ) 0, K j c γ j Y true Ω R n, c R Objective Function Common Constraints Disjunction Constraints Fied Charges Logic Propositions Continuous Variables Y j true, false Boolean Variables Properties: a) Every GDP can be transformed into an MINLP b) Every bounded MINLP can be transformed into GDP 7

8 Process Networ with fied charges GDP model Min Z cc c d st Y Y p 0 c c 0 Y Y 5 p c c 0 Y Y 7 p c c 0 Y Y Y Y Y Y Y Y Y Y Y Y Y Y 0 U Y, Y, Y, Y, Y, Y rue, False c, c, c 8

9 Generalized Disjunctive Programming GDP) Raman and Grossmann 994) min jj Z s.t. r c ) f 0 ) Y j g j ) 0 K c γ j Y true Ω Y j R n, c true, R false Objective Function Common Constraints Disjunction Constraints Fied Charges Logic Propositions Continuous Variables Boolean Variables Relaation of GDP? 9

10 Big-M MINLP BM) MINLP reformulation of GDP min Z f ) K jj st.. r ) 0 g ) M ), j J, K j j j j J j j j, K A a 0, {0,} j Big-M Parameter Logic constraints Williams 990) NLP Relaation 0 => Lower bound to optimum of GDP j 0

11 Hull Relaation Formulation Consider Disjunction K j J g c j Y j ) 0 j heorem: Conve Hull of Disjunction Lee, Grossmann, 000) Disaggregated variables j {, c) j v jj 0 v jj g j j j, c U j j / ) 0, jj, 0 j v j j j, j J j j, j J j - weights for linear combination - Generalization of Balas 979) - Stubbs and Mehrotra 999), } => Conve Constraints Hull relaation: intersection of conve hull of each disjunction

12 Remars. Perspective function h v, ) g v / ) If g) is a bounded conve function, h v, ) is a bounded conve function Hiriart-Urruty and Lemaréchal 99) h,0) 0 for bounded g) j g j j / j ) 0 0 j U j R. Replace where by: ) ) g / ) ))) g 0) ) 0 j j j j j j a. Eact approimation of the original constraints as ε 0. Furman, Sawaya & Grossmann 009) b. he constraints are eact at j = 0 and at j = regardless of value of ε. if 0, ) g 0)) g 0) 0 0 j j j if, ) g / )) g 0)0) ) g / )) 0 j j j j j j j j c. he LHS of the new constraint is conve.

13 Hull Relaation Problem HRP) HRP: 0 ν g j min Z j j K jj j s. t. r ) 0 U jj ν j j j jj j ν j j, K, j J, K / ) j f ), K 0, j J A a, K j, ν 0, 0, j J, K j Conve Hull each disjunction Logic constraints Property: he NLP HRP) yields a lower bound to optimum of GDP).

14 Strength Lower Bounds heorem: he relaation of HRP) yields a lower bound that is greater than or equal to the lower bound that is obtained from the relaation of problem BM Grossmann, Lee 00) Conve hull relaation Big-M relaation Conve Hull of a set of disjunctions is smallest conve set that includes set of disjunctions. Projected relaation of CH) onto the space of BM) is as tight or tighter than that of BM) 4

15 Methods Generalized Disjunctive Programming GDP Branch and bound Lee & Grossmann, 000) Logic based methods Decomposition Outer-Approimation Generalized Benders uray & Grossmann, 997) Reformulation MINLP Branch and Bound Outer-Approimation Generalized Benders Etended Cutting Plane Hull relaation Big-M 5

16 GDP Eample ) ) min c Z 0 4) ) 0 ) 4) 0 ) ).. Y Y Y s t 0 4) ) 0 ) 4) 0 ) ) c c c Y Y Y,,. },, { 0, 8,, 0 j false true Y c j rue Y Y Y j 6

17 GDP Eample Find 0, S ) S ) S ) to minimize Z = -) + -) + c S Contour of f ) Local solutions Global Optimum.9,.707) Z* =.7 S S 0,0) 7

18 Eample : conve hull S Conve hull = convus j ) S S 8

19 Eample: CRP solution S Conve hull = convus j ) = 0.06 = = Conve combination of z j * S z j = v j / j Local solution point 0,0) S Conve hull optimum, Z L =.54 Lower Bound L =.59,.797) Infeasible for GDP 9

20 Eample : branch and bound First Node: S Optimal solution: Z U =.7 S Optimal Solution.9,.707) Z* =.7 0,0) S S 0

21 Eample : branch and bound Second Node: convs U S ) Optimal solution: Z L =.7 S Lower Bound Z L =.7 Upper Bound Z U =.7 S S 0,0)

22 Eample: Search ree Branching Rule: j - the weight of disjunction Fi Y j as true: fi j as. Root Node Y First Node Fi =, Z =.7 [, ] = [.9,.707] = [0,,0] Conve hull of all S i Z =.5454 = [0.06,0.955,0.09] Z L =.54 Branch on Y Y Second Node Conve hull of S and S Z =.7 = [ 0.7,0,0.6] Z U =.7 Bactrac Z L =.7 > Z U Stop

23 Process Networ with Fied Charges üray and Grossmann 997) 8 Boolean variables, 5 continuous, constraints 8 disjunctions,5 nonlinear) E Y A Y 4 6 Y 7 Y Y B 4 Y F D j : Unit 7 C Y i Y j Specifications

24 Optimal solution Minimum Cost: $ 68.0M/year A Raw Material Reactor B E 4 4 F 5 Reactor Products D j : Unit 7 4

25 MINLP- Branch and Bound Method Hull-Rel Z L = 6.48 = [0.,0.69,0.0,.0,,0,] 0 Z L = 5.08 Big-M 0 Y 4 = 0 Y 4 = Fi = Fi = 0 Y 6 = 0 Y 6 = Z L = Y =0 =0 = [0,,0.0,.0,,0,] 8 Y 8 = Y 8 Y 8 = Fi = Fi = 0 Stop Z L = 75.0 > Z U = [,0,0.0,.0,,0,] 0000] Y = 0 Y = 9 0* Z U = 7.79 Z U = 68.0 = Z* = [0,,,.0,,0,] = [0,,0,0,.0,,0,] Feasible Solution Optimal Solution Y = 0 Y = Y = Y = 0 Y = 8 5 nodes vs. 7 nodes of Big-M lower bound = 5.08) 5

26 Question Can we obtain stronger relaations than with Hull-Relaation? Etend Disjunctive Programming heory to Nonlinear Conve Sets DP: Linear programming with disjunctions Balas 974, 979, 985, 988) 6

27 Equivalence between GDP and DP Sawaya 007) GDP DP 0 he integrality of is guaranteed Proposition: GDP and DP have equivalent solutions. 7

28 Equivalent Conve Disjunctive Programs Regular Form: Form represented by the intersection of the union of conve sets Fisinregularform in Balas 985) 8

29 Illustrative Eample: Basic Steps F S S S S PP) S P P) S P P) hen F can be brought to DNF through basic steps. Apply Basic Step to: S S P P ) P P ) S P P ) P P ) P P ) P P ) We can then rewrite F S S S as F S S Apply Basic Step to: S S P P ) P P ) P P ) P P )) P P ) S P P P) P P P) P P P) P P P) P P P ) P P P ) P P P) P P P ) We can then rewrite F S S as F S which is its equivalent DNF 9

30 Hierarchy of Relaations for Conve Disjunctive Programs Illustration: F 0 P P) P P) P P P P 0

31 Hierarchy of Relaations for Conve Disjunctive Programs Illustration: clconv P P) F 0 P P) P P) P P clconv P P P) P

32 Hierarchy of Relaations for Conve Disjunctive Programs Illustration: h rel F 0 ) F P P) P ) 0 P P P P P No Basic Step Applied => HR

33 Hierarchy of Relaations for Conve Disjunctive Programs Illustration: clconv F 0 ) F P P) P ) 0 P P F P P ) P P ) P P ) P P P ) P P P P P P No Basic Step Applied => HR Basic Step Applied

34 Hierarchy of Relaations for Conve Disjunctive Programs Illustration: clconv F 0 ) F P P) P ) 0 P h rel F ) P P F P P) P P) P P) P P) P P P P ighter relaation! P No Basic Step Applied => HR P Basic Step Applied => CH 4

35 Conve nonlinear program equivalent to a conve disjunctive program NLPDP: Objective as he solution of the constraint NLP relaation leads to the solution of the DP! Generalizes conveification results of MILPs 5

36 Conve nonlinear program equivalent to a conve disjunctive program Illustrative Eample Solution of the relaation Disjunctive Program Solution of the DP Solution of the relaed program is different from solution of the disjunctive program 6

37 Conve nonlinear program equivalent to a conve disjunctive program Illustrative Eample Disjunctive Program Place objective as constraint and intersect with disjunction Solution of the program and its relaation Z =.7.9,.707) Solution of the hull relaation of DNF is the same as the solution of the disjunctive program heorem.8) 7

38 Summary of practical rules to apply basic steps Apply basic steps between those disjunctions with at least one variable in common. he more variables in common two disjunctions have the more the tightening epected. A basic step between a half space and a disjunctions with two disjuncts one of which is a point contained in the facet of the half space will not tighten the relaation. A smaller increase in the size of the formulation is epected when basic steps are applied between improper disjunctions and proper disjunctions. 8

39 Process Networ Revisited Illustrative Eample A E Y 4 6 Y 7 Y Y Y 4 Y 5 B F D C Y i Y j j : Unit 7 Specifications We can obtain a tighter relaation by applying basic steps between the improper disjunctions and the proper disjunctions Optimal Solution Z rel = obtained from Hull Relaation with basic steps Solves as an NLP! 9

40 Sizes of Formulations 40

41 Numerical Results All problems were solved using NLP branch-and-bound SBB/CONOP.4 GAMS) able: Performance using different reformulation strategies Poor lower bounds 4

42 Numerical Results All problems were solved using NLP branch-and-bound SBB/CONOP.4 GAMS) able: Performance using different reformulation strategies Improved lower bounds 50%probs 4

43 Numerical Results All problems were solved using NLP branch-and-bound SBB/CONOP.4 GAMS) able: Performance using different reformulation strategies Proposed vs BM: faster 0 out of Proposed vs HR: faster 8 out of Improved lower bounds 00%probs 4

44 Etension to Nonconve GDP Basic idea: strengthen th lower bound of global loptimum Phase Phase Nonconve GDP Conve GDP ight Conve GDP Initial lower bound Stronger lower bound Relaation Under/over estimating functions Conve envelopes Strengthen relaation Apply basic steps Remars. Since transformation to DNF impractical special rules are applied to identify promising basic steps. Stronger relaation can also be used to infer tighter bounds for variables 44

45 Illustrative Eample: Optimal reactor selection I A B F Demand F: Flow X: Conversion Objective GDP Formulation Min Z = -qfx + gf + CP Function - Profit) Demand constraint 8 I II Conversion s.t. FX d Y F X LO X X X CP Cp UP Y F X LO X X X CP Cp UP Selection Reactor Y Y = rue Feasible Region CP,X,F R F LO F F UP Y,,Y {rue, False} 0 0 X Optimum Z* =

46 Illustrative Eample: Optimal reactor selection I Lee & Grossmann 00) Relaation F Bilinear erms 8 Relaation No Basic Steps) I II Conve Envelopes Conve Hull Relaation X Ma Min Z = -qfx qp - gf + gf CP + CP s.t. FX P d P F.X LO + F UP.X - F UP.X LO P F.X UP + F LO.X F LO.X UP P F.X LO + F LO.X F LO.X LO P F.X UP + F UP.X F UP.X UP Y F X LO X X X CP Cp UP Y Y = rue CP,X,F R F LO F F UP 0 Y 0,,Y {rue, False} Y F X LO X X X CP Cp UP Lower bound Z* = -.8 <

47 Illustrative Eample: Optimal reactor selection I Proposed Relaation F Bilinear erms 8 Relaation Basic Steps) I II Conve Basic Hull Step Relaation Conve Envelopes Min Z = -P -FX F + + CP CP s.t. FX P d Y P F.X LO + F UP.X - F UP.X LO Y P P d F.X UP + F LO.X F LO.X UP P d UP LO UP LO P F. X F P. XF.X F LO LO+. XF UP LO.X F LO P.X LO F. X F. X F LO UP P F.X LO UP P F. X F. X F UP UP+ FLO. X UP.X F UP.X P UP F. X F. X F LO LO LO LO LO LO P F. X F. X F. X P F. X F. X F Y UP UP UP UP Y UP UP P F. F. X F P F. X F. X F. X X F X F X LO UP F X X X X LO UP X X X LO UP LO UP X X CP X Cp X X X CP Cp CP Cp CP Cp Y Y = rue CP,X,F R F LO F F UP 0 Y 0,,Y {rue, False} X LO UP LO UP. X. X. X. X UP LO LO UP Lower bound Z* = -. < -.0 and tighter than -.8! 47

48 Illustrative Eample: Optimal reactor selection I Comparison of Relaations F 8 Relaation No Basic Steps) Actual Feasible Region 0 0 Relaation Basic Steps) X he application of basic step prior to the discrete relaation leads to a tighter relaed feasible region = > stronger lower bounds 48

49 Dimensions of est Problems Bilinear/Concave Bilinear erms Concave Functions Discrete Variables Continuous Variables Eample 0 Eample 0 5 Eample Eample Eample Eamples - Optimal Reactor selection I - Optimal Reactor selection II - HEN with investment cost - multiple size Regions uray & Grossmann, 996) 4- Water reatment Networ Design problem Galan & Grossmann, 998) 5- Pooling Networ Design problem Lee & Grossmann, 00) Strong linear relaations eist for bilinear and concave functions 49

50 Dimension of Case Studies Linear Fractional, Posynomial, Eponential Cont. Vars. Boolean Vars. Logic Const. Disj. Const. Global Const. PROC 5 PROC 5 RXN 4 6 RXN 4 6 HEN 8 Reference PROC, PROC : Optimal Process Networ Problem RXN, RXN : Optimal Reactor Networ Problem HEN : Optimal lheat Echanger Networ Problem Strong nonlinear relaations eist for linear fractional and posynomial functions 50

51 Heat Echanger Networ Problem,.. min 4 U Q A U Q A s t C C A c A c A c A c Z C Heat Echanger Networ Generalized Disjunctive Program Q Y Q Y C A Q Y C U Q A Y U U C H H ) ), ) ), 0 0 0,,,, FCP Q FCP Q FCP Q FCP Q C A Q C U Q A in H H in C C out H H out H H 4 C 4 H ) ) ) ), ) ),,,, 4, 4 4 Q Q Q EMA EMA FCP Q FCP Q i C H C i C H C total in C in C in H H in C C Linear Fractional erms in constraints, ) ), ) ) ) ), ) ), 4,,, 4,, 4,,,,,,,, in H out H in H out H in H in C in H in C in C out H out C in C out H out C 5,...,4 0,, 4,, EMA I Q i i in C in C

52 Prediction of Lower Bounds Global Optimum Bilinear Concave Global Optimum Lower Bound Hull Relaation Lower Bound Basic Steps DNF Lower Bound React React HEN Water Pool Process Linear Process Fractional, Posynomial, Eponential RXN RXN HEN Lower bounds improved in all cases Ave. increase % 8 out of 0 achieved theoretically best lower bound DNF)! 5

53 Global Optimization Methodology GDP reformulation Apply basic steps following the rules presented Disjunctive B&B Bound Contraction Zamora & Grossmann, 999) Y j Y j Feasible discrete Spatial B&B Spatial Branch and Bound Lee & Grossmann, 00) 5

54 Remars Computational Performance- Bilinear/Concave Global Optimum Global Optimization echnique using Hull Relaation Nodes Bound contract. % Avg) CPU ime sec) Global Optimization echnique using Proposed Relaation Nodes Bound contract. % Avg) CPU ime sec) Eample Eample Eample Eample Eample Proposed relaation led to a significant bound contraction at the root node. - 44% reduction number of nodes, % reduction CPU time tighter relaation but increased size of proposed relaation Size of the LP Relaation Hull Relaation) Size of the LP Relaation Proposed) Constraints Variables Constraints Variables Eample Eample Eample Eample Eample

55 Computational Performance- Nonlinear Global Optimum Global Optimization echnique using Lee & Grossmann Relaation Nodes Bound contract. % Avg) CPU ime sec) Global Optimization echnique using Proposed Relaation Nodes Bound contract. % Avg) CPU ime sec) PROCN PROCN RXN RXN HEN Remars -Proposed relaation led to a significant bound contraction at the root node. -he reduced number of nodes is a further indication of tighter relaation - Modest savings compared to bilinear/concve due to small size 55

56 Conclusions -Proposed an etension of disjunctive programming theory to nonlinear conve sets that tyields hierarchy h of relaations concept tbasic steps) -ightest of these relaations allows in theory the solution of the DP as an NLP - Applied the proposed framewor to several instance obtaining significant improvements in the performance tighter lower bounds) - Proposed framewor can be applied dto nonconve GDP problems yielding tighter lower bounds on global optimum bilinear, concave, linear fractional) and can be etended to nonlinear conve envelopes 56

57 57

58 Rules to apply basic steps Proposition 4. Let S and S be two disjunctions in which the variables appearing in S do not appear in S and viceversa. hen the system satisfies cl conv S S ) = cl conv S )) cl conv S ) Basic step Hull relaation S S cl conv S S ) cl conv S ) cl conv S ) S S cl conv S ) S S cl conv S ) RULE : Apply basic steps between disjunctions with at least one variable in common. RULE : Apply basic steps between disjunctions with more variables in common. 58

59 Rules to apply basic steps cont.) Proposition 6. When a basic step is applied between an improper disjunction with a proper disjunction the number of conve sets in the resulting disjunction is not increased Intersection proper disjunction conve sets with one improper disjunction conve set F = S S S = P P S = P Number of conve sets after the intersection of proper disjunctions : F = S = P P ) P P ) RULE 4: Apply basic steps between improper and proper disjunctions 59

60 Illustrative Eample Nonconve GDP Nonconve GDP Objective function and feasible region of MGDP NC Z*=

61 Illustrative Eample Conve GDP Relaation Phase No basic steps) Lower bound Z*=-.58 Relaation of MGDP NC without basic step leads to a significant ifi gap with the optimal solution 6

62 Illustrative Eample Conve GDP Relaation Phase Basic steps) Lower bound Z*=-.55 Eact! he objective function is added to the set of constraints before the application of basic steps Z*=-.55 he relaation of MGDP NC after basic steps leads to the optimal solution! 6

63 Numerical Results Instances: CirclesD0,CirclesD6, CirclesD6 CirclesD6 Problem Illustration Generalized Disjunctive Program 6

64 Process Networ Problem Instances: Proc08, Proc0, Proc Problem Illustration Generalized Disjunctive Program

65 Farm Layout Land Problem Instances: Flay0,Flay0,Flay04 Problem Illustration Generalized Disjunctive Program Length A j Width W i A i L i 65

66 Constrained Layout Problem Instances: Clay00,Clay00,Clay004 Problem Illustration Generalized Disjunctive Program del ij L i H i i,y i ) Bo i dely ij L i H i j,y j ) Bo j r,y ) Circular area 66

67 Nonconve Generalized Disjunctive Programs Min Z f ) Objective Function s.t. c K g ) 0 Global Constraints i D r c i Y i ) 0 i ) Disjunctions K Bilinear Concave R Linear fractional ΩY)= rue n,c R,Y i {rue,false} Logic Propositions i D, K Goal : Develop efficient algorithms for bilinear, concave, linear fractional GDP with improved relaations => stronger lower bounds global optimum 67

68 Proposed framewor to obtain stronger relaations for nonconve GDP Bilinear, Concave, Linear fractional GDP) Ruiz, Grossmann 0) Basic idea: Based on relaation of the nonconve GDP as a conve GDP eploit the theory behind DP for conve sets to obtain stronger relaations. he framewor consists of two main phases: - Generate a valid Conve Generalized Disjunctive Program relaation for the nonconve GDP problem bilinear, concave: linear conve envelopes linear fractional: conve nonlinear envelopes - Strengthen the continuous relaation of the conve GDP obtained in phase by applying a set of basic steps he hierarchy of relaations obtained by the application of basic steps is valid for the original nonconve GDP problem 68

69 Proposed Framewor Phase ) Bilinear, concave Nonconve GDP Linear GDP Relaation Polyhedral Relaation e.g. McCormic conve Envelopes, secant) awarmalani & Sahinidis, 00 GDP RLP is a valid Linear GDP relaation for GDP NC Carnegie Mellon Remar: For linear fractional conve nonlinear GDP relaation 69

70 Proposed Framewor Phase ) Valid Hierarchy of Relaations Equivalence in LGDP representations Inclusion of nonconve GDP in LGDP Hence applying heorem 4. Balas, 979) Valid Hierarchy of Relaations for GDP NC 70

71 Dimensions of est Problems Bilinear erms Concave Functions Discrete Variables Continuous Variables Eample 0 Eample 0 5 Eample Eample Eample Eamples - Optimal Reactor selection I - Optimal Reactor selection II - HEN with investment cost - multiple size Regions uray & Grossmann, 996) 4- Water reatment Networ Design problem Galan & Grossmann, 998) 5- Pooling Networ Design problem Lee & Grossmann, 00) Carnegie Mellon 7

72 Eample : Optimal Reactor selection I Bilinear GDP Superstructure of Reactor Networ Generalized Disjunctive Program R Min Z FX F CP) R No. of cont. vars. : No. of disc. vars. : No. of bilinear terms: No. of concave functions: 0 s. t. FX d Y F X lo up X X X CP Cp Y Y Y F X lo up X X X CP Cp lo up X, F, CP R, F F F, Y, Y { rue, False} Optimal selection R Z* = -.0 Carnegie Mellon 7

73 Eample : Optimal Reactor selection II Concave GDP Superstructure of Reactor Networ Generalized Disjunctive Program R Min Z Cp R s.t. Fa Fc Fc No. of cont. vars. : 5 No. of disc. vars. : No. of bilinear terms: 0 No. of concave functions: Y Fa Fb 0.7 Cp Fa LO UP Fa Fa Fa Y Fa Fb 0.7 Cp Fa LO Fa Fa Fa UP Optimal selection Y Y LO Fc Dc Dc UP R LO Fc Dc Dc Fa LO Fa Fa UP UP Carnegie Mellon Z* = 6. 7

74 Eample : Heat echanger networ design Bilinear/Concave GDP Heat Echanger Networ HX HX HX No. of cont. vars. : 8 No. of disc. vars. : 9 No. of bilinear terms: 4 No. of concave functions: 9 Generalized Disjunctive Program Min Z CP FCP ) C s.t. FCP FCP FCP i i h outh ) Ccu FCPc outc inh ) ) AU incw outh) ) AU s ) s ) AU h ih inh h inh c outc FCP ) FCP ) h inh c inc Yi CPi i Ai lo A Ai Yi i CPi i Ai up lo A A Ai inc ) i CPi i Ai i up lo up A i i A A i A outc ) outcw ) hu Y i Optimal Objective Y i Yi Y i rue i,, Z*= lo up lo up Carnegie Mellon 74

75 S S S Eample 4: Water reatment Networ design Bilinear GDP bilinear outside disjunction) Process superstructure M M M D/E/F A/B/C G/H/I S4 S5 S6 M4 Generalized Disjunctive Program Min Z = CP PU st s.t. f j is is f j i im f f i j i j j j SU MU SU No. of cont. vars. : 4 No. of disc. vars. : 9 No. of bilinear terms: 6 Optimal structure S M A S4 f j i fi D h i f j h YP j jh j fi' j F fi, j i 0 j, i OPU CP i OPU i F i S, i' IPU j,, j SU PU S M D S5 M4 0 f, f i, j, j i j 0 CP S Z* = 4 YP h { true, false} h D PU 75 Carnegie Mellon

76 Eample 5: Pooling networ design Bilinear GDP bilinear inside disjunction) Process superstructure Generalized Disjunctive Program Min Z = CPj CSi cij fijw d Stream i Pool j Product S S S S4 S5 S P P P P4 No. of cont. vars. : 76 No. of disc. vars. : 9 No. of bilinear terms: 4 Optimal structure Stream i Pool j Product S S5 Carnegie Mellon P P Z* = s.t. fijw f ii ww jj ww f jw jj ii jj ii ww K jj ww K ww S f ijw f f iw ijw' w' W Z 0 jw w jj jj w' W f lo YS i j J w W CS i i f ijw f jw jw f jw j J K i I, j J, ww ' 0 K, ww YSi fijw 0 CS i 0 YPj lo f fijw ii ww f jw fijw, ww K ii f f f jw j fijw, ww, K ii j K CP j j j 0 ;0 f, f jw ijw f up ijw jw 0 CS, CP ; YS, YP { true, false} i j i j YP j 0, i I, w W 0, K, ww CPj j i I j J 76

77 Relaation Results Global Optimum Lower Bound Hull Relaation Lower Bound Basic Steps DNF Lower Bound Eample Eample Eample Eample Eample Remars -Proposed methodology leads to improvements in the lower bounds => % gap reduction -he lower bound of the DNF is the best lower bound attainable for a given LGDP. - Note in eamples, and 4 the lower bound is the same as the lower bound of the DNF Often, it is not necessary to reach the DNF form to have good lower bounds. Carnegie Mellon 77

78 Linear fractional terms Nonlinear Conve Envelopes Size reformulations Predicted lower bounds - In all cases lower bound is the same as the lower bound of the DNF the best lower bound attainable. Carnegie Mellon 78

79 Global Optimization Methodology GDP reformulation Apply basic steps following the rules presented Disjunctive B&B Bound Contraction Zamora & Grossmann, 999) Y j Y j Feasible discrete Spatial B&B Spatial Branch and Bound Lee & Grossmann, 00) Carnegie Mellon

80 Remars Computational Performance Global Optimum Global Optimization echnique using Hull Relaation Nodes Bound contract. % Avg) CPU ime sec) Global Optimization echnique using Proposed Relaation Nodes Bound contract. % Avg) CPU ime sec) Eample Eample Eample Eample Eample Proposed relaation led to a significant bound contraction at the root node. - 44% reduction number of nodes, % reduction CPU time tighter relaation but increased size of proposed relaation Size of the LP Relaation Hull Relaation) Size of the LP Relaation Proposed) Carnegie Mellon Constraints Variables Constraints Variables Eample Eample Eample Eample Eample

81 MINLP Reformulations Hull reformulation Specify in HRP as 0- variables 0 ).. ) min r t s f Z K J j j j,, 0, K J j U K j j j J j j ν ν, 0, ) /, K J j g K j j j j J j j ν, 0, 0,, K J j a A j j j j j ν 8

82 S S S Eample 4: Water reatment Networ design Bilinear GDP bilinear outside disjunction) Process superstructure M M M D/E/F A/B/C G/H/I S4 S5 S6 M4 Generalized Disjunctive Program Min Z = CP PU st s.t. f j i S i S f j i i M f f i j i j j j SU MU SU No. of cont. vars. : 4 No. of disc. vars. : 9 No. of bilinear terms: 6 Optimal structure S M A S4 f j i fi D h j i f j jh F f j i' h YP j, i OPU j CP f j i, i 0 i OPU i F i S, i' IPU j,, j SU PU S M D S5 M4 0 f, f i, j, j i j 0 CP S Z* = 4 YP h { true, false} h D PU 8