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Cutting Plane Algorithm for Convex
Generalized Disjunctive Programming
Francisco Trespalacios and Ignacio E. Grossmann
May 23, 2017
Center for Advanced Process Decision-making
Department of Chemical Engineering Carnegie Mellon University
Pittsburgh, PA 15213
Motivation and goals
Discrete-continuous optimization problems can be formulated in many
ways as mixed-integer programming problems
Motivation • Some formulations are good, some not
• How to derive “good” formulations?1
Use Generalized Disjunctive Programming (GDP)2 as a general modeling
framework to derive mixed-integer programming formulations
Goals
Derive cutting planes for a weak formulation using a stronger one
• Stronger formulation obtained through the application of basic steps
2
1. Hooker,J., Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, John Wiley & Sons (2000)
1. Wlliams H.P. Model Building in Mathematical Programming, John Wiley (1999)
1. Jeroslow, R.G. anf J.K. Lowe, Modelling with integer variables, Mathematical Programming StudiesVolume 22, 1984, pp 167-184
2. Raman R., Grossmann I.E., Modelling and Computational Techniques for Logic-Based Integer Programming, Computers and Chemical Engineering, 18, 563, 1994
Modeling mixed-integer programs
The quality of the formulation
depends on the modeler’s skills
Classic
Modeler
MILP/ MINLP
More control on the model:
(BM) is smaller, while (HR)
has a tighter relaxation
Higher level of modeling – Using
not only algebraic formulations,
but also logic arguments
Classic
GDP
Approach
Modeler
Big M
(BM)
MILP/ MINLP
GDP model
Hull Ref.
(HR)
MILP/ MINLP
Modeler
GDP model
+ cuts
Proposed
Improve the “weak” (BM) formulation by adding
cuts, derived from a very strong formulation
3
Strengthened
r-(HR) region
min distance1
Solve r-(BM)
(BM) + cuts
MILP/ MINLP
(HR) strengthened
through basic steps
1. Stubbs, Robert A. and Mehrotra, Sanjay: “A branch-and-cut method for 0-1 mixed convex Programming”, Math. Program., Ser. A 86: 515–532 (1999)
Modeling mixed-integer programs
Classic
Classic
GDP
Approach
Modeler
MILP/ MINLP
Big M
(BM)
MILP/ MINLP
Modeler
GDP model
Hull Ref.
(HR)
MILP/ MINLP
Modeler
GDP model
+ cuts
Proposed
Strengthened
r-(HR) region
min distance1
Solve r-(BM)
The performance of the algorithms strongly depend on
the size of the problem and tightness of relaxation
4
(BM) + cuts
MILP/ MINLP
Modeling mixed-integer programs
Classic
Classic
GDP
Approach
Modeler
MILP/ MINLP
Big M
(BM)
MILP/ MINLP
Modeler
GDP model
Hull Ref.
(HR)
MILP/ MINLP
5
Modeler
GDP model
+ cuts
Proposed
Strengthened
r-(HR) model
min distance1
Solve r-(BM)
(BM) + cuts
MILP/ MINLP
GDP is a higher level of representation for MILP/MINLP
Optimization problem with algebraic expressions, disjunctions & logic propositions
General
Convex
formGDP
of GDP1
Illustration – Process network
F2
F1
min 𝑧 = 𝑓 𝑥
F7
Objective Function
F3
Convex
Convex
s.t. 𝑔 𝑥 ≤ 0
⋁ 𝑖 ∈ 𝐷𝑘
𝑟𝑘𝑖
F6
R1
𝑌𝑘𝑖
𝑘∈𝐾
𝑥 ≤0
𝛺 𝑌 = 𝑇𝑟𝑢𝑒
F4
S2
F5
S1
Global Constraints
⊻ 𝑖 ∈ 𝐷𝑘 𝑌𝑘𝑖
R2
Disjunctions
𝑘∈𝐾
Logic Propositions
max 𝑧 = 𝑃7 𝐹7 − 𝑃1 𝐹1 − 𝑐𝑅 − 𝑐𝑆
Objective function
𝐹1 = 𝐹2 + 𝐹3
𝐹7 = 𝐹5 + 𝐹6
Global Constraints
𝑌𝑅1
𝑌𝑅2
𝐹 = 𝐹2 = 0
𝐹6 = 𝛽𝑅1 𝐹2
⋁ 6
𝐹3 = 𝐹4 = 0
𝐹4 = 𝛽𝑅2 𝐹3
𝑐𝑅 = 𝛾𝑟1
𝑐𝑅 = 𝛾𝑟2
𝑌𝑆_𝑁𝑂
𝑌𝑆1
𝑌𝑆2
𝐹5 = 𝛽𝑆1 𝐹4 ⋁ 𝐹5 = 𝛽𝑆2 𝐹4 ⋁ 𝐹5 = 0
𝑐𝑆 = 𝛾𝑠2
𝑐𝑆 = 𝛾𝑠2
𝑐𝑆 = 0
Disjunctions
𝑌𝑅1 ⊻ 𝑌𝑅2
𝑥 ∈ ℝ𝑛
𝑌𝑘𝑖 ∈ 𝑇𝑟𝑢𝑒, 𝐹𝑎𝑙𝑠𝑒
6
𝑌𝑆1 ⊻ 𝑌𝑆2 ⊻ 𝑌𝑆_𝑁𝑂
𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘
𝑌𝑅1 ⇔ 𝑌𝑆_𝑁𝑂
1. Raman R. and Grossmann I.E., “Modelling and Computational Techniques for Logic-Based Integer Programming”, Computers and Chemical Engineering, 18, 563, 1994.
Logic
Modeling mixed-integer programs
Classic
Classic
GDP
Approach
Modeler
MILP/ MINLP
Big M
(BM)
MILP/ MINLP
Modeler
GDP model
Hull Ref.
(HR)
MILP/ MINLP
7
Modeler
GDP model
+ cuts
Proposed
Strengthened
r-(HR) model
min distance1
Solve r-(BM)
(BM) + cuts
MILP/ MINLP
GDPs can be reformulated using either the Big-M (BM)
or the Hull Reformulation (HR)
GDP
(BM)
𝑓 𝑥
min 𝑧 = 𝑓 𝑥
Large, positive M.
Constraint relaxes
when yik = 0
s.t. 𝑔 𝑥 ≤ 0
˅ 𝑖 ∈ 𝐷𝑘
(HR)
𝑟𝑘𝑖
𝑌𝑘𝑖
𝑥 ≤0
𝑥 𝑙𝑜 ≤ 𝑥 ≤ 𝑥 𝑢𝑝
𝑔 𝑥 ≤0
𝑥=
𝑖𝑘
𝑟𝑘𝑖 𝑥 ≤ 𝑀 (1 − 𝑦𝑖𝑘 )
𝑖 ∈𝐷𝑘 𝑦𝑖𝑘
⊻ 𝑖 ∈ 𝐷𝑘 𝑌𝑘𝑖
𝛺 𝑌 = 𝑇𝑟𝑢𝑒
x is disaggregated. Constraints
transformed with perspective function2
If 𝑦𝑘𝑖 = 0, 𝑦𝑘𝑖 𝑟𝑘𝑖 𝜈 𝑘𝑖 /𝑦𝑘𝑖 = 0
Linear case: 𝐴𝑘𝑖 𝜈 𝑘𝑖 ≤ 𝑎𝑘𝑖 𝑦𝑘𝑖
Reformulation of the
logic constraints in
the discrete space1
𝑌𝑘𝑖 ∈ 𝑇𝑟𝑢𝑒, 𝐹𝑎𝑙𝑠𝑒
8
2.
𝑘𝑖
𝑦𝑘𝑖 𝑟𝑘𝑖 𝜈 𝑘𝑖 /𝑦𝑘𝑖 ≤ 0
=1
𝑖 ∈𝐷𝑘 𝑦𝑘𝑖
=1
𝐻𝑦 ≥ ℎ
𝑥 𝑙𝑜 ≤ 𝑥 ≤ 𝑥 𝑢𝑝
𝑥 𝑙𝑜 𝑦𝑘𝑖 ≤ 𝜈 𝑘𝑖 ≤ 𝑥 𝑢𝑝 𝑦𝑘𝑖
𝑦𝑘𝑖 ∈ 0,1
Logic variables
into binary vars
1.
𝑖 ∈𝐷𝑘 𝜈
Easily accomplished as described in: Williams H.P., Mathematical Building in Mathematical Programming, John Wiley, 1985
𝑦𝑘𝑖 𝑟𝑘𝑖 𝜈 𝑘𝑖 /𝑦𝑘𝑖 is convex if if 𝑟𝑘𝑖 𝑥 ≤ 0 is convex.
(HR) provides a tighter relaxation than (BM)
Illustration of (BM) and (HR) relaxations
(BM)
x2
(HR)
Variables: 6 (4 binary)
Constraints: 18
F:
BM( 𝑨𝟏 ⊻ [𝑨𝟐])
BM( 𝑩𝟏 ⊻ [𝑩𝟐])
x2
Variables: 14 (4 binary)
Constraints: 36
F:
HR( 𝑨𝟏 ⊻ [𝑨𝟐])
HR( 𝑩𝟏 ⊻ [𝑩𝟐])
(HR) is the intersection of the
convex hull of each disjunction
𝑨𝟏
𝑨𝟏
𝑩𝟏
𝑩𝟏
𝑩𝟐
𝑩𝟐
(HR) is tighter than (BM), but
does not yield convex hull
𝑨𝟐
𝑨𝟐
x1
Can we further improve the tightness of the (HR)?
9
x1
Modeling mixed-integer programs
Classic
Classic
GDP
Approach
Modeler
MILP/ MINLP
Big M
(BM)
MILP/ MINLP
Modeler
GDP model
Hull Ref.
(HR)
MILP/ MINLP
Modeler
GDP model
+ cuts
Proposed
Strengthened
r-(HR) model
Using basic steps
10
min distance1
Solve r-(BM)
(BM) + cuts
MILP/ MINLP
The HR after the application of a Basic Step (BS), is at
least as tight as the previous
Hull Reformulation
F:
F:
x2
Application of a Basic Step
Variables: 14 (4 binary)
Constraints: 36
𝑨𝟏 ⊻ [𝑨𝟐]
𝑩𝟏 ⊻ [𝑩𝟐]
x2
𝑨𝟏
𝑨𝟏
𝑨𝟐
𝑨𝟐
⊻
⊻
⊻
𝑩𝟏
𝑩𝟐
𝑩𝟏
𝑩𝟐
Variables: 14 (4 binary)
Constraints: 51
Tighter feasible region!
(Better Bounds)
𝑨𝟏
𝑨𝟏
𝑩𝟏
𝑩𝟏
𝑩𝟐
𝑩𝟐
𝑨𝟐
𝑨𝟐
x1
11
x1
Basic Steps improve tightness but increase problem size
Improper disjunction: No OR
statement (i.e. global constraints)
𝐴
Original
Problem
𝐵1 ∨ [𝐵2 ]
# of
disj. |K|
Improper Basic Step: Between
proper and improper disjunction
𝐶1 ∨ [𝐶2 ]
The (HR) after each Basic
Step is as tight as the
previous formulation
Proper disjunction
𝐷1 ∨ 𝐷2 ∨ [𝐷3 ]
BS 1:
A, B
𝐴
𝐴
∨
𝐵1
𝐵2
# ofBasic
termsStep:
does Between
not increase,
Proper
but
terms
increase
proper disjunctions in size
𝐶1 ∨ [𝐶2 ]
𝐷1 ∨ 𝐷2 ∨ [𝐷3 ]
BS 2:
A,B,C
4
𝐴
𝐴
𝐴
𝐴
𝐵1 ∨ 𝐵1 ∨ 𝐵 2 ∨ 𝐵 2
𝐶1
𝐶2
𝐶1
𝐶2
# of disj. terms increases
exponentially, and terms
increase in size
(HR) of DNF is a
perfect formulation!
3
2
𝐷1 ∨ 𝐷2 ∨ [𝐷3 ]
BS 3:
DNF
12
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐴
𝐵1
𝐵1
𝐵1
𝐵1
𝐵1
𝐵1
𝐵2
𝐵2
𝐵2
𝐵2
𝐵2
𝐵2
⊻
⊻
⊻
⊻
⊻
⊻
⊻
⊻
⊻
⊻
⊻
𝐶1
𝐶1
𝐶1
𝐶2
𝐶2
𝐶2
𝐶1
𝐶1
𝐶1
𝐶2
𝐶2
𝐶2
𝐷1
𝐷2
𝐷3
𝐷1
𝐷2
𝐷3
𝐷1
𝐷2
𝐷3
𝐷1
𝐷2
𝐷3
Source: Balas E., Disjunctive Programming and a hierarchy of relaxations for discrete optimization problems, SIAM J. Alg. Disc. Meth., 6, 466-486, 1985
Juan P. Ruiz, Ignacio E. Grossmann, A hierarchy of relaxations for nonlinear convex generalized disjunctive programming
1
Modeling mixed-integer programs
Classic
Classic
GDP
Approach
Modeler
MILP/ MINLP
Big M
(BM)
MILP/ MINLP
Modeler
GDP model
Hull Ref.
(HR)
MILP/ MINLP
13
Modeler
GDP model
+ cuts
Proposed
Strengthened
r-(HR) model
min distance
Solve r-(BM)
(BM) + cuts
MILP/ MINLP
Deriving cuts to (BM) using stronger formulations
Generation of cuts in (BM) formulation using (HR)
Max Z
x2
s.t.
𝑨𝟏 ∨ [𝑨𝟐 ]
𝑩𝟏 ∨ [𝑩𝟐 ]
𝑪𝟏 ∨ [𝑪𝟐 ]
Z*
(BM)
x1
14
Deriving cuts to (BM) using stronger formulations
Generation of cuts in (BM) formulation using (HR)
x2
𝒙𝑩𝑴
Z*
𝒙𝒔𝒆𝒑
(BM) + 1 (HR)
cut
(HR)
Solve: min 𝑓(𝑥)
𝑥∈(𝐵𝑀)
Yields: 𝑥 𝐵𝑀
Solve: min 𝜑 𝑥 = 𝑥 − 𝑥 𝐵𝑀
𝑥∈(𝐻𝑅)
𝑝
(BM)
Yields: 𝑥 𝑠𝑒𝑝
Obtaining a valid cut:
𝜉 𝑇 (𝑥 − 𝑥 𝑠𝑒𝑝 ) ≥ 0
𝜉 is a subgradient of 𝜑(𝑥) at 𝑥 𝑠𝑒𝑝
x1
15
Deriving cuts to (BM) using stronger formulations
Generation of cuts in (BM) formulation using basic steps
x2
𝒙𝑩𝑴
Z*
Stronger
cutting plane
(HR) after
BS A and B
Solve: min 𝑓(𝑥)
𝑥∈(𝐵𝑀)
𝒙𝒔𝒆𝒑
(BM) + 1 basic step cut
Yields: 𝑥 𝐵𝑀
Solve: min 𝜑 𝑥 = 𝑥 − 𝑥 𝐵𝑀
𝑥∈(𝑆𝐸𝑃)
𝑝
(BM)
Yields: 𝑥 𝑠𝑒𝑝
Obtaining a valid cut:
𝜉 𝑇 (𝑥 − 𝑥 ∗ ) ≥ 0
𝜉 is a subgradient of 𝜑(𝑥) at 𝑥 𝑠𝑒𝑝
x1
16
Deriving cuts to (BM) using stronger formulations
Generation of cuts in (BM) formulation using (HR) vs. basic steps
Max Z
x2
𝒛𝑩𝑴𝟑
𝒛𝑩𝑴𝟐
𝒛𝑩𝑴
Z*
(BM) + 1 (HR)
cut
s.t.
𝑨𝟏 ∨ [𝑨𝟐 ]
𝑩𝟏 ∨ [𝑩𝟐 ]
𝑪𝟏 ∨ [𝑪𝟐 ]
(BM)
(BM) + 1 basic step cut
x1
17
Deriving cuts to (BM) using stronger formulations
Generation of each cut requires solution of two NLP
NLP : 𝑥 − 𝑥 𝐵𝑀
NLP: r-(BM)
min 𝑥𝑛
Solve relaxed (BM)
s.t.
Constraints where no
basic step was applied
Solution
𝑥 𝐵𝑀
𝑔 𝑥 ≤0
𝑖𝑘
𝑟𝑘𝑖 𝑥 ≤ 𝑀 (1 − 𝑦𝑖𝑘 ) 𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘
𝑖∈𝐷𝑘 𝑦𝑖𝑘
=1
0 ≤ 𝑦𝑘𝑖 ≤ 1
min
𝑗(𝑥𝑗
− 𝑥𝑗𝐵𝑀 )2
2
2;
𝑥 ∈ (𝑆𝐸𝑃)
2
2
s.t.
is convex and gradient simple.
For linear cases
1 or
∞
𝑔𝑒 𝑥 ≤ 0
𝑥 = 𝑖∈𝐷𝑘 𝜈 𝑘𝑖
𝑒 ∈ 𝐸\𝐸
𝑘 ∈ 𝐾\𝐾
𝑦𝑘𝑖 𝑟𝑘𝑖 𝜈 𝑘𝑖 /𝑦𝑘𝑖 ≤ 0
𝑘 ∈ 𝐾\𝐾, 𝑖 ∈ 𝐷𝑘
𝑖∈𝐷𝑘 𝑦𝑘𝑖
=1
𝑘 ∈ 𝐾\𝐾
𝑘∈𝐾
𝑥 𝑙𝑜 𝑦𝑘𝑖 ≤ 𝜈 𝑘𝑖 ≤ 𝑥 𝑢𝑝 𝑦𝑘𝑖 𝑘 ∈ 𝐾\𝐾, 𝑖 ∈ 𝐷𝑘
𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘
𝐻𝑦 ≥ ℎ
Constraints where basic
steps were applied
𝜉 𝑇 (𝑥 − 𝑥 𝑠𝑒𝑝 ) ≥ 0
Add cuts
𝜉 𝑇 (𝑥 − 𝑥 𝑠𝑒𝑝 ) ≥ 0, where
𝜉 = 2(𝑥 𝑠𝑒𝑝 − 𝑥 𝐵𝑀 ) is the subgradient
0 ≤ 𝑦𝑘𝑖 ≤ 1
𝑥=
Solution
𝑥 𝑠𝑒𝑝
𝑖∈𝐷𝑘 𝜈
𝑘 ∈ 𝐾\𝐾, 𝑖 ∈ 𝐷𝑘
𝑘𝑖
𝑘∈𝐾
𝑦𝑘𝑖 𝑔𝑒 𝜈 𝑘𝑖 /𝑦𝑘𝑖 ≤ 0
𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘 ; 𝑒 ∈ 𝐸𝑘
𝑦𝑘𝑖 𝑟𝑘𝑖 𝜈 𝑘𝑖 /𝑦𝑘𝑖 ≤ 0
𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘 ; 𝑘𝑖 ∈ 𝐾𝐼𝑘𝑖
𝑖∈𝐷𝑘 𝑦𝑘𝑖
=1
𝑘∈𝐾
𝑥 𝑙𝑜 𝑦𝑘𝑖 ≤ 𝜈 𝑘𝑖 ≤ 𝑥 𝑢𝑝 𝑦𝑘𝑖 𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘
0 ≤ 𝑦𝑘 𝑖 ≤ 1
18
𝑘 ∈ 𝐾, 𝑖 ∈ 𝐷𝑘
Algorithm generates cutting planes for (BM)
There are many alternative heuristics and strategies
Add cut
𝜉 𝑇 (𝑥 − 𝑥 𝑠𝑒𝑝 ) ≥ 0, where 𝜉 = 2(𝑥 𝑠𝑒𝑝 − 𝑥 𝐵𝑀 )
Algorithm
Decisions in the algorithm
Continue adding cuts?
• Number of cuts
• Value of minimum distance problem
Solve r-(BM) to obtain 𝑥 𝐵𝑀
No
Add
cut?
Yes
Solve: min
𝑥∈(𝑆𝐸𝑃)
𝑥 − 𝑥 𝐵𝑀
and obtain 𝑥 𝑠𝑒𝑝
2
2
Solve (BM)
+ cuts
Proper basic steps
• Number of “key disjunctions” (one, five, as most as
possible?)
• Number of basic steps in each “key”
• Which disjunctions to intersect in each “key”
Improper basic steps
• Which global constraints intersect with which
disjunctions?
• Repeat some global constraints to increase the
number of intersections?
Minimization norm
2
•
2 is convex and its gradient simple. For linear
cases
1 or
∞ provide LP min problem
GDP presolve1
• Eliminate infeasible terms, find better bounds, and
provide a value that characterizes each disjunction
1.
19
Francisco Trespalacios and Ignacio E. Grossmann, Algorithmic approach for improved mixed-integer reformulations of convex Generalized Disjunctive Programs, submitted for
publication Informs JOC
Examples and results
20
Different strategies were tested for the algorithm
Heuristic for selection of intersecting disjunctions is based on three key concepts
Global
constraints
“key disjunctions”
Tested strategies
21
Strategy
# of key disjunctions
K0
0
K1
1
K5
5
KK
Max
Strategy
# of key disjunctions
I1
No-redundant
I2
Max
Selection of intersecting disjunctions
Only with disjunctions that share variables
in common
For improper
Basic steps First priority: “key disjunctions”
No-redundant
Second: order of disjunction
Between disjunctions that share variables
in common
For proper
Basic steps
Consider # of terms in resulting disjunction
= (# of terms in D1)*(# of terms in D2)
The disjunction with highest char. value
from the presolve (when presolve is active)
Algorithm was tested with convex GDP problems
Constrained Layout (Clay)
Process Flowsheet Type A (Proc-1)
8 instances
4 instances
y
x19
6
x1
1
x12
1
3
x3
x2
4
2
2
x4
x11
5
x6
3
x16
x25
x17
𝐽
𝑗=1
4 instances
x22
x18
8
Non linear term
𝐽
𝑑𝑖𝑗 𝑒
Farm Layout (Flay)
7
x9
Unit function is nonlinear:
x
𝑥𝑗 𝑡𝑖𝑗
−1 −
𝑠𝑖𝑗 𝑥𝑗 ≤ 0
𝑗=1
Type A: Select at most one unit
𝐿
A2
Process Flowsheet Type B (Proc-2)
A3
𝐻
3 instances
A4
Type B: Possible to select more than one unit
A1
x
22
x23
x21
x8
y
x13
x15
6
∆𝒙𝟐𝟓
x24
4
x5
∆𝒚𝟒𝟔
5
x20
Algorithm solves problems faster than (BM) and (HR)
19 test instances
Algorithm with 5 “key disjunctions” nonredundant improper basic steps and 3 cuts
Percentage of problem solved vs. time
% of problems
100
90
(BM)
80
70
(HR)
60
K5-I1-3cuts
(BM)
50
(HR)
40
30
20
10
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 5,500 6,000 6,500 7,000
Time (s)
Implemented in GAMS 24.1.3. SBB solver
23
The algorithm improves the relaxation gap
Relaxation does not improve much after the first cut
Average relaxation gap vs. number of cuts
Relaxation gap
1.0
(BM)
0.9
0.8
(HR)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
Number of cuts
24
K0-I1
K1-I1
K5-I1
KK-I1
(HR)
K0-I2
K1-I2
K5-I2
KK-I2
(BM)
7
8
9
10
Algorithm improves solution times using any strategy
Using ~3 cuts seems to provide the best solution times
Accumulated solution time vs. number of cuts
Acc. Solution time (s)
40,000
(HR)
35,000
30,000
(BM)
25,000
20,000
15,000
10,000
5,000
0
0
1
2
3
4
5
6
Number of cuts
25
K0-I1
K1-I1
K5-I1
KK-I1
(HR)
K0-I2
K1-I2
K5-I2
KK-I2
(BM)
Implemented in GAMS 24.1.3. SBB solver
7
8
9
10
Conclusions and future work
Proposed new cutting plane algorithm for convex GDP that relies on concept of basic
steps and GDP-to-MINLP reformulations
Results show that the proposed algorithm improves formulations in convex GDPs
• Solved larger problems in less time using the algorithm
• Obtained stronger relaxations than BM and HR
Future work will explore improvements for the algorithm and test additional problems
26