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Ignacio E. GrossmannCenter for Advanced Process Decision-making
Carnegie Mellon University Pittsburgh, PA 15213, USA
Overview of Generalized Disjunctive Programming
EWO SeminarJanuary 22, 2009
2
• EWO Problems involve Discrete/Continuous Optimization Linear/Nonlinear models0-1 and continuous decisions
• Optimization Models Mixed-Integer Linear Programming (MILP)Mixed-Integer Nonlinear Programming (MINLP)
Motivation
• OutlineModeling with GDP (LOGMIP)Solution methods (relaxations)Example supply chain with contracts for purchasesNew frontiers:
- Improved relaxations (basic steps) (Strip packing problem)- Nonconvex GDPs
Alternative approach:Logic-based: Generalized Disjunctive Programming (GDP)
3
• Mixed-Integer Linear/Nonlinear Programming
Objective Function
Equality Constraints
mn yRxyxgyx hts
yxfZ
1,0,0,
0),(..),(min
∈∈
≤=
=
)(
MINLP: f(x,y) and g(x,y) – commonly assumed to be convex and boundedoften not true in practice => nonconvex
f(x,y) and g(x,y) commonly linear in y often true in practice
Inequality Constraints
MI(N)LP
MILP: f(x,y) and g(x,y) linear in x and y
4
Codes MILP: Commercial : CPLEX, XPRESS, (GUROBI ?) XA, OSLOpen Source (COIN-OR) : CBC, SYMPHONY Codes MINLP: SBB GAMS simple B&B MINLP-BB (AMPL)Fletcher and Leyffer (1999)
Bonmin (COIN-OR) Bonami et al (2006) FilMINT Linderoth and Leyffer (2006)
DICOPT (GAMS) Viswanathan and Grossman (1990) AOA (AIMSS)
α−ECP Westerlund and Peterssson (1996) MINOPT Schweiger and Floudas (1998)
BARON Sahinidis et al. (1998) Global solvers Couenne (COIN-OR) Belottti et al. (2008)
Mixed-integer Nonlinear Programming
5
Generalized Disjunctive Programming
Motivation
1. Facilitate modeling of discrete/continuous optimization problems through use algebraic constraints andsymbolic logic expressions
2. Improve combinatorial search effort3. Improve handling nonlinearities
6
Definition of GDPRaman, R. and I.E. Grossmann, "Modeling and Computational Techniques for Logic Based IntegerProgramming," Computers and Chemical Engineering, 18, 563 (1994).
Logic-based Outer-ApproximationTurkay, M. and I.E. Grossmann, "Logic-Based MINLP Algorithms For the Optimal Synthesis ofProcess Networks," Computers and Chemical Engineering , 20, 959-978 (1996).
Convex hull nonlinear disjunctionsLee, S. and I.E. Grossmann, "New Algorithms for Nonlinear Generalized Disjunctive Programming,”Computers and Chemical Engineering, 24, pp.2125-2141 (2000).
Established relation with DPSawaya, N. and I.E. Grossmann, “Reformulations, Relaxations and Cutting Planes for Linear Generalized Disjunctive Programming,” submitted for publication Math. Prog. (2008).
Basic References
LOGMIPVecchietti, A. and I.E. Grossmann, "LOGMIP: A Disjunctive 0-1 Nonlinear Optimizer for Process Systems Models, Computers and Chemical Engineering 23, 555-565 (1999).
7
Background reading
Balas E., “Disjunctive Programming and a Hierarchy of Relaxations for Discrete ContinuousOptimization Problems”, SIAM J. Alg. Disc. Meth., Vol. 6, No. 3, 1985.
Jeroslow R.G., “Logic-Based Decision Support: Mixed-Integer Model Formulation”, Annals of Discrete Mathematics, 40, North Holland (Amsterdam), 1989.
Hooker J., “Logic-based methods for optimization: combining optimization and constraintsatisfaction”, John Wiley & Sons, 2000.
8
“Typical” disjunction for scheduling
Product A and Product B must be processed on same line
Sequencing decision:A before B OR B before ALet TA, TB be starting times (variables)Let pA, pB be processing times (parameters)
[ ] [ ]A A B B B AT p T T p T+ ≤ + ≤∨
Disjunction
A before B OR B before A
TA TB TB TA
9
Generalized Disjunctive Programming (GDP)
( )
1
min ( )
( ) 0
( ) 0
Ω
kk
jk
jkk
k jk
nk
jk
Z c f x
s.t. r x
Y
g x k K j J
c γ
Y true
x R , c RY true, false
= +
≤
⎡ ⎤⎢ ⎥
≤ ∈⎢ ⎥∈ ⎢ ⎥=⎣ ⎦=
∈ ∈
∈
∑
∨
Objective Function
Common Constraints
Continuous Variables
Boolean Variables
Logic Propositions
OR operator
Disjunction
Fixed Charges
Constraints
Quantitative/Symbolic Optimization Model
•Raman and Grossmann (1994) (Extension Balas, 1979)
10
1 2
1 2
1 2 2
3 3
1 2 1
1 2 3
2 3
1 2
min 2. .
2 0 2 05 7
1 0
0 5 0 5, 0
,j
Z c x xs t
Y Yx x x
c c
Y Yx x x
Y Y Y(Y Y )
x , x c Y true fa
= + +
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− + + ≤ ∨ − ≤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎣ ⎦ ⎣ ⎦
¬⎡ ⎤ ⎡ ⎤∨⎢ ⎥ ⎢ ⎥− ≤ =⎣ ⎦ ⎣ ⎦
∧ ¬ ⇒ ¬¬ ∧
≤ ≤ ≤ ≤ ≥∈ , 1,2,3.lse j =
Small GDP Problem
11
SET I /1*3/;SET J /1*2/;BINARY VARIABLES Y(I);POSITIVE VARIABLES X(J), C;VARIABLE Z;EQUATIONS EQUAT1, EQUAT2, EQUAT3, EQUAT4,
EQUAT5, EQUAT6, DUMMY, OBJECTIVE;
EQUAT1.. X('2')- X('1') + 2 =L= 0;EQUAT2.. C =E= 5;EQUAT3.. 2 - X('2') =L= 0;EQUAT4.. C =E= 7;EQUAT5.. X('1')-X('2') =L= 1;EQUAT6.. X('1') =E= 0;DUMMY.. SUM(I, Y(I)) =G= 0;OBJECTIVE.. Z =E= C + 2*X('1') + X('2');X.UP(J)=20;C.UP=7;
Input file LOGMIP (GAMS)$ONTEXT BEGIN LOGMIPDISJUNCTION D1, D2;D1 ISIF Y('1') THEN
EQUAT1;EQUAT2;
ELSIF Y('2') THENEQUAT3;EQUAT4;
ENDIF;
D2 ISIF Y('3') THEN
EQUAT5;ELSE
EQUAT6;ENDIF;Y('1') and not Y('2') -> not Y('3');Y('2') -> not Y('3') ;Y('3') -> not Y('2') ;$OFFTEXT END LOGMIPOPTION MIP=LOGMIPM;MODEL PEQUE2 /ALL/;SOLVE PEQUE2 USING MIP MINIMIZING Z;
Big-MFormulation
12
LogMIP
Part of GAMS Modeling System-Disjunctions specified with IF Then ELSE statementsDISJUNCTION D1(I,K,J);D1(I,K,J)
with (L(I,K,J)) ISIF Y(I,K,J) THEN
NOCLASH1(I,K,J);ELSE
NOCLASH2(I,K,J);ENDIF;
-Logic can be specified in symbolic form (⇒, OR, AND, NOT )or special operators (ATMOST, ATLEAST, EXACTLY)
-Linear case: MILP reformulation big-M, convex hull-Nonlinear: Logic-based OA
http://www.ceride.gov.ar/logmip/
Aldo Vecchietti, INGAR
13
Generalized Disjunctive Programming (GDP)
( )
1
min ( )
( ) 0
( ) 0
Ω
kk
jk
jkk
k jk
nk
jk
Z c f x
s.t. r x
Y
g x k K j J
c γ
Y true
x R , c RY true, false
= +
≤
⎡ ⎤⎢ ⎥
≤ ∈⎢ ⎥∈ ⎢ ⎥=⎣ ⎦=
∈ ∈
∈
∑
∨
Objective Function
Common Constraints
Disjunction
Fixed Charges
Continuous Variables
Boolean Variables
Logic Propositions
OR operator Constraints
Solution methods: Relaxation?
14
Systematic Procedure to Derive LinearInequalities for Logic Propositions
Goal is to Convert Logical Expression intoConjunctive Normal Form (CNF): Q1 ∧ Q2 ∧ . . . ∧ Qs
where clause Qi : P1 ∨ P2 ∨ . . . ∨ Prand Pi is a literal
3. Recursively distribute or over and(P1 ∧ P2) ∨ P3 ⇔ (P1 ∨ P3) ∧ (P2 ∨ P3)
2. Move negation inward applying De Morgan’s theorem¬ (P1 ∧ P2) ⇔ ¬ P1 ∨ ¬ P2¬ (P1 ∨ P2) ⇔ ¬ P1 ∧ ¬ P2
1)1( ≥−+⇒ ∑∑∈∈
iNegiNonnegi
i yy
Linear inequality
Steps to obtain CNF (Clocksin, Melish, 1981)
1. Replace implication by disjunctionP1 ⇒ P2 ⇔ ¬ P1 ∨ P2 ¬ negation
15
ExampleIf prod A or prod B implies reactor 3 or reactor 4
P1 P2 P3 P4
P1 ∨ P2 ⇒ P3 ∨ P4
1. Remove implication¬(P1 ∨ P2) ∨ (P3 ∨ P4)
2. Apply De Morgan’s(¬ P1 ∧ ¬ P2 )∨ (P3 ∨ P4)
3. Distribute OR over AND(¬ P1 ∨ P3 ∨ P4) ∧ (¬ P2 ∨ P3 ∨ P4) => CNF !
clause 1 clause 2from clause 1 from clause 21 – y1 + y3 + y4 ≥ 1 1 – y2 + y3 + y4 ≥ 1
y3 + y4 ≥ y1y3 + y4 ≥ y2
16
Big-M MI(N)LP (BM)
• MINLP reformulation of GDP
min ( )
. . ( ) 0 ( ) (1 ) , ,
1,
0, 0,1
k
k
jk jkk K j J
jk jk jk k
jkj J
jk
Z f x
s t r xg x M j J k K
k K
A a x
γ λ
λ
λ
λλ
∈ ∈
∈
= +
≤≤ − ∈ ∈
= ∈
≤≥ ∈
∑ ∑
∑
Big-M Parameter
Logic constraints
LP/NLP Relaxation 0 1jkλ≤ ≤
17
Convex Hull Formulation (CH)
• Consider Disjunction k ∈ K( ) 0
k
jk
jkj J
jk
Y
g x
c γ∈
⎡ ⎤⎢ ⎥
≤⎢ ⎥⎢ ⎥=⎢ ⎥⎣ ⎦
∨
Theorem: Convex Hull of Disjunction k (Lee, Grossmann, 2000)Disaggregated variables ν j
λj - weights for linear combination
, 0)/(
1,0 ,1
,0
, ,|),(
kjk
jk
jkjk
jkJj
jk
kjkjk
jk
Jjjkjk
Jj
jk
Jjvg
JjUv
cvxcx
k
k
∈≤
≤<=∑
∈≤≤
∑=∑=
∈
∈∈
λλ
λλ
λ
γλ
Convex Constraints=>
k
Convex for g(x) convex
18
Remark I
kjk jkj J
A x b∈
⎡ ⎤≤⎣ ⎦∨For linear disjunctions
Convex-hull
1
0 1
k
k
jkj J
jk jk jk jk k
jkj J
jk k
x
A b j J
j J
ν
ν λ
λ
λ
∈
∈
=
≤ ∈
=
≤ ≤ ∈
∑
∑
( )jk jk jkg x A x b= −set
Balas (1985)
19
Remark II
For special cases disaggregated variables can be eliminated
0Y Y
L x U x¬⎡ ⎤ ⎡ ⎤
∨⎢ ⎥ ⎢ ⎥≤ ≤ =⎣ ⎦ ⎣ ⎦Example
x0 L U
Convex hull formulation1 2
1
2 0*(1 )0,1
xL U
ν ν
λ ν λ
ν λλ
= +
≤ ≤
= −=
0,1L x Uλ λλ
≤ ≤=
Since ν2 = 0 ⇒
w-MIP representable (Raman, G, 1994)
See also John Hooker EWO SeminarTour Modeling Techniques
20
Remark III
0, ( )( (0)) (0) 0 0jk jk jkif g gλ ε ε= ⇒ − = ≤
1, ((1)( ( / (1)) (0)(0) (1) ( / (1)) 0jk jk jk jk jk jkif g g gλ ν ε ν= ⇒ − = ≤
a. Exact approximation of the original constraints as ε → 0.
c. The LHS of the new constraint is convex.
b. The constraints are exact at λjk = 0 and at λjk = 1 regardless of value of ε.
Replace by:( / ) 0jk jk jk jkgλ ν λ ≤ 0 jk jkUν λ≤ ≤where
((1 ) )( ( / ((1 ) ))) (0)(1 ) 0jk jk jk jk jk jkg gε λ ε ν ε λ ε ε λ− + − + − − ≤
Furman, Sawaya & Grossmann (2009)
( / ) 0jk jk jk jkgλ ν λ ≤How to implement for zero ?jkλ
Nonlinear disjunctions
21
MI(N)LP Reformulation (CH)
, 1,0 ,0 ,
, ,0)/(
, 1
, ,0
,
0)( ..
)( min
Kk JjxaA
Kk Jjg
Kk
Kk JjU
Kk x
xrts
xfZ
kjk
jk
kjk
jk
jkjk
Jjjk
kjkjk
jk
Jj
jk
Kk Jjjkjk
k
k
k
∈∈=≥≤
∈∈≤
∈=∑
∈∈≤≤
∈∑=
≤
+∑ ∑=
∈
∈
∈ ∈
λλ
λλ
λ
λ
λγ
ν
ν
ν
ν
LP/NLP Relaxation 0 1jkλ≤ ≤
Convex Hull vs Big-M
Proposition: The projected relaxation of (CH) onto the space of (BM) is always as tight or tighter than that of (BM) (Grossmann I.E. , S. Lee, 2003)
Trade-off: Big-M fewer vars/weaker relaxation vs Convex-Hull tighter relaxation/more vars
(CH) Relaxed Projected Feasible Region
.
(BM) Relaxed
Feasible Region
23
Logic based methods
Branch and bound(Lee & Grossmann, 2000)
DecompositionOuter-Approximation
(Turkay & Grossmann, 1997)
Methods Generalized Disjunctive Programming
Convex-hull Big-M
Reformulation MI(N)LP
LP-based B&BOuter-Approximation
Extended Cutting Plane
GDP
24
A Branch and Bound Algorithm for GDP
• Tree Search NLP subproblem at each node
• Solve Relaxation GDP (CRP)lower bound
CRP
+ fix a term indisjunction
CRPCRP
+ convex hullof remaining
terms
• Branching Rule Set the largest λj as 1 Dichotomy rule
• Logic inferenceCNF unit resolution (Raman & Grosmann, 1993)
• Depth first searchWhen all the terms are fixed
upper bound • Repeat Branching until ZL > ZU.
25
Process Network with Fixed Charges• Türkay and Grossmann (1997)
Superstructure of the process
1
2
6
7
4
3
5 8
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
: Unitj
Y1 ∨ Y2
Y6 ∨ Y7
Y4 ∨ Y5
C
D
F
E
Yi ∨ Yj
Specifications
26
Minimum Cost: $ 68.01M/year
2
6
4
8
x1
x4
x19
x13
x14
x11
x12
x18
x20
x23 x24x5A
B
: Unitj
D
F
E
RawMaterial ProductsReactor Reactor
Optimal solution
x7
x6x10
x17
x25
x8
27
Proposed BB MethodZL = 62.48
λ = [0.31,0.69,0.03,1.0,1,0,1]
ZU = 68.01 = Z*λ = [0,1,0,0,1.0,1,0,1]
Optimal Solution
ZU = 71.79λ = [0,1,1,1.0,1,0,1]
Feasible Solution
ZL = 75.01 > ZU
λ = [1,0,0.022,1.0,1,0,1]
ZL = 65.92λ = [0,1,0.022,1.0,1,0,1]
0
32
41
Fix λ2 = 1
Fix λ3 = 1 Fix λ3 = 0
Fix λ2 = 0
Stop
5 nodes vs. 17 nodes of Standard BB (lower bound = 15.08)
Proposed BB
0
ZL = 15.08 Big-M Std. BB
1 2
43
1413 5 6
812111615 7
10*9
Y4 = 0 Y4 = 1
Y6 = 0 Y6 = 1
Y8 = 0Y8 = 1
Y1 = 0 Y1 = 1
Y8 = 0 Y8 = 1
Y2 = 0 Y2 = 1 Y1 = 1
Y3 = 0 Y3 = 1
28
Logic-based Outer ApproximationMain point: avoids solving MINLP in full space
Turkay, Grossmann (1997)
SDkiiDifalseYforc
xB
SDkDitrueYforc
xhxgts
xfcZ
kikk
i
kkiikk
ik
SDkk
∈≠∈=⎭⎬⎫
==
∈∈=⎭⎬⎫
=≤
≤
+= ∑∈
,ˆ,0
0
,ˆ0)(0)(..
)(min
ˆγ (NLPD)
x ∈ Rn, ci ∈ Rm,
NLP Subproblem:(reduced)
α+= ∑k
kcZMin
(MGDP)1,...,= 0)()()()()()(..
Lxxxgxg
xxxfxftsT
T
llll
lll
⎪⎭
⎪⎬⎫
≤−∇+
−∇+≥α
SDk
cL
xxxhxh
Y
ikk
ik
Tikik
ik
Di k
∈
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=∈
≤−∇+∨∈
γl
lll 0)()()(
Ω (Y) = True α ∈ R, x ∈ Rn, c ∈ Rm, Y ∈ true, falsem
Master Problem:
Proceed as OA. Requires initialization several NLPs to cover all disjunctions
Redundant constraints are eliminated with falsevalues
Master problem solved with disjunctive branch and bound orwith MILP reformulation
29
Process Network with Fixed Charges• Türkay and Grossmann (1997)
Superstructure of the process
1
2
6
7
4
3
5 8
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
: Unitj
Y1 ∨ Y2
Y6 ∨ Y7
Y4 ∨ Y5
C
D
F
E
Yi ∨ Yj
Specifications
30
11
22
66
77
44
33
55 88
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
C
D
F
E
NLP1 = 73.7
11
22
66
77
44
33
55 88
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
C
D
F
E
NLP2 = 103.6
11
22
66
77
44
33
55 88
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
C
D
F
E
NLP3 = 133.8 MIP = 67.9Lower bound
LOGMIP- Logic Based OA
11
22
66
77
44
33
55 88
x1
x4
x6
x21
x19
x13
x14
x11
x7
x8
x12
x15
x9
x16 x17
x25x18
x10
x20
x23x22 x24x5
x3x2
A
B
C
D
F
E
NLP4 = 68.0
Optimum!
31
1
2
5
7
9
10
11
13
14
15
17
18
19
23
25
26
27
28
29
30
3
4
6
8
12
16
20
21
22
24
31
32 32"
33
34
35
36 36"37
38
ACRYLONITRILE
ACETYLENE
PROPYLENE
NAPHTA
CUMENE
ISOPROPANOL
CHLOROBENZENE
ETHYLOBENZENE
ACETONE
STYRENE
PHENOL
ACETALDEHYDE
ETHANOL
BENZENE
ETHYLENE
ACETIC ACID
ETHYLENE
CHLOROHYDRIN
ESTERS
ACETIC ANHYDRIDE
ETHYLENE GLYCOL
VINYL ACETATE
CARBON MONOXIDE
FORMALDEHYDE
METHANOL
GLYCOLIC ACID
ACRYLONITRILE
Local marketInternational
HCN
BYPRODUCTS
ETHYLENE DICHLORIDE
LP Multiperiod PlaningModel38 processes28 chemicals10 monthsPossibility contracts:Naphtha, Ethylene,Acetylene
Example: Optimal Production Planning with Contracts
Purchases:Fixed priceDiscount after amountBulk discount after amountContracts fixed time
Park, Park, Melle, Grossmann (2006)I&EC Res., 45, 5013-5026
32
Multiperiod Production Planning LP Model
i
jt jt jt jtj J t T j J t T
it ijt jt jt jt jti I j JM t T j J t T j J t T
Max PROFIT S P
W V SF
ψ φ
δ ξ θ∈ ∈ ∈ ∈
∈ ∈ ∈ ∈ ∈ ∈ ∈
= −
− − −
∑∑ ∑∑
∑ ∑ ∑ ∑∑ ∑∑
TtJMjJjIiWW iitijijijt ∈∈∈∈= ,',,'μ
TtJMjIiQW iitijt ∈∈∈≤ ,,
TtJjdSd
aPaUjtjt
Ljt
Ujtjt
Ljt
∈∈⎪⎭
⎪⎬⎫
≤≤
≤≤,
, 1 ,j j
j t ijt jt jt ijt jti O i I
V W P V W S j J t T−∈ ∈
+ + = + + ∈ ∈∑ ∑
TtJjSdSF jtU
jt jt∈∈−≥ ,
TtJjSFSF Ujtjt ∈∈≤≤ ,0
TtJjVV Ujtjt ∈∈≤ ,
, , , 0jt jt it jtS P W V ≥
Mass balance process
Capacity
Mass balance chemicals
Shortfalls
Purchases
Sales
Limit inventory
Fixed price purchases
What if prices not fixedbut given by contracts?
33
. Discount after djtσ amount.
TtJRjPPCOST djt
djt
djt
djt
djt ∈∈+= ,2211 ϕϕ
TtJRj
P
P
y
P
P
y
djt
djt
djt
djt
djt
djt
djt
djt
∈∈
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
≥
=∨
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
≤≤ ,
00
02
1
2
2
1
1
σσ
TtJRjPPP djt
djt
djt ∈∈+= ,21
Disjunctive modelTtJRjPPCOST d
jtdjt
djt
djt
djt ∈∈+= ,2211 ϕϕ
TtJRjPPP djt
djt
djt ∈∈+= ,21
TtJRjPPP djt
djt
djt ∈∈+= ,12111
TtJRjyP djt
djt
djt ∈∈≤≤ ,0 111 σ
TtJRjyP djt
djt
djt ∈∈= ,212 σ
TtJRjUyP djt
djt
djt ∈∈≤≤ ,0 22
TtJRjyyy djt
djt
djt ∈∈=+ ,21
1,0, 21 ∈djt
djt yy
MILP model (Convex hull)
Price drops after amount σ
34
Bulk discount
TtJRj
P
PCOST
y
P
PCOST
y
bjt
bjt
bjt
bjt
bjt
bjt
bjt
bjt
bjt
bjt
bjt
bjt
∈∈
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
≥
=∨
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
≤≤
= ,
0
2
2
1
1
σ
ϕ
σ
ϕ
Disjunctive Model
TtJRjPPCOST bjt
bjt
bjt
bjt
bjt ∈∈+= ,2211 ϕϕ
TtJRjPPP bjt
bjt
bjt ∈∈+= ,21
TtJRjyP bjt
bjt
bjt ∈∈≤≤ ,0 11 σ
TtJRjyUPy bjt
bjt
bjt
bjt
bjt ∈∈≤≤ ,222σ
TtJRjyyy bjt
bjt
bjt ∈∈=+ ,21
1,0, 21 ∈bjt
bjt yy
MILP Model (Convex Hull)
Price for total amount drops after amount σ
35
Fixed-duration contracts
TtJRj
P
P
P
PCOST
PCOST
PCOST
y
P
P
PCOST
PCOST
y
P
PCOST
y
ljt
ltj
ljt
ltj
ljt
ljt
ltj
ljt
ltj
ltj
ljt
ltj
ljt
ljt
ljt
ljt
ljt
ltj
ljt
ljt
ltj
ljt
ltj
ljt
ljt
ljt
ljt
ljt
ljt
ljt
ljt
ljt
ljt
∈∈
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
≥
≥
≥
=
=
=
∨
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
≥
≥
=
=
∨
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
≥
=
+
+
++
++
+
++ ,
32,
31,
3
2,3
2,
1,3
1,
3
3
21,
2
1,2
1,
2
2
1
1
1
σ
σ
σ
ϕ
ϕ
ϕ
σ
σ
ϕ
ϕ
σ
ϕ
Disjunctive ModelTtJRjPCOST
LCp T
lptj
lpj
ljt
pt
∈∈= ∑ ∑∈ ∈
,τ
ττϕ
TtJRjPPLCp T
lptj
ljt
pt
∈∈= ∑ ∑∈ ∈
,τ
τ
LCpTTTTtJRjyUPy pt
plpj
lj
lptj
lpj
lpj ∈⊂∈⊂∈∈≤≤ ,,, τσ τττττ
TJRjyy lj
LCp
lpj ∈∈≤∑
∈
τττ ,
1,0∈lpjy τ , 3,2,1=LC
MILP Model
Price gradually drops ifamount σ purchased fixednumber of months
36
22,073.06100.9546,00240,6066,1602
18,085.95100.1813,41612,60601
Profit[103 $]
Time periods
CPU time [s]ConstraintsCont.
variables0-1
variablesCase
Computational Results
0
100
200
300
400
500
600
700
800
900
1000
1 2 3 4 5 6 7 8 9 10
Time periods
Qau
ntiti
es [k
ton]
fixed Ethyfixed Acetfixed Naphdiscount Ethydiscount Acetdiscount Naphbulk Ethybulk Acetbulk Naphlength Ethylength Acetlength Naph
Purchases raw materials
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
PROFIT REVENUES PURCHASES OPERATION STORAGE*100 SHORTFALLS
1E5
$ Without ContractsWith Contracts
Comparison without/with contracts
37
Computational Results
ILOG CPLEX Dec 1, 2008 22.9.2 LNX 7311.8080 LX3 x86/Linux Cplex 11.2.0, GAMS Link 34 MIP Presolve eliminated 44425 rows and 38571 columns. Reduced MIP has 909 rows, 1367 columns, and 3267 nonzeros. Reduced MIP has 270 binaries, 0 generals, 0 SOSs, and 0 indicators.
Implied bound cuts applied: 3 Flow cuts applied: 40 Gomory fractional cuts applied: 15 MIP Solution: 22073.060039 (959 iterations, 21 nodes)
MODEL STATISTICS BLOCKS OF EQUATIONS 47 SINGLE EQUATIONS 46,002 BLOCKS OF VARIABLES 30 SINGLE VARIABLES 40,606 NON ZERO ELEMENTS 85,033 DISCRETE VARIABLES 6,160
S O L V E S U M M A R Y MODEL MULT OBJECTIVE NPV TYPE MIP DIRECTION MAXIMIZE SOLVER CPLEX FROM LINE 878 **** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 1 OPTIMAL **** OBJECTIVE VALUE 22073.0600 RESOURCE USAGE, LIMIT 0.470 10000.000 ITERATION COUNT, LIMIT 1437 1000000
38
Carnegie Mellon
1
. .
( )
, ,
jk
k
jk
k
Tk
k K
jk jk
j Jk j k
j J
L U
jk k
k
Min Z c d x
s t Bx bY
A x a k Kc
Y k K
Y Truex x xY True False j J k K
c
γ
∈
∈
∈
= +
≥
⎡ ⎤⎢ ⎥
≥ ∈⎢ ⎥⎢ ⎥=⎣ ⎦
∨ ∈
Ω =
≤ ≤∈ ∈ ∈
∈
∑
∨
R k K∈
Linear Generalized Disjunctive ProgrammingLGDP Model
Objective function
Common constraints
Disjunctive constraints
Logic constraints
Boolean variables
Logical OR operator
Continuous variables
Can we obtain stronger relaxations?How to deal with Boolean and logic constraints in Disjunctive Programming?
Sawaya N.W. and Grossmann I.E. (2008)
39
Carnegie Mellon
Reformulating LGDP into DisjunctiveProgramming Formulation
1
. .
( )
, ,
jk
k
jk
k
Tk
k K
jk jk
j Jk j k
j J
L U
jk k
k
Min Z c d x
s t Bx bY
A x a k Kc
Y k K
Y Truex x xY True False j J k K
c
γ
∈
∈
∈
= +
≥
⎡ ⎤⎢ ⎥
≥ ∈⎢ ⎥⎢ ⎥=⎣ ⎦
∨ ∈
Ω =
≤ ≤∈ ∈ ∈
∈
∑
∨
R k K∈
LGDP
1
. . 1
1
0 1 ,
jk
k
k
Tk
k K
jk jk
j Jk j k
jkj J
L U
jk k
k
Min Z c d x
s t Bx b
A x a k Kc
k K
H hx x x
j J k K
c
λ
γ
λ
λ
λ
∈
∈
∈
= +
≥
=⎡ ⎤⎢ ⎥
≥ ∈⎢ ⎥⎢ ⎥=⎣ ⎦
= ∈
≥
≤ ≤≤ ≤ ∈ ∈
∈
∑
∨
∑
R k K ∈
LDP => Integrality λ guaranteed
Proposition. LGDP and LDP have equivalent solutions.
40
Carnegie Mellon
Equivalent Forms in DP Through Basic Steps
tt TF S
∈= ∩
,t T∈ , a polyhedron, .t
t i i ti QS P P i Q
∈= ∪ ∈
Thus the RF is:
where for
There are many forms between CNF and DNF that are equivalent
Regular Form (RF): form represented by intersection of unions of polyhedra
Proposition 1 (Theorem 2.1 in Balas (1979)). Let F be a disjunctive set in RF. Then F
can be brought to DNF by | | 1T − recursive applications of the following basic steps,
which preserve regularity:
For some , , ,r s T r s∈ ≠ bring r sS S∩ to DNF, by replacing it with:
( ).rs
rs i ti Qt Q
S P P∈∈
= ∪ ∩
⇒ as basic steps are performed tighter relaxations are obtained
41
Carnegie Mellon
Illustrative Example: Hierarchy of Relaxations
1 2
1 2
1 1
2 2
0.5 01 0
0 10 1 0 1
x xx xx x
x x
− + ≥− − + ≥
= =⎡ ⎤ ⎡ ⎤∨⎢ ⎥ ⎢ ⎥≤ ≤ ≤ ≤⎣ ⎦ ⎣ ⎦
1 2 1 2
1 2 1 2
1 1
2 2
0.5 0 0.5 01 0 1 0
0 10 1 0 1
x x x xx x x x
x xx x
− + ≥ − + ≥⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− − + ≥ − − + ≥⎢ ⎥ ⎢ ⎥∨⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥
≤ ≤ ≤ ≤⎣ ⎦ ⎣ ⎦
Application of 2 Basic Steps
Convex Hull of disjunction
Convex Hull of disjunction
1x
2x
TighterRelaxation!
LP Relaxation
42
Carnegie Mellon
y
xL = ?
W
(0,0)
Set of small rectangles
ij
ji
j
(xi,yi)
j
Numerical Example:Strip-packing problem
Problem statement: Hifi (1998)Given a set of small rectangles with width Hi and length Li.Large rectangular strip of fixed width W and unknown length L.Objective is to fit small rectangles onto strip without overlap and rotation while minimizing length L of the strip.
43
Carnegie Mellon
. . i i
Min ltst lt x L≥ +
1 2 3 4
, ,
ij ij ij ij
i i j j j i i i j j j i
i i i
i N
Y Y Y Yi j N i j
x L x x L x y H y y H y
x UB L
∀ ∈
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∨ ∨ ∨ ∀ ∈ <⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
+ ≤ + ≤ − ≥ − ≥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦≤ −
i i
i NH y W i N
∀ ∈
≤ ≤ ∀ ∈1 1 2 3 4
, , , , , , , , , i i ij ij ij ijlt x y Y Y Y Y True False i j N i j+∈ ∈ ∀ ∈ <R
GDP/DP Model forStrip-packing problem
Objective functionMinimize length
Disjunctive constraintsNo overlap between rectangles
Bounds on variables
44
Carnegie Mellon
25 Rectangle Problem Optimal solution= 31
Original CH1,112 0-1 variables4,940 cont vars7,526 constraintsLP relaxation = 9
Strengthened1,112 0-1 variables5,783 cont vars8,232 constraintsLP relaxation = 27!
=>
31 Rectangle Problem Optimal solution= 38
Original CH2,256 0-1 variables9,716 cont vars14,911 constraintsLP relaxation = 10.64
Strengthened2,256 0-1 variables11,452 cont vars15,624 constraintsLP relaxation = 33!
=>
45
Nonconvex GDP
( )
1
m in ( )
( ) 0
( ) 0
Ω
kk
jk
jkk
k jk
nk
jk
Z c f x
s.t. r x
Y
g x k K j J
c γ
Y true
x R , c RY true, fa lse
= +
≤
⎡ ⎤⎢ ⎥
≤ ∈⎢ ⎥∈ ⎢ ⎥=⎣ ⎦=
∈ ∈
∈
∑
∨
Objective Function
Common Constraints
Disjunctions
Logic Propositions
OR operator
f, g and r: nonconvex
46
• Introducing convex underestimators
Convex Underestimator GDP (R)
( )
1
m in ( )
( ) 0
( ) 0
Ω
, :
kk
jk
jkk
k jk
nk
jk
j
Z c f x
s .t. r xY
g x k K J
c γ
Y tru e
x R , c RY tru e , fa lse
f r a n d g co n vex
∈
= +
≤
⎡ ⎤⎢ ⎥
≤ ∈⎢ ⎥⎢ ⎥=⎢ ⎥⎣ ⎦
=
∈ ∈
∈
∑
∨
Convex underestimatorsBilinear: LinearMcCormick (1976), Al-Khayyal (1992)
Linear fractional: Convex nonlinearQuesada and Grossmann (1995)
Concave separable: Linear secant
Problem (R) yields a valid lower bound to Problem (GDP)
47
Convex envelopesConcave function
xba
Secant g(x)
[ ( ) ( )]( ) ( ) ( )f b f ag x f a x ab a
−= + −
−
f(x)
48
Bilinearw = xy
L U L Ux x x y y y≤ ≤ ≤ ≤
L L L L
U U U U
L U L U
U L U L
w x y y x x y
w x y y x x y
w x y y x x y
w x y y x x y
≥ + −
≥ + −
≤ + −
≤ + −
McCormick convex envelopes
For other convex envelopes/underestimators see:Tawarmalani, M. and N. V. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Vol. 65, Nonconvex Optimization And Its Applications series, Kluwer Academic Publishers, Dordrecht, 2002
49
Carnegie Mellon
Proposed framework to obtainstronger relaxations for nonconvex GDP
(Bilinear and Concave GDP)
The framework consists of two main phases:
1- Generate a valid Linear Generalized Disjunctive Program (LGDP) relaxation for the nonconvex GDP problem (e.g. bilinear and concave).
2- Strengthen the continuous relaxation of the LGDP obtained in phase 1 by applying a set of basic steps
(Ruiz & Grossmann, 2008)
50
Carnegie Mellon
Relaxation Results
-5241-5468-5515-4640Example 5
431.90431.90400.661214.87Example 4
97858.8694925.7791671.18114384.78Example 3
6.086.085.656.31Example 2
-1.10-1.10-1.28-1.01Example 1
DNF Lower Bound
Lower Bound (Proposed
Relaxation)
Lower Bound (Lee & Grossmann
Relaxation)Global
Optimum
Remarks
-Often, it is not necessary to reach the DNF form to have good lower bounds. Note that examples 1, 2 and 4 show that the lower bound is the same as the lower bound of the DNF
-Proposed methodology leads to improvements in the lower bounds.
-The lower bound of the DNF is the best lower bound attainable for a given LGDP.
51
Carnegie Mellon
Conclusions
Unified Linear GDP with Disjunctive Programming- Developed DP equivalent formulation for GDP- Numerical results have shown great improvement in lower bound
for strip packing problem
Nonconvex GDPs- Tighter lower bounds can be obtained in bilinear and concave
problems by applying basic steps
GDP modeling framework- Provides a logic-based framework for linear and nonlinear
discrete-continuous optimization- big-M and convex hull alternative formulations different relaxations- Solution methods: reformulation, branch and bound, decomposition- Numerical example of planning problem with contracts has shown
the impact of the convex hull formulation (with help of CPLEX)