A tighter piecewise McCormick relaxation for bilinear...
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A Tighter Piecewise McCormick
Relaxation for Bilinear Problems
Pedro M. CastroCentro de Investigação Operacional
Faculdade de Ciências
Universidade de Lisboa
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MINLP 2014
Problem definition (NLP)
June 3, 2014 Tighter Piecewise McCormick Relaxation 2
𝑎𝑖𝑗𝑞- parameters
𝒙 – vector of continuous variables𝐵𝐿𝑞 - (𝑖, 𝑗) index set
𝑖 ≠ 𝑗 – strictly bilinear problems𝑖 = 𝑗 – can be allowed (quadratic problems)ℎ𝑞(𝑥)- linear function in 𝑥
Relaxation provides a lower bound
min 𝑓0(𝑥)
𝑓𝑞(𝑥) =
(𝑖,𝑗)∈𝐵𝐿𝑞
𝑎𝑖𝑗𝑞𝑥𝑖𝑥𝑗 + ℎ𝑞(𝑥) 𝑞 ∈ 𝑄
• Bilinear program
0 ≤ 𝑥𝐿 ≤ 𝑥 ≤ 𝑥𝑈
𝑥 ∈ ℝ𝑚
𝑓𝑞(𝑥) ≤ 0 𝑞 ∈ 𝑄\{0}
(P)
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MINLP 2014
Introduction
• Bilinear problems occur in a variety of applications– Process network problems (Quesada & Grossmann, 1995)
– Water networks (Bagajewicz, 2000)
– Pooling and blending (Haverly, 1978)
• Non-convex, leading to multiple local solutions– Gradient based solvers unable to certify optimality
• Need for global optimization algorithms– What do they have in common?
• Linear (LP) or mixed-integer linear (MILP) relaxation of (P) → LB– LP: Standard McCormick envelopes (1976)
– MILP: Piecewise McCormick envelopes (Bergamini et al. 2005)
– MILP: Multiparametric disaggregation (Teles et al. 2013; Kolodziej et al. 2013)
• Solution of (P) with fast local solver → UB– Using LB as starting point
• Tight relaxation critical to ensure convergence– Relative optimality gap (UB-LB)/LB<
June 3, 2014 3Tighter Piecewise McCormick Relaxation
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MINLP 2014
Generalized Disjunctive Program (Balas, 1979; Raman & Grossmann, 1994)
Piecewise McCormick relaxation
• Domain of divided into 𝑁 uniform partitions
June 3, 2014 4Tighter Piecewise McCormick Relaxation
𝑤𝑖𝑗 = 𝑥𝑖𝑥𝑗
Optimization will
pick single partition. . .
𝑥𝑗𝐿
𝑥𝑗𝑈
𝑥𝑗,1𝐿 𝑥𝑗,2
𝐿 𝑥𝑗,𝑁𝐿
𝑥𝑗,𝑁𝑈𝑥𝑗,𝑁−1
𝑈𝑥𝑗,2𝑈𝑥𝑗,1
𝑈𝑛 = 1 𝑛 = 2 𝑛 = 𝑁
𝑦𝑗,1 = 𝑡𝑟𝑢𝑒 𝑦𝑗,2 = 𝑡𝑟𝑢𝑒 𝑦𝑗,𝑁 = 𝑡𝑟𝑢𝑒∨ ∨ ∨. . .
𝑥𝑗𝑈𝑥𝑗
𝐿 𝑥𝑗
𝑥𝑖
Convex
hull
1.25
1.45
1.65
1.85
2.05
2.25
2.45
200 250 300 350 400
𝑁 = 2
1.25
1.45
1.65
1.85
2.05
2.25
2.45
200 250 300 350 400
𝑁 = 20
Standard McCormick𝑁 = 1
∨𝑛
𝑦𝑗𝑛
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝐿 ⋅ 𝑥𝑗𝑛
𝐿
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑈 ⋅ 𝑥𝑗𝑛
𝑈
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑈 ⋅ 𝑥𝑗𝑛
𝐿
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝐿 ⋅ 𝑥𝑗𝑛
𝑈
𝑥𝑗𝑛𝐿 ≤ 𝑥𝑗 ≤ 𝑥𝑗𝑛
𝑈
𝑥𝑖𝐿 ≤ 𝑥𝑖 ≤ 𝑥𝑖
𝑈
𝑥𝑗𝑛𝐿 = 𝑥𝑗
𝐿 + (𝑥𝑗𝑈 − 𝑥𝑗
𝐿) ⋅ (𝑛 − 1)/𝑁
𝑥𝑗𝑛𝑈 = 𝑥𝑗
𝐿 + (𝑥𝑗𝑈 − 𝑥𝑗
𝐿) ⋅ 𝑛/𝑁
(PR-MILP)
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MINLP 2014
New tighter piecewise relaxation
• Basic idea– Use partition-dependent
bounds also for variable 𝑥𝑖• 𝑥𝑖𝑗𝑛
𝐿 /𝑥𝑖𝑗𝑛𝑈 lower/upper bound
when 𝑥𝑗 constrained to 𝑛
– Better bounds tighter relaxation lower gap
June 3, 2014 5Tighter Piecewise McCormick Relaxation
∨𝑛
𝑦𝑗𝑛
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖𝑗𝑛
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝑗𝑛𝐿 ⋅ 𝑥𝑗𝑛
𝐿
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖𝑗𝑛
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑗𝑛𝑈 ⋅ 𝑥𝑗𝑛
𝑈
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖𝑗𝑛
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑗𝑛𝑈 ⋅ 𝑥𝑗𝑛
𝐿
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖𝑗𝑛
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝑗𝑛𝐿 ⋅ 𝑥𝑗𝑛
𝑈
𝑥𝑖𝑗𝑛𝐿 ≤ 𝑥𝑖 ≤ 𝑥𝑖𝑗𝑛
𝑈
𝑥𝑗𝑛𝐿 ≤ 𝑥𝑗 ≤ 𝑥𝑗𝑛
𝑈
𝑥𝑗𝑛𝐿 = 𝑥𝑗
𝐿 + (𝑥𝑗𝑈 − 𝑥𝑗
𝐿) ⋅ (𝑛 − 1)/𝑁
𝑥𝑗𝑛𝑈 = 𝑥𝑗
𝐿 + (𝑥𝑗𝑈 − 𝑥𝑗
𝐿) ⋅ 𝑛/𝑁
(PRT-MILP)
Convex
hull
min 𝑧𝑅 = 𝑓0 𝑥 =
𝑖,𝑗 ∈𝐵𝐿
𝑎𝑖𝑗0𝑤𝑖𝑗 + ℎ0 𝑥
𝑓𝑞 𝑥 =
𝑖,𝑗 ∈𝐵𝐿
𝑎𝑖𝑗𝑞𝑤𝑖𝑗 + ℎ𝑞 𝑥 ≤ 0 ∀𝑞 ∈ 𝑄\{0}
𝑥𝑗 = 𝑛 𝑥𝑗𝑛 𝑛 𝑦𝑗𝑛 = 1
∀ {𝑗|(𝑖, 𝑗) ∈ 𝐵𝐿}
𝑥𝑗𝑛𝐿 = 𝑥𝑗
𝐿 +𝑥𝑗𝑈 − 𝑥𝑗
𝐿 ∙ 𝑛 − 1
𝑁
𝑥𝑗𝑛𝑈 = 𝑥𝑗
𝐿 +𝑥𝑗𝑈 − 𝑥𝑗
𝐿 ∙ 𝑛
𝑁𝑥𝑗𝑛𝐿 ⋅ 𝑦𝑗𝑛 ≤ 𝑥𝑗𝑛 ≤ 𝑥𝑗𝑛
𝑈 ⋅ 𝑦𝑗𝑛
∀ 𝑗 𝑖, 𝑗 ∈ 𝐵𝐿 , 𝑛
𝑤𝑖𝑗 ≥ 𝑛
𝑥𝑖𝑗𝑛 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖𝑗𝑛
𝐿 ⋅ 𝑥𝑗𝑛 − 𝑥𝑖𝑗𝑛𝐿 ⋅ 𝑥𝑗𝑛
𝐿 ⋅ 𝑦𝑗𝑛
𝑤𝑖𝑗 ≥ 𝑛
𝑥𝑖𝑗𝑛 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖𝑗𝑛
𝑈 ⋅ 𝑥𝑗𝑛 − 𝑥𝑖𝑗𝑛𝑈 ⋅ 𝑥𝑗𝑛
𝑈 ⋅ 𝑦𝑗𝑛
𝑤𝑖𝑗 ≤ 𝑛
𝑥𝑖𝑗𝑛 ⋅ 𝑥𝑗𝑛𝐿 + 𝑥𝑖𝑗𝑛
𝑈 ⋅ 𝑥𝑗𝑛 − 𝑥𝑖𝑗𝑛𝑈 ⋅ 𝑥𝑗𝑛
𝐿 ⋅ 𝑦𝑗𝑛
𝑤𝑖𝑗 ≤ 𝑛
𝑥𝑖𝑗𝑛 ⋅ 𝑥𝑗𝑛𝑈 + 𝑥𝑖𝑗𝑛
𝐿 ⋅ 𝑥𝑗𝑛 − 𝑥𝑖𝑗𝑛𝐿 ⋅ 𝑥𝑗𝑛
𝑈 ⋅ 𝑦𝑗𝑛
𝑥𝑖 = 𝑛 𝑥𝑖𝑗𝑛
∀ (𝑖, 𝑗)
𝑥𝑖𝑗𝑛𝐿 ⋅ 𝑦𝑗𝑛 ≤ 𝑥𝑖𝑗𝑛 ≤ 𝑥𝑖𝑗𝑛
𝑈 ⋅ 𝑦𝑗𝑛 ∀ (𝑖, 𝑗) ∈ 𝐵𝐿, 𝑛 ∈ {1, … , 𝑁}
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MINLP 2014
How to generate bounds 𝑥𝑖𝑗𝑛𝐿 & 𝑥𝑖𝑗𝑛
𝑈 ?
• Optimality bound contraction with McCormick envelopes– Solve multiple instances of LP problem (BC), two per:
• Non-partitioned variable 𝑥𝑖• Partitioned variable 𝑥𝑗• Partition 𝑛
– Focus on regions that can actually improve current UB• 𝑧´ obtained from solving (P) with local solver
– (BC) infeasible? Remove partition 𝑛∗ of 𝑗∗ from (PRT-MILP)
– Computation can be time consuming
June 3, 2014 6Tighter Piecewise McCormick Relaxation
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝐿 + 𝑥𝑖
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝐿 ⋅ 𝑥𝑗
𝐿
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑈 + 𝑥𝑖
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑈 ⋅ 𝑥𝑗
𝑈
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝐿 + 𝑥𝑖
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑈 ⋅ 𝑥𝑗
𝐿
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑈 + 𝑥𝑖
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝐿 ⋅ 𝑥𝑗
𝑈
∀ (𝑖, 𝑗) ∈ 𝐵𝐿
𝑥𝑗∗𝐿 = 𝑥𝑗∗𝑛∗
𝐿 ≤ 𝑥𝑗∗ ≤ 𝑥𝑗∗𝑛∗𝑈 = 𝑥𝑗∗
𝑈
𝑥𝐿 ≤ 𝑥 ≤ 𝑥𝑈
𝑓𝑞(𝑥) =
𝑖,𝑗 ∈𝐵𝐿𝑞
𝑎𝑖𝑗𝑞𝑤𝑖𝑗 + ℎ𝑞(𝑥) ≤ 0 ∀𝑞 ∈ 𝑄\{0}
𝑥𝑖∗𝑗∗𝑛∗𝐿 ∶= min 𝑥𝑖∗ (𝑥𝑖∗𝑗∗𝑛∗
𝑈 ∶= max 𝑥𝑖∗
(BC)
𝑓0 𝑥 =
𝑖,𝑗 ∈𝐵𝐿0
𝑎𝑖𝑗0𝑤𝑖𝑗 + ℎ0(𝑥) ≤ 𝑧´
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MINLP 2014
New optimization algorithm
• Key features
– User selects:
• 𝑁 partitions
• Partitioned
variables 𝑥𝑗
– Preliminary bound
contraction stage
– Bounds updated at
different levels
– LB from
MILP relaxation
– UB from NLP (single
starting point)
– Computes an
optimality gap
June 3, 2014 Tighter Piecewise McCormick Relaxation 7
Given:
partitions
, , , , ,
SOLVE (P)with local solver
upper bound
SOLVE (standard bound contract)
update, , ,
Initialization
SOLVE (BC)
Feasible?remove partitionNOtighter bounds
,
YES
YESupdate bounds
NO
YESupdate bounds
NO
NO
SOLVE(PRT-MILP)
YES
OutputOptimal solution
Optimality gap
Time Elapsed
lower bound
SOLVE (P)with local solver
upper bound
Given:
partitions
, , , , ,
SOLVE (P)with local solver
upper bound
SOLVE (standard bound contract)
update, , ,
Initialization
SOLVE (BC)
Feasible?remove partitionNOtighter bounds
,
YES
YESupdate bounds
NO
YESupdate bounds
NO
NO
SOLVE(PRT-MILP)
YES
OutputOptimal solution
Optimality gap
Time Elapsed
lower bound
SOLVE (P)with local solver
upper bound
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MINLP 2014
Illustrative example P1
June 3, 2014 Tighter Piecewise McCormick Relaxation 8
min 𝑓0 𝑥 = −𝑥1 + 𝑥1𝑥2 − 𝑥2
−6𝑥1 + 8𝑥2 ≤ 33𝑥1 − 𝑥2 ≤ 3
0 ≤ 𝑥1, 𝑥2 ≤ 1.5 Standard optimality bound contraction
0.357143 ≤ 𝑥1 ≤ 1.375
0 ≤ 𝑥2 ≤ 1.26
Partitioned variable: 𝑥1𝐵𝐿 = {(2,1)}
Non-partitioned variable: 𝑥2
Partition 𝑛 𝑥1𝒏𝐿 𝑥1𝒏
𝑼 𝑥21𝒏𝐿 𝑥21𝒏
𝑼
1 0.357143 0.696429 LP is infeasible
2 0.696429 1.035714 0 1.137609
3 1.035714 1.375 0.107143 1.195150
Partition-dependent bound contraction 𝑁 = 3
𝑧 = 𝑓0 𝑥 = −1.083333
Global optimum
𝑧𝑅 = −1.185207
(PR-MILP) relaxation
𝑧𝑅 = −1.169619
(PRT-MILP) relaxation
Gap
8.60%
7.38%
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MINLP 2014
How about using more partitions?
• Up to two orders of magnitude reduction in optimality gap– Major increase in total computational time
– Still, new approach performs best!
June 3, 2014 9Tighter Piecewise McCormick Relaxation
Relaxation Partitions (𝑁) 15 150 1500 7500
(PR-MILP)Total CPUs 0.93 1.60 3.63 302
Optimality gap 1.67% 0.188% 0.0189% 0.0038%
(PRT-MILP)Optimality gap 0.77% 0.010% 0.0002%
Total CPUs 3.63 27.8 267
P1
Relaxation Partitions (𝑁) 9 90 900 2700
(PR-MILP)Total CPUs 0.83 1.2 7.76 478
Optimality gap 8.41% 0.84% 0.0830% 0.0278%
(PRT-MILP)Optimality gap 0.84% 0.02% 0.0002%
Total CPUs 3.33 24.0 223
P2
min 6𝑥12 + 4𝑥2
2 − 2.5𝑥1𝑥2
𝑥1𝑥2 − 8 ≥ 0
1 ≤ 𝑥1, 𝑥2 ≤ 10
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MINLP 2014
Results for larger test problems
• 16 easiest water-using design problems– New (PRT-MILP) approach leads to lower gaps at termination (3600 CPUs)
– Faster in 56% of the cases (finest partition level)
– Single failure when finding best-known solution for (P)
June 3, 2014 10Tighter Piecewise McCormick Relaxation
Number of variables (PR-MILP) (PRT-MILP)
Non-partitioned |𝐼| Partitioned |𝐽| Partitions (𝑁) Solution Total CPUs Gap Gap Total CPUs Solution
Ex2 20 1610
74.469918.8 0.1250% 0.0484% 141
74.469950 3609 0.1245% 0.0103% 1536
Ex6 30 2510
142.081644.7 0.0435% 0.0110% 205
142.081650 3614 0.1379% 0.0041% 1090
Ex8 30 2010
164.489818.8 0.7134% 0.2132% 207
164.4898100 3613 1.9508% 0.0130% 1803
Ex10 15 910
169.11736.69 0.0160% 0.0008% 45.3
169.1173100 809 0.0023% 0.0000% 323
Ex14 32 2410 329.9689 282 1.2259% 1.0499% 447
329.569820 329.9566 3615 2.3368% 0.9206% 4270
Ex15 40 3010 361.5177 231 0.7963% 0.7427% 723 361.5177
20 361.6856 3618 0.9361% 0.7779% 4486 361.6856
Ex16 24 1610
285.934317.4 1.1105% 0.8861% 212
285.934350 3610 1.5755% 0.0526% 2042S
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MINLP 2014
How to determine the number of partitions?
• Quality of relaxation increases with 𝑁– Careful not to generate intractable MILPs (Gounaris et al. 2009)
• Several works identify most appropriate value for a particular problem(Misener et al. 2011; Wittmann-Holhbein and Pistikopoulos, 2013; 2014)
– No method to estimate 𝑁 as a function of problem complexity
– Needed to provide a fair comparison with commercial solvers
• Proposed formula meets important criteria:– Returns an integer value
– Ensures minimum of two partitions• So as to benefit from piecewise relaxation scheme
– Reduces 𝑁 with increase in problem size• Roughly inversely proportional (r2=0.76)
June 3, 2014 11Tighter Piecewise McCormick Relaxation
𝑁 = 1 +𝛼
|𝐼| ∙ |𝐽|𝛼 = 1.8𝐸4
![Page 12: A tighter piecewise McCormick relaxation for bilinear problemsminlp.cheme.cmu.edu/2014/papers/castro.pdfA Tighter Piecewise McCormick Relaxation for Bilinear Problems Pedro M. Castro](https://reader034.fdocuments.net/reader034/viewer/2022042123/5e9ec91a16229736fe280656/html5/thumbnails/12.jpg)
MINLP 2014
Comparison to commercial solvers
• Performance profiles(Dolan & Moré, 2002)
– KPI: Optimality gap
June 3, 2014 12Tighter Piecewise McCormick Relaxation
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
Cum
ula
tive d
istr
ibution f
unction
(PR-MILP)
(PRT-MILP)
GloMIQO 2.3
BARON 12.3.3
GAMS 24.1.3, CPLEX 12.5.1, GloMIQO 2.3,
BARON 12.3.3, Intel Core i7-3770 (3.07 GHz),
8 GB RAM, Windows 7 64-bit
Termination criteria: gap=0.0001%, Time=3600 CPUs
• (PRT-MILP) better (PR-MILP)
• GloMIQO outperforms BARON
# Failures finding best-known solution (34 problems)
Algorithm GloMIQO (PRT-MILP) BARON (PR-MILP)
Suboptimal 3 5 2 7
No solution 0 0 4 0
Total 3 5 6 7
![Page 13: A tighter piecewise McCormick relaxation for bilinear problemsminlp.cheme.cmu.edu/2014/papers/castro.pdfA Tighter Piecewise McCormick Relaxation for Bilinear Problems Pedro M. Castro](https://reader034.fdocuments.net/reader034/viewer/2022042123/5e9ec91a16229736fe280656/html5/thumbnails/13.jpg)
MINLP 2014
Why not bivariate partitioning?
• Domain of 𝑥𝑖 and 𝑥𝑗 known a priori for each 2-D partition
– Slightly better performance than univariate partitioning for P1
– Gap for 1252 partitions one order of magnitude larger vs. (PRT-MILP)
June 3, 2014 13Tighter Piecewise McCormick Relaxation
𝑛=1
𝑁
𝑛´=1
𝑁
𝑦𝑖𝑛𝑗𝑛´
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛´𝐿 + 𝑥𝑖𝑛
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝑛𝐿 ⋅ 𝑥𝑗𝑛´
𝐿
𝑤𝑖𝑗 ≥ 𝑥𝑖 ⋅ 𝑥𝑗𝑛´𝑈 + 𝑥𝑖𝑛
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑛𝑈 ⋅ 𝑥𝑗𝑛´
𝑈
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛´𝐿 + 𝑥𝑖𝑛
𝑈 ⋅ 𝑥𝑗 − 𝑥𝑖𝑛𝑈 ⋅ 𝑥𝑗𝑛´
𝐿
𝑤𝑖𝑗 ≤ 𝑥𝑖 ⋅ 𝑥𝑗𝑛´𝑈 + 𝑥𝑖𝑛
𝐿 ⋅ 𝑥𝑗 − 𝑥𝑖𝑛𝐿 ⋅ 𝑥𝑗𝑛´
𝑈
𝑥𝑖𝑛𝐿 ≤ 𝑥𝑖 ≤ 𝑥𝑖𝑛
𝑈
𝑥𝑗𝑛´𝐿 ≤ 𝑥𝑗 ≤ 𝑥𝑗𝑛´
𝑈
∀(𝑖, 𝑗)
(PR
-BV
-MIL
P)
Relaxation Partitions (𝑁) 16 144 2500 10000
Univariate
(PR-MILP)
Total CPUs 0.88 1.42 9.11 429
Optimality gap 1.70% 0.197% 0.0114% 0.0029%
𝑵 4 12 50 100 125
Bivariate
(PR-BV-MILP)
Optimality gap 1.28% 0.188% 0.0078% 0.0029% 0.0018%
Total CPUs 0.93 0.91 7.55 217 469
![Page 14: A tighter piecewise McCormick relaxation for bilinear problemsminlp.cheme.cmu.edu/2014/papers/castro.pdfA Tighter Piecewise McCormick Relaxation for Bilinear Problems Pedro M. Castro](https://reader034.fdocuments.net/reader034/viewer/2022042123/5e9ec91a16229736fe280656/html5/thumbnails/14.jpg)
MINLP 2014
Conclusions
• Bilinear problems tackled through piecewise relaxation (PR)
• Novel approach with partition-dependent boundsfor all bilinear variables– Significant improvement in relaxation quality
– Need for extensive optimality-based bound contraction (BC)• Total computational time is sometimes lower
• New algorithm outperforms state-of-the-art global solvers– Specific class of water-using network design problems
• PR and BC schemes should be used to greater extent– Integrating with spatial B&B subject of future work
• See CACE paper for further details
• Acknowledgments:– Luso-American Foundation (2013 Portugal-U.S. Research Networks)
– Fundação para a Ciência e Tecnologia (Investigador FCT 2013)
June 3, 2014 14Tighter Piecewise McCormick Relaxation