Reformulations, Relaxations and Cutting Planes for Linear Generalized Disjunctive Programming

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Reformulations, Relaxations and Cutting Planes
for Linear Generalized Disjunctive Programming

• Ignacio E. Grossmann
• Nicolas Sawaya, Juan Pablo Ruiz
• Department of Chemical Engineering
• Center for Advanced Process Decision-making
• Carnegie Mellon University
• Pittsburgh, PA 15213, U.S.A.
• MIP-2008
• Columbia University, New York, NY

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Discrete/Continuous Optimization Models
Mixed Integer Program (MIP) - Most common
non-linear discrete/continuous optimization
model. - Purely equation-based. - If all
functions in MIP are linear ? MILP
(nonlinear ? MINLP).

• Disjunctive Programming (DP)
• - Developed by Balas E. (1979, 1985, 1998
  (1974))
• - Linear programming (LP) with
  disjunctive constraints.

Generalized Disjunctive Program (GDP)
- Linear Raman, Grossmann (1994) Nonlinear
Turkay, Grossmann (1996) - Combination of
algebraic equations, disjunctions and logic
propositions. - Natural representation of
engineering problems.
Mixed-Logic Linear Programming Hooker and
Osorio (1999)
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Goal To unify GDP with DP in order to develop
MILP reformulations with improved relaxations for
linear GDP

• To unify linear GDP with DP in order to develop
• - A hierarchy of LP relaxations for linear GDP
• - A family of disjunctive cutting planes for
  linear GDP

• Brief extension to Nonlinear and Bilinear GDPs
• - Approximation of convex hull and
  cutting plane algorithm
• - Tightening bounds of bilinear GDPs
  through basic steps

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Linear Generalized Disjunctive ProgrammingLGDP
Model
Raman R. and Grossmann I.E. (1994) (LGDP)
Objective function
Common constraints
Disjunctive constraints
Logic constraints
Boolean variables
Logical OR operator
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Process Network with fixed charges
LOGMIP-GAMS
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LGDP to MILP Reformulations
Big-M Reformulation
Note (Mjk maxLB ? x ? UB (Ajk x - ajk), j? Jk
and k? K ).
Relaxation ?jk ? 0,1 becomes 0 ? ?jk ? 1 in
(BM)
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LGDP to MILP Reformulations Convex Hull
Reformulation Lee S. and Grossmann I.E. (2000)

• Min Z dTx
• s.t. Bx ? b
• Ajk ?jk ? ajk ?jk j? Jk , k? K
• x ?jk k? K (CH)
• 0 ? ?jk ? ?jk Ujk j? Jk , k?
  K
• 1 k? K
• H? ? h
• x? Rn, ?jk? Rn , ?jk? 0,1 j? Jk , k?
  K

Disaggregated variables
Relaxation ?jk ? 0,1 becomes 0 ? ?jk ? 1 in
(CH)
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Proposition The Convex Hull of a set of
disjunctions is the smallest convex set that
includes that set of disjunctions.
Furthermore, the projected relaxation of (CH)
onto the space of (BM) is always as tight or
tighter than that of (BM) (Grossmann Lee ,
2002)
1. Tighter feasible region/lower bound ? less
nodes ? decrease in computational solution time.
2. More variables and constraints ? more
iterations ? increase in computational solution
time.
Is Convex Hull best relaxation?
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Disjunctive Programming
Constraint set of a DP can be expressed in two
equivalent extreme forms
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Linear Generalized Disjunctive Programming LGDP
Model
Raman R. and Grossmann I.E. (1994) (LGDP)
Objective function
Common constraints
Disjunctive constraints
Logic constraints
How to deal with Boolean and logic constraints in
Disjunctive Programming?
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Reformulating LGDP into Disjunctive Programming
Formulation
Sawaya N.W. and Grossmann I.E. (2007)

gt Integrality ? guaranteed
Proposition. LGDP and LDP have equivalent
solutions.
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Equivalent Forms in DP Through Basic Steps
There are many forms between CNF and DNF that are
equivalent
Regular Form (RF) form represented by
intersection of unions of polyhedra
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Illustrative Example Basic Steps
Then F can be brought to DNF through 2 basic
steps.
which is its equivalent DNF
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Equivalent Forms for GDP
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Converting LDP to MIP reformulations
gt Convex Hull
gt MIP representation
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Family of MIP Reformulations For GDP
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Particular case Convex Hull Reformulation of LGDP
Lee S. and Grossmann I.E. (2000)
(CH)
Disaggregated variables
While this MILP formulation has stronger
relaxation than big-M, it is not strongest!!
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A Hierarchy of Relaxations for DP
Hull Relaxation (Balas, 1985)
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A Hierarchy of Relaxations for GDP
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Illustrative Example Hierarchy of Relaxations
LP Relaxation
Tighter Relaxation!
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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.

i
j
j
j
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GDP/DP Model forStrip-packing problem
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DP Model For4 Rectangle Strip-packing Problem
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25 Rectangle Problem Optimal solution 31
Original CH 1,112 0-1 variables 4,940 cont
vars 7,526 constraints LP relaxation 9
Strengthened 1,112 0-1 variables 5,783 cont
vars 8,232 constraints LP relaxation 27!
gt
31 Rectangle Problem Optimal solution 38
Original CH 2,256 0-1 variables 9,716 cont
vars 14,911 constraints LP relaxation 10.64
Strengthened 2,256 0-1 variables 11,452 cont
vars 15,624 constraints LP relaxation 33!
gt
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Motivation for Cutting Plane Method
1. Tighter feasible region/lower bound ? less
nodes ? decrease in computational solution time.
2. More variables and constraints ? more
iterations ? increase in computational solution
time.
Feasible region as tight as full space AND fewer
variables and constraints
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DERIVATION OF CUTTING PLANESDual perspective
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DERIVATION OF CUTTING PLANESDual perspective
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CUT GENERATION PROBLEMDual perspective
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CUT GENERATION PROBLEMPrimal perspective
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Cut Generation Problem for Lee
Grossmann(Separation Problem)Primal perspective
separation problem Sawaya, Grossmann (2006)
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Derivation of Cutting PlanesPrimal perspective
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Cutting Plane Method Derivation of Cutting Planes
Propositions 12, 13, 14 (1) Let ? (z) ? z
- zbm2 ? (z zbm)T(z zbm). Then,
? ? ? ? (z zbm)

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CUTTING PLANE METHOD
1. Solve relaxed Big-M MILP.
2. Solve separation problem. Feasible region
corresponds to relaxed hull relaxation.
3. Cutting plane is generated and added to
relaxed big-M MILP.
4. Solve strengthened relaxed Big-M MILP. Go to
2.
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NUMERICAL RESULTS21-RECTANGLE STRIP PACKING
PROBLEM
Table 3 Results for twenty one-rectangle
strip-packing problem (CPLEX v. 8.1, default MIP
options turned on)
Relaxation Optimal Solution Gap () Total Nodes in MIP Solution Time for Cut Generation (sec) Total Solution Time (sec) Number of Nodes per sec
Convex Hull 9.1786 --- --- 968 652 0 gt10 800 89.69
Big-M 9 24 62.5 1 416 137 0 4 093.39 345.95
Big-M 20 cuts 9.1786 24 61.75 306 029 3.74 917.79 334.80
Big-M 40 cuts 9.1786 24 61.75 547 828 7.48 1 063.51 518.76
Big-M 60 cuts 9.1786 24 61.75 28 611 11.22 79.44 419.32
Big-M 62 cuts 9.1786 24 61.75 32 185 11.59 91.4 403.27
Total solution time includes times for relaxed
MIP(s) LP(s) from separation problem MIP
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NUMERICAL RESULTS21-RECTANGLE STRIP PACKING
PROBLEM
Table 3 Results for twenty one-rectangle
strip-packing problem (CPLEX v. 8.1, default MIP
options turned on)
Relaxation Optimal Solution Gap () Total Nodes in MIP Solution Time for Cut Generation (sec) Total Solution Time (sec) Number of Nodes per sec
Convex Hull 9.1786 --- --- 968 652 0 gt10 800 89.69
Big-M 9 24 62.5 1 416 137 0 4 093.39 345.95
Big-M 20 cuts 9.1786 24 61.75 306 029 3.74 917.79 334.80
Big-M 40 cuts 9.1786 24 61.75 547 828 7.48 1 063.51 518.76
Big-M 60 cuts 9.1786 24 61.75 28 611 11.22 79.44 419.32
Big-M 62 cuts 9.1786 24 61.75 32 185 11.59 91.4 403.27
Total solution time includes times for relaxed
MIP(s) LP(s) from separation problem MIP
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Non-linear Discrete/Continuous Optimization GDP
Model
Lee Grossmann I.E. (2000)
(GDP)
Objective function
Common constraints
Disjunctive constraints
Logic constraints
convex functions
Logical OR operator
Boolean variables
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Cutting Plane Method
3. Cutting plane is generated and added to
relaxed (BM) problem.
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Convex Hull Formulation

• Consider Disjunction k ? K

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Remarks

1.

If g(x) is a bounded convex function,
is a bounded convex function
Hiriart-Urruty and Lemaréchal (1993)
2.
for bounded g(x)
3. For linear constraints convex hull reduces to
result by Balas (1985)
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Cutting Plane Method Separation Problem
CONVEX NLP
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Computational Implementationof Separation Problem
where
1. The divisibility by 0 problem is avoided.
2. The new constraints are an exact approximation
of the original constraints as e ? 0.
3. The new constraints are an exact approximation
of the original constraints at yjk 0 and
at yjk 1 regardless of value of e.
4. The LHS of the new constraints are convex.
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Numerical ExampleDesign of Multi-product Batch
Plant

• Problem statement Ravemark (1995)
• Design of batch plant with multiple units in
  parallel and intermediate storage tanks.

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Numerical ResultsDesign of 10 Unit/Product Batch
Plant
Table 1 Results for design of 10 stage/product
batch plant using traditional BB (SBB)
Relaxation Optimal Solution Gap () Total Nodes in MINLP Solution Time for Cut Generation (sec) Total Solution Time (sec) Number of Nodes per sec
Convex Hull 650 401. 14 729 948.49 10.9 5 359 0 711.76 7.53
Big-M 641 763.19 729 948.49 12.1 12 449 0 787.98 15.80
Big-M 58 cuts 650 401. 14 729 948.49
10.9 7 528
8.7 610.00
12.52
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Global Optimization of Bilinear Generalized
Disjunctive Programs
Juan Ruiz
Min
Objective Function
s.t.
Global Constraints
Disjunctions
k K
O(Y) True
Logic Propositions
Bilinearities may lead to multiple local minima ?
Global Optimization techniques are required
Relaxation of Bilinear terms using McCormick
envelopes leads to a LGDP ? Improved relaxations
for Linear GDP has recently been obtained
(Sawaya Grossmann, 2007)
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Guidelines for applying basic steps in Bilinear
GDP

• Replace bilinear terms in GDP by McCormick convex
  envelopes (LGDP)
• Apply basic steps between those disjunctions with
  at least one variable in common.
• The more variables in common two disjunctions
  have the more the tightening can be expected
• If bilinearities are outside the disjunctions
  apply basic steps by introducing them in the
  disjunctions previous to the relaxation.
• If bilinearities are inside the disjunctions a
  smaller tightening effect is expected.
• A smaller increase in the size of the formulation
  is expected when basic steps are applied between
  improper disjunctions and proper disjunctions.

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Case Study I Water treatment network design
Process superstructure
M1
S4
S1
A/B/C
M4
M2
S5
S2
D/E/F
M3
S6
S3
G/H/I
N of cont. vars. 114 N of disc. vars. 9 N of
bilinear terms 36
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Case Study II Pooling network design
N of cont. vars. 76 N of disc. vars. 9 N of
bilinear terms 24
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Performance
    Global Optimization Technique using Lee Grossmann relaxation Global Optimization Technique using proposed relaxation Relative Improvement
Example 1 Initial Lower Bound 400.66 499.86 24.90
Bound contraction 99.7
Nodes 399 204 51
    Global Optimization Technique using Lee Grossmann relaxation Global Optimization Technique using proposed relaxation Relative Improvement
Example 2 Initial Lower Bound -5515 -5468 0.90
Bound contraction 8
Nodes 748 683 9
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Conclusions
Unified GDP with Disjunctive Programming -
Developed DP equivalent formulation for GDP
- Developed a family of MIP
reformulations for GDP - Developed a hierarchy
of relaxations for GDP
Developed framework for obtaining improved LP
relaxations - Demonstrated improved relaxations
can be obtained compared to
convex hull formulation Lee Grossmann (2000)
- Numerical results have shown great
improvement in lower bound for
strip packing problem
Cutting Planes - Showed equivalence dual and
primal cut-generation problems. - Developed a
primal cut-and-branch algorithm where cutting
planes were generated from the
primal separation problem
Nonlinear GDPs - Cutting planes can be readily
extended - Concept basic steps improves
relaxation in bilinear GDPs
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CONCLUSIONS AND CONTRIBUTIONS
1-) We established novel connections between
disjunctive programming and linear GDP.
2-) We extended Balas theory of equivalent forms
to linear GDP.
3-) We developed a family of MIP reformulations
that encompasses all possible MIP formulations
for linear GDP, including the Lee and
Grossmann formulation.
4-) We developed a hierarchy of relaxations for
linear GDP that mirror those developed by
Balas for disjunctive programs
5-) We described the facets of the
hull-relaxation in the dual space, and showed
that every facet of the hull-relaxation can
be obtained from the convex hull of some
individual disjunction.
6-) We showed that the equivalence between the
dual and primal cut-generation problems.
7-) We developed a primal cut-and-branch
algorithm where cutting planes were generated
from the primal separation problem and were
added to the root node of a BB tree.
8-) We proceeded to derive different families of
cutting planes in the primal space (from
different norms) that obtain from the
aforementioned cut-generation problem,
9-) We established rigorous approximations for
reformulations of nonlinear GDP problems
10-) We extended the primal cut-and-branch
algorithm developed for the linear case to the
nonlinear case.
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NUMERICAL RESULTS21-RECTANGLE STRIP PACKING
PROBLEM
Table 3 Results for twenty one-rectangle
strip-packing problem (CPLEX v. 8.1, default MIP
options turned on)
Relaxation Optimal Solution Gap () Total Nodes in MIP Solution Time for Cut Generation (sec) Total Solution Time (sec) Number of Nodes per sec
Convex Hull 9.1786 --- --- 968 652 0 gt10 800 89.69
Big-M 9 24 62.5 1 416 137 0 4 093.39 345.95
Big-M 20 cuts 9.1786 24 61.75 306
029 3.74 917.79 334.80 Big-M 40
cuts 9.1786 24 61.75 547 828 7.48 1
063.51 518.76 Big-M 60 cuts 9.1786 24 61.75 28
611 11.22 79.44 419.32 Big-M 62
cuts 9.1786 24 61.75 32 185 11.59 91.4 403.27
Total solution time includes times for relaxed
MIP(s) LP(s) from separation problem MIP
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Convex Hull
The Convex Hull of a set of disjunctions is the
smallest convex set that includes that set of
disjunctions.
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DERIVATION OF CUTTING PLANESDual perspective