Is NSGAII Ready for LargeScale MultiObjective Optimization?
Abstract
:1. Introduction
2. Related Work
3. Materials and Methods
3.1. ComponentBased NSGAII
Algorithm 1 Pseudocode of an evolutionary algorithm. 

3.2. Parameter Space for AutoConfiguring NSGAII
 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. AutoConfiguration 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  ${G}_{tx}$  (A2)  12  
f  (A5)  5 GHz (BW = 500 MHz)  
$\alpha $  (A8)  15  
$\beta $  (A8)  10000  
$\delta $  (A8)  1  
$\rho \left[W\right]$  (A8)  1  
${n}_{tx}$  8  
${n}_{rx}$  2  
${\lambda}_{P}^{micro}$$(Cells/k{m}^{2})$  300  300  300  600  600  600  900  900  900  
Pico  ${G}_{tx}$  (A2)  20  
f  (A5)  20 GHz (BW = 2000 MHz)  
$\alpha $  (A8)  9  
$\beta $  (A8)  6800  
$\delta $  (A8)  0.5  
$\rho \left[W\right]$  (A8)  1  
${n}_{tx}$  64  
${n}_{rx}$  4  
${\lambda}_{P}^{pico}$$(Cells/k{m}^{2})$  1500  1500  1500  1800  1800  1800  2100  2100  2100  
Femto  ${G}_{tx}$  (A2)  28  
f  (A5)  68 GHz (BW = 6800 MHz)  
$\alpha $  (A8)  5.5  
$\beta $  (A8)  4800  
$\delta $  (A8)  0.2  
$\rho \left[W\right]$  (A8)  1  
${n}_{tx}$  256  
${n}_{rx}$  8  
${\lambda}_{P}^{femto}$$(Cells/k{m}^{2})$  3000  3000  3000  6000  6000  6000  9000  9000  9000  
UEs  ${\lambda}_{P}^{UE}$$(UE/k{m}^{2})$  1000  2000  3000  1000  2000  3000  1000  2000  3000 
Problem Formulation and Objectives
References
 Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A Fast and Elitist Multiobjective Genetic Algorithm: NSGAII. IEEE Trans. Evol. Comput. 2002, 6, 182–197. [Google Scholar] [CrossRef] [Green Version]
 Li, H.; Zhang, Q. Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGAII. IEEE Trans. Evol. Comput. 2009, 13, 284–302. [Google Scholar] [CrossRef]
 Reyes Sierra, M.; Coello Coello, C.A. Improving PSOBased Multiobjective Optimization Using Crowding, Mutation and ϵDominance. In Evolutionary MultiCriterion Optimization; Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 505–519. [Google Scholar]
 Nebro, A.J.; Durillo, J.J.; GarcíaNieto, J.; Coello, C.A.C.; Luna, F.; Alba, E. SMPSO: A new PSObased metaheuristic for multiobjective optimization. In Proceedings of the 2009 IEEE Symposium on Computational Intelligence in MultiCriteria DecisionMaking (MCDM 2009), Nashville, TN, USA, 30 March–2 April 2009; pp. 66–73. [Google Scholar] [CrossRef]
 Zavala, G.R.; Nebro, A.J.; Luna, F.; Coello, C.A.C. A survey of multiobjective metaheuristics applied to structural optimization. Struct. Multidiscip. Optim. 2014, 49, 537–558. [Google Scholar] [CrossRef]
 Becerra, D.; Sandoval, A.; RestrepoMontoya, D.; Nino, L.F. A parallel multiobjective Ab initio approach for protein structure prediction. In Proceedings of the 2010 IEEE International Conference on Bioinformatics and Biomedicine, Houston, TX, USA, 9–12 December 2010; pp. 137–141. [Google Scholar]
 Fang, W.; Guan, Z.; Su, P.; Luo, D.; Ding, L.; Yue, L. MultiObjective Material Logistics Planning with Discrete Split Deliveries Using a Hybrid NSGAII Algorithm. Mathematics 2022, 10, 2871. [Google Scholar] [CrossRef]
 Turkson, R.F.; Yan, F.; Ahmed Ali, M.K.; Liu, B.; Hu, J. Modeling and multiobjective optimization of engine performance and hydrocarbon emissions via the use of a computer aided engineering code and the NSGAII genetic algorithm. Sustainability 2016, 8, 72. [Google Scholar] [CrossRef] [Green Version]
 AdensoDíaz, B.; Laguna, M. Finetuning of algorithms using fractional experimental designs and local search. Oper. Res. 2006, 54, 99–114. [Google Scholar] [CrossRef] [Green Version]
 Durillo, J.; Nebro, A. jMetal: A Java framework for multiobjective optimization. Adv. Eng. Softw. 2011, 42, 760–771. [Google Scholar] [CrossRef]
 Nebro, A.; Durillo, J.J.; Vergne, M. Redesigning the jMetal MultiObjective Optimization Framework. In Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation (GECCO Companion ’15), Madrid, Spain, 11–15 July 2015; ACM: New York, NY, USA, 2015; pp. 1093–1100. [Google Scholar] [CrossRef]
 LópezIbáñez, M.; DuboisLacoste, J.; Pérez Cáceres, L.; Stützle, T.; Birattari, M. The irace package: Iterated Racing for Automatic Algorithm Configuration. Oper. Res. Perspect. 2016, 3, 43–58. [Google Scholar] [CrossRef]
 Zitzler, E.; Deb, K.; Thiele, L. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evol. Comput. 2000, 8, 173–195. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Blot, A.; Hoos, H.H.; Jourdan, L.; KessaciMarmion, M.É.; Trautmann, H. MOParamILS: A Multiobjective Automatic Algorithm Configuration Framework. In Learning and Intelligent Optimization; Festa, P., Sellmann, M., Vanschoren, J., Eds.; Springer International Publishing: Cham, Switzerland, 2016; pp. 32–47. [Google Scholar]
 Bezerra, L.C.T.; LópezIbáñez, M.; Stützle, T. Automatic ComponentWise Design of Multiobjective Evolutionary Algorithms. IEEE Trans. Evol. Comput. 2016, 20, 403–417. [Google Scholar] [CrossRef] [Green Version]
 Bezerra, L.C.T.; LópezIbáñez, M.; Stützle, T. Automatically Designing StateoftheArt Multi and ManyObjective Evolutionary Algorithms. Evol. Comput. 2020, 28, 195–226. [Google Scholar] [CrossRef] [PubMed]
 Nebro, A.J.; LópezIbáñez, M.; BarbaGonzález, C.; GarcíaNieto, J. Automatic Configuration of NSGAII with jMetal and Irace; Association for Computing Machinery, Inc.: New York, NY, USA, 2019; pp. 1374–1381. [Google Scholar] [CrossRef] [Green Version]
 Huband, S.; Barone, L.; While, R.; Hingston, P. A Scalable Multiobjective Test Problem Toolkit. In Proceedings of the Third International Conference on Evolutionary MultiCriterion Optimization, EMO 2005, Guanajuato, Mexico, 9–11 March 2005; Coello, C., Hernández, A., Zitler, E., Eds.; Springer: Berlin, Germany, 2005. Lecture Notes in Computer Science. Volume 3410, pp. 280–295. [Google Scholar]
 Deb, K.; Thiele, L.; Laumanns, M.; Zitzler, E. Scalable Test Problems for Evolutionary Multiobjective Optimization. In Evolutionary Multiobjective Optimization. Theoretical Advances and Applications; Abraham, A., Jain, L., Goldberg, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 105–145. [Google Scholar]
 Durillo, J.J.; Nebro, A.J.; Coello, C.A.C.; GarciaNieto, J.; Luna, F.; Alba, E. A Study of Multiobjective Metaheuristics When Solving Parameter Scalable Problems. IEEE Trans. Evol. Comput. 2010, 14, 618–635. [Google Scholar] [CrossRef] [Green Version]
 Tian, Y.; Si, L.; Zhang, X.; Cheng, R.; He, C.; Tan, K.C.; Jin, Y. Evolutionary LargeScale MultiObjective Optimization: A Survey. ACM Comput. Surv. 2021, 54, 174. [Google Scholar] [CrossRef]
 Nebro, A.J.; Luna, F.; Alba, E.; Dorronsoro, B.; Durillo, J.J.; Beham, A. AbYSS: Adapting Scatter Search to Multiobjective Optimization. IEEE Trans. Evol. Comput. 2008, 12, 439–457. [Google Scholar] [CrossRef]
 Bohli, A.; Bouallegue, R. How to Meet Increased Capacities by Future Green 5G Networks: A Survey. IEEE Access 2019, 7, 42220–42237. [Google Scholar] [CrossRef]
 LopezPerez, D.; Ding, M.; Claussen, H.; Jafari, A.H. Towards 1 Gbps/UE in Cellular Systems: Understanding UltraDense Small Cell Deployments. IEEE Commun. Surv. Tutorials 2015, 17, 2078–2101. [Google Scholar] [CrossRef] [Green Version]
 González González, D.; Mutafungwa, E.; Haile, B.; Hämäläinen, J.; Poveda, H. A Planning and Optimization Framework for Ultra Dense Cellular Deployments. Mob. Inf. Syst. 2017, 2017, 9242058. [Google Scholar] [CrossRef]
 Luna, F.; LuqueBaena, R.; Martínez, J.; ValenzuelaValdés, J.; Padilla, P. Addressing the 5G Cell Switchoff Problem with a Multiobjective Cellular Genetic Algorithm. In Proceedings of the IEEE 5G World Forum, 5GWF 2018—Conference Proceedings, Silicon Valley, CA, USA, 9–11 July 2018; pp. 422–426. [Google Scholar] [CrossRef] [Green Version]
 Luna, F.; ZapataCano, P.H.; GonzálezMacías, J.C.; ValenzuelaValdés, J.F. Approaching the cell switchoff problem in 5G ultradense networks with dynamic multiobjective optimization. Future Gener. Comput. Syst. 2020, 110, 876–891. [Google Scholar] [CrossRef]
 Zille, H.; Ishibuchi, H.; Mostaghim, S.; Nojima, Y. Mutation operators based on variable grouping for multiobjective largescale optimization. In Proceedings of the 2016 IEEE Symposium Series on Computational Intelligence (SSCI), Athens, Greece, 6–9 December 2016; pp. 1–8. [Google Scholar] [CrossRef]
 Knowles, J. A summaryattainmentsurface plotting method for visualizing the performance of stochastic multiobjective optimizers. In Proceedings of the 5th ISDA, Washington, DC, USA, 8–10 September 2005; pp. 552–557. [Google Scholar]
 Vucetic, B.; Yuan, J. Performance Limits of MultipleInput MultipleOutputWireless Communication Systems. In SpaceTime Coding; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2005; chapter 1; pp. 1–47. [Google Scholar]
 Piovesan, N.; Fernandez Gambin, A.; Miozzo, M.; Rossi, M.; Dini, P. Energy sustainable paradigms and methods for future mobile networks: A survey. Comput. Commun. 2018, 119, 101–117. [Google Scholar] [CrossRef]
 Son, J.; Kim, S.; Shim, B. Energy Efficient UltraDense Network Using Long ShortTerm Memory. In Proceedings of the 2020 IEEE Wireless Communications and Networking Conference (WCNC), Seoul, Republic of Korea, 25–28 May 2020; pp. 1–6. [Google Scholar]
Solutions Creation  Evaluation  Termination 

 Random  Latin hypercube sampling  Scatter search   Sequential  Multithreaded   By evaluations  By time  By keyboard  By quality indicator 
Selection  Variation  Replacement 
 Nary tournament  Random  Neighbour  Differential evolution   Crossover and mutation  Differential evolution   Ranking and density estimator  $(\mu +\lambda )$  $(\mu ,\lambda )$ 
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 
Realcoded 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 
Binarycoded variables  
createInitialSolutions  random 
variation  crossoverAndMutationVariation 
crossover  {singlePoint, HUX, uniform} 
crossoverProbability  [0.0, 1.0] 
mutation  {bitflip} 
mutationProbabilityFactor  [0.0, 2.0] 
Default Settings for NSGAII  Settings of AutoNSGAII 

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  NSGAII  AutoNSGAII  NSGAII  AutoNSGAII 
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 NSGAII  Settings of AutoNSGAII 

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 
NSGAII  AutoNSGAII  

LL  ${0.733}_{\pm 0.074}$  ${0.857}_{\pm 0.041}$ 
LM  ${0.726}_{\pm 0.077}$  ${0.834}_{\pm 0.044}$ 
LH  ${0.707}_{\pm 0.116}$  ${0.814}_{\pm 0.071}$ 
ML  ${0.619}_{\pm 0.084}$  ${0.871}_{\pm 0.030}$ 
MM  ${0.659}_{\pm 0.098}$  ${0.843}_{\pm 0.048}$ 
MH  ${0.685}_{\pm 0.099}$  ${0.823}_{\pm 0.067}$ 
HL  ${0.699}_{\pm 0.080}$  ${0.868}_{\pm 0.034}$ 
HM  ${0.644}_{\pm 0.128}$  ${0.792}_{\pm 0.119}$ 
HH  ${0.725}_{\pm 0.103}$  ${0.812}_{\pm 0.086}$ 
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Nebro, A.J.; GaleanoBrajones, J.; Luna, F.; Coello Coello, C.A. Is NSGAII Ready for LargeScale MultiObjective Optimization? Math. Comput. Appl. 2022, 27, 103. https://doi.org/10.3390/mca27060103
Nebro AJ, GaleanoBrajones J, Luna F, Coello Coello CA. Is NSGAII Ready for LargeScale MultiObjective Optimization? Mathematical and Computational Applications. 2022; 27(6):103. https://doi.org/10.3390/mca27060103
Chicago/Turabian StyleNebro, Antonio J., Jesús GaleanoBrajones, Francisco Luna, and Carlos A. Coello Coello. 2022. "Is NSGAII Ready for LargeScale MultiObjective Optimization?" Mathematical and Computational Applications 27, no. 6: 103. https://doi.org/10.3390/mca27060103