Is NSGA-II Ready for Large-Scale Multi-Objective Optimization?
Abstract
:1. Introduction
2. Related Work
3. Materials and Methods
3.1. Component-Based NSGA-II
Algorithm 1 Pseudo-code of an evolutionary algorithm. |
|
3.2. Parameter Space for Auto-Configuring NSGA-II
- random: the variable takes a random value within the bounds.
- bounds: if the value is lower/higher than the lower/upper bound, the variable is assigned the lower/upper bound.
- round: if the value is lower/higher than the lower/upper bound, the variable is assigned the upper/lower bound.
3.3. Experimental Methodology
3.3.1. Scenarios
3.3.2. Auto-Configuration and Performance Assessment
- A file describing the parameter space included in Table 2.
- A set of problems used for training.
- An executable program that, for each combination of problem and configuration selected by irace, returns an indicator value so that irace can compare different configurations.
- The total number of different configurations to generate. The default value is 100,000.
3.3.3. Computing Environments
4. Results
4.1. ZDT Benchmark
4.2. The CSO Problem
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. UDN Modeling and Instances
Cell | Parameter | Equation | LL | LM | LH | ML | MM | MH | HL | HM | HH |
---|---|---|---|---|---|---|---|---|---|---|---|
Micro | (A2) | 12 | |||||||||
f | (A5) | 5 GHz (BW = 500 MHz) | |||||||||
(A8) | 15 | ||||||||||
(A8) | 10000 | ||||||||||
(A8) | 1 | ||||||||||
(A8) | 1 | ||||||||||
8 | |||||||||||
2 | |||||||||||
300 | 300 | 300 | 600 | 600 | 600 | 900 | 900 | 900 | |||
Pico | (A2) | 20 | |||||||||
f | (A5) | 20 GHz (BW = 2000 MHz) | |||||||||
(A8) | 9 | ||||||||||
(A8) | 6800 | ||||||||||
(A8) | 0.5 | ||||||||||
(A8) | 1 | ||||||||||
64 | |||||||||||
4 | |||||||||||
1500 | 1500 | 1500 | 1800 | 1800 | 1800 | 2100 | 2100 | 2100 | |||
Femto | (A2) | 28 | |||||||||
f | (A5) | 68 GHz (BW = 6800 MHz) | |||||||||
(A8) | 5.5 | ||||||||||
(A8) | 4800 | ||||||||||
(A8) | 0.2 | ||||||||||
(A8) | 1 | ||||||||||
256 | |||||||||||
8 | |||||||||||
3000 | 3000 | 3000 | 6000 | 6000 | 6000 | 9000 | 9000 | 9000 | |||
UEs | 1000 | 2000 | 3000 | 1000 | 2000 | 3000 | 1000 | 2000 | 3000 |
Problem Formulation and Objectives
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Solutions Creation | Evaluation | Termination |
---|---|---|
- Random - Latin hypercube sampling - Scatter search | - Sequential - Multithreaded | - By evaluations - By time - By keyboard - By quality indicator |
Selection | Variation | Replacement |
- N-ary tournament - Random - Neighbour - Differential evolution | - Crossover and mutation - Differential evolution | - Ranking and density estimator - - |
Parameter | Domain |
---|---|
algorithmResult | {externalArchive, population} |
populationSizeWithArchive | [10, 200] s.t. algorithmResult == externalArchive |
externalArchive | crowdingDistanceArchive s.t. algorithmResult == externalArchive |
offspringPopulationSize | [1, 400] |
selection | {tournament, random} |
selectionTournamentSize | (2, 10) s.t. selection == tournament |
Real-coded variables | |
createInitialSolutions | {random, latinHypercubeSampling, scatterSearch} |
variation | crossoverAndMutationVariation |
crossover | {SBX, BLX_ALPHA} |
crossoverProbability | [0.0, 1.0] |
crossoverRepairStrategy | {random, round, bounds} |
sbxDistributionIndex | [5.0, 400.0] s.t. crossover == SBX |
blxAlphaCrossoverAlphaValue | [0.0, 1.0] s.t. crossover == BLX_ALPHA |
mutation | {uniform, polynomial, linkedPolynomial, nonUniform} |
mutationProbabilityFactor | [0.0, 2.0] |
mutationRepairStrategy | {random, round, bounds} |
polynomialMutationDistributionIndex | [5.0, 400.0] s.t. mutation ∈ {polynomial, linkedPolinomial} |
uniformMutationPerturbation | [0.0, 1.0] s.t. mutation == uniform |
nonUniformMutationPerturbation | [0.0, 1.0] s.t. mutation == nonUniform |
Binary-coded variables | |
createInitialSolutions | random |
variation | crossoverAndMutationVariation |
crossover | {singlePoint, HUX, uniform} |
crossoverProbability | [0.0, 1.0] |
mutation | {bitflip} |
mutationProbabilityFactor | [0.0, 2.0] |
Default Settings for NSGA-II | Settings of AutoNSGA-II |
---|---|
algorithmResult: population | algorithmResult: externalArchive |
populationSize: 100 | populationSizeWithArchive: 56 |
offspringPopulationSize: 100 | offspringPopulationSize: 14 |
variation: crossoverAndMutationVariation | variation: crossoverAndMutationVariation |
crossover: SBX | crossover: BLX_ALPHA |
crossoverProbability: 0.9 | crossoverProbability: 0.88 |
crossoverRepairStrategy: random | crossoverRepairStrategy: bounds |
sbxDistributionIndexValue: 20.0 | blxAlphaCrossoverAlphaValue: 0.94 |
mutation: polynomial | mutation: nonUniform |
mutationProbabilityFactor: 1 | mutationProbabilityFactor: 0.45 |
mutationRepairStrategy: random | mutationRepairStrategy: round |
polynomialMutationDistributionIndex: 20.0 | nonUniformMutationPerturbation: 0.3 |
selection: tournament | selection: tournament |
selectionTournamentSize: 2 | selectionTournamentSize: 9 |
Time (h) | Evaluations | ||||
---|---|---|---|---|---|
Problem | Variables | NSGA-II | AutoNSGA-II | NSGA-II | AutoNSGA-II |
ZDT1 | 2048 | 0.13 | 0.02 | 1,250,500 | 182,356 |
4096 | 0.51 | 0.12 | 2,906,100 | 484,356 | |
8192 | 2.40 | 0.50 | 6,622,600 | 1,039,156 | |
16,384 | 11.19 | 2.15 | 14,741,200 | 2,180,656 | |
32,768 | - | 9.04 | - | 4,605,556 | |
65,356 | - | 31.66 | - | 9,494,556 | |
131,072 | - | 120.02 | - | 19,359,356 | |
ZDT2 | 2048 | 0.14 | 0.02 | 1,472,800 | 164,756 |
4096 | 0.62 | 0.10 | 3,433,100 | 429,156 | |
8192 | 2.77 | 0.49 | 7,676,600 | 986,556 | |
16,384 | 12.30 | 2.28 | 17,059,600 | 2,358,056 | |
32,768 | - | 9.28 | - | 4,736,056 | |
65,356 | - | 39.19 | - | 10,081,856 | |
131,072 | - | 138.85 | - | 21,703,556 | |
ZDT3 | 2048 | 0.10 | 0.03 | 1,089,800 | 253,356 |
4096 | 0.47 | 0.16 | 2,514,200 | 610,956 | |
8192 | 2.08 | 0.62 | 5,463,000 | 1,267,656 | |
16,384 | 9.18 | 2.68 | 11,877,500 | 2,820,556 | |
32,768 | - | 11.39 | - | 6,158,256 | |
65,356 | - | 40.69 | - | 11,912,856 | |
131,072 | - | - | - | - | |
ZDT4 * | 2048 | - | 2.62 | - | 21,746,882 |
ZDT6 | 2048 | 0.45 | 0.04 | 5,401,100 | 291,856 |
4096 | 1.82 | 0.16 | 11,482,400 | 659,956 | |
8192 | 7.16 | 0.66 | 24,897,300 | 1,374,056 | |
16,384 | - | 3.08 | - | 3,221,156 | |
32,768 | - | 15.51 | - | 7,941,156 | |
65,356 | - | 63.79 | - | 17,685,556 | |
131,072 | - | - | - | - |
Default Settings for NSGA-II | Settings of AutoNSGA-II |
---|---|
algorithmResult: population | algorithmResult: externalArchive |
populationSize: 100 | populationSizeWithArchive: 93 |
offspringPopulationSize: 100 | offspringPopulationSize: 32 |
variation: crossoverAndMutationVariatio | variation: crossoverAndMutationVariation |
crossover: singlePoint | crossover: singlePloint |
crossoverProbability: 0.90 | crossoverProbability: 0.89 |
mutation: bitFlip | mutation: bitFlip |
mutationProbabilityFactor: 1 | mutationProbabilityFactor: 1.7 |
selection: tournament | selection: tournament |
selectionTournamentSize: 2 | selectionTournamentSize: 9 |
NSGA-II | AutoNSGA-II | |
---|---|---|
LL | ||
LM | ||
LH | ||
ML | ||
MM | ||
MH | ||
HL | ||
HM | ||
HH |
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Nebro, A.J.; Galeano-Brajones, J.; Luna, F.; Coello Coello, C.A. Is NSGA-II Ready for Large-Scale Multi-Objective Optimization? Math. Comput. Appl. 2022, 27, 103. https://doi.org/10.3390/mca27060103
Nebro AJ, Galeano-Brajones J, Luna F, Coello Coello CA. Is NSGA-II Ready for Large-Scale Multi-Objective Optimization? Mathematical and Computational Applications. 2022; 27(6):103. https://doi.org/10.3390/mca27060103
Chicago/Turabian StyleNebro, Antonio J., Jesús Galeano-Brajones, Francisco Luna, and Carlos A. Coello Coello. 2022. "Is NSGA-II Ready for Large-Scale Multi-Objective Optimization?" Mathematical and Computational Applications 27, no. 6: 103. https://doi.org/10.3390/mca27060103