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Multiobjective and Many-Objective Optimization using Evolutionary Algorithms
2019-4-16 1
Lin Qiuzhen College of Computer Science and Software Engineering,
Shenzhen University, Shenzhen
Outline
2019-4-16 2
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble MOEA Framework
4 A Clustering-based MaOEA
5 Conclusions & Future Work
Outline
2019-4-16 3
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble MOEA Framework
4 A Clustering-based MaOEA
5 Conclusions & Future Work
Multi/Many-objective Optimization Problem (MOP/MaOP)
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1 2Minimize ( ) ( ( ), ( ),..., ( ))subject to :
mF x f x f x f xx
m objective vectors
search space
Dominance & Pareto Optimal Solutions
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y dominates x if and only if:Ø y is no worse than x in any obj, and Ø y is better than x in at least one obj.For examples:• B dominates C• A and C are not comparable
x is Pareto optimal if no other solution dominates it.
Pareto set (PS) = the set of all Pareto optimal solutions in the decision space (x-space).
Pareto front (PF) = the image of the PS in the objective space (F-space).
Outline
2019-4-16 6
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble MOEA Framework
4 A Clustering-based MaOEA
5 Conclusions & Future Work
Multiobjective Evolutionary Algorithms (MOEAs)
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Ø Approximate the PS (PF) in a single run
Ø Main procedures:
Mating Selection:
select parent pairs to constitute a mating pool
Reproduction:
create offspring based on evolutionary operators
Environmental Selection:
preserve a set of non-dominated solutions to approximate the PS (PF)
Ø Three typical MOEAs: Pareto-dominance-based, indicator-based and decomposition-based MOEAs.
General framework of MOEAs
Pareto-dominance-based MOEAs
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The most famous MOEA: NSGA-II [1]
Other MOEAs:SPEA2,PAES2,
…NSGA-III,
VaEAθ-DEA
[1] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, 2002.
Pareto-dominance-based MOEAs(need to consider diversity)
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Non-dominated sorting in NSGA-II
Indicator-based MOEAs
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Ø Selection based on performance indicator (hypervolume,inverted generational distance, Hausdorff distance, etc.)
Ø Relevant algorithms:IBEA, SMS-EMOA, HyPE,…
Ø Due to the high computational cost, this type of MOEAs is less considered than others, especially for MaOPs.
Decomposition-based MOEAs
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Ø Decompose a MOP into multiple single optimization problems or multiple simple MOPs
Ø A set of weight vectors to determine diversity and convergence measured by weighted aggregation objective values
Ø Relevant algorithms: MOEA/D, MOEA/D-M2M, MOEA/D-STM, MOEA/D-IR,
…
[2] Q. F. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposition,” IEEE Trans. Evol. Comput., vol. 11 , no. 6, pp. 712–731, 2007.
Decomposition-based MOEAs
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Ø Three common aggregation approaches: Weighted sum approach (WS), Tchbycheff approach (TCH) and penalty-based boundary intersection approach (PBI)
(a) WS, (b) TCH, (c) PBIPF: Pareto frontIR: improvement region
Outline
2019-4-16 13
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble Framework for MOPs
4 A Clustering-based MaOEA
5 Conclusions & Future Work
An effective ensemble framework for MOPs
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Ø Motivation: Make full use of the advantages of three typical MOEAsØ Basic ideal: by combining the advantages of various evolutionary operators
and selection criteria that are run on multiple populations.
• 50 offspring solutions get by SBX and DE • solutions from SBX arecentralized around four corner points • solutions from DE are distributed more evenly
An effective ensemble framework for MOPs
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Ø Motivation: Make full use of the advantages of three typical MOEAsØ Basic ideal: by combining the advantages of various evolutionary operators
and selection criteria that run on multiple populations.
• decomposition-based selection criterion with uniform weight vectors is not so good at tackling the problems with discontinuous and irregular PFs
• Pareto-based selection criterion is unable to provide strong convergence pressure on UF2
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• Two mechanisms, namely competition and cooperation, are employed to drive the running of the ensembles
• Competition on evolutionary operators
• Cooperation on selection criteria
• A simple algorithm based on the idea of ensemble framework is introduced by employing Pareto and decomposition-based populations, termed EF-PD
An effective ensemble framework for MOPs
• EF-PDCompetition on SBX and DE
Cooperation on Pareto-based anddecomposition-based criterion
PP indicates the population evolvedby SBX and selected by Pareto-based criterion.DP denotes the population evolvedby DE and selected by decomposition-based criterion
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An effective ensemble framework for MOPs
Competition on SBX and DEBoth SBX and DE are adaptively run according to their credits (FSBX and FDE)
where at generation g and denote the index sets of subproblems enhanced by SBX and DE. and are the number of executions of SBX and DE indicates the enhancement of the i-th subproblem by TCH function, as:
the enhancement brought by the new solution y associated to the i-th subproblem over the original associated solution x
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EF-PD
gSBXI g
DEIgSBXN g
DENif
Competition on SBX and DEThe normalized credits (FS and FD) can be obtained by
Based on FS and FD, the number of executions of DE and SBX at generation g+1 can be calculated by
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EF-PD
Cooperation on Pareto-based and decomposition-based criterion
SDE (the offspring set generated by DE), SSBX (the offspring set generated by SBX)2019-4-16 20
EF-PD
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EF-PDCooperation on Pareto-based and decomposition-based criterion
and respectively denote the selectionmechanisms associated to the use of the decomposition-based and Pareto-based approaches
• Experimental results on ZDT and WFG problems
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EF-PD
Performance indicator: Inverted Generational Distance (IGD)
• Experimental results on DTLZ and UF problems
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EF-PD
Performance indicator: Inverted Generational Distance (IGD)
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EF-PD
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EF-PD
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EF-PD
[3] W. Wang, S. Yang, Q, Lin, Q. Zhang, K. Wong, C. A. Coello Coello, J. Chen, “An Effective Ensemble Framework for Multi-objective Optimization,” IEEE Transactions on Evolutionary Computation, in press, 2018.
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A generalized ensemble framework for MOPs
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A generalized ensemble framework for MOPs
[3] W. Wang, S. Yang, Q, Lin, Q. Zhang, K. Wong, C. A. Coello Coello, J. Chen, “An Effective Ensemble Framework for Multi-objective Optimization,” IEEE Transactions on Evolutionary Computation, in press, 2018.
Outline
2019-4-16 29
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble MOEA Framework
4 A Clustering-based MaOEA
5 Conclusions & Future Work
Many-objective evolutionary algorithms (MaOEAs)
2019-4-16 30
The major issue for solving MaOPs
inefficiency of Pareto
dominance
difficulty of diversity
maintenance
high computation
al cost
inefficiency of variation operators
Pareto-dominance-based
MOEAs
Decomposition-based MOEAs
Indicator-based MOEAs For all MOEAs
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Main ideal:Ø Without considering uniformly distributed weight vectors, which are used in
decomposition-based algorithms to maintain diversity.
Ø Without considering Pareto dominance relationship, which is inefficiency for MaOPs
Ø Using clustering method to classify the population into a number of clusters, which is expected to solve the difficulty of balancing convergence and diversity in high dimensional objective space.
A clustering-based Many-objective evolutionary algorithms (MaOEA/C)
2019-4-16 32
iq iaiq iaia
MaOEA/C
Two clustering methods are considered:Ø Partitional clustering method (PCM)
Ø Hierarchical clustering method (HCM)
Ø The adopted two-step clustering strategy (PCM followed HCM) aims to efficiently classify the union of parent and offspring populations into N clusters, requiring a computational cost similar to that of most state-of-the-art MaOEAs
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MaOEA/C
Partitional clustering method (PCM)• divide a population S into m clusters• m axes serve as the clustering centers, m is the number of objectives• angle value between solutions and axes as the similarity metric
Run Algorithm 1 to divide the solutions set S into m clusters and each cluster is also a solutions set
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MaOEA/CHierarchical clustering method (HCM)• divide a population S into k clusters, k is less than the size of S
• initialize each solution in S as the center, and each solution in S is regarded as a cluster
• Finally, only k clusters are remain
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MaOEA/C
Hierarchical clustering method (HCM)• angle value between two different centroids as the similarity metric• each time, combine two clusters (with the minimum
angle value between their centroids) into a new cluster , and update its centroid by
• all the computation of angle values and update of centroids are run in the normalized objective space.
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MaOEA/C
Pseudo-code of environmentalselection inMaOEA/C
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MaOEA/C
Pseudo-code of MaOEA/C
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A simple example with bi-objective optimization to illustrate the environmental selection process, using (a) PCM and then followed (b) HCM.
MaOEA/CTwo-step clustering strategy (PCM followed HCM)
In each final cluster, a solution with the best convergence performance is preserved.
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MaOEA/C
SimulationResults onMaFs
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MaOEA/C
SimulationResults onWFGs
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MaOEA/C(a) Average performance rank on 5-, 8-, 10-, 13- and 15-objectives
(b) Average performance rank on MaF1-7 and WFG1-9 problems
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MaOEA/C
2.094
3.8884.125
4.381 4.400 4.493 4.619
MaOEA/C EFR-RR Ɵ-DEA VaEA NSGA-III MOEA/D-DU SRA
Average ranking of Friedman’s test for the compared MaOEAs
All considered problems (MaF, WFG) with different objectives (5, 8, 10, 13, 15) are included
[4] Q. Lin,S. Liu, K. Wong, M. Gong, C. A. Coello Coello, J. Chen, J. Zhang, “A Clustering-based Evolutionary Algorithm for Many-objective Optimization Problems,” IEEE Transactions on Evolutionary Computation, in press, 2018
2019-4-16 43
The average running times of the selected seven MaOEAs on MaF1-MaF7 (M1-M7) and WFG1-WFG9 (W1-W9) with 10 objectives
MaOEA/C
Outline
2019-4-16 44
1 Basic Concepts
Petri网研究现状2 Three Typical MOEAs
3 An Ensemble MOEA Framework
4 A Clustering-based MaOEA
5 Conclusions & Future Work
Conclusions
Ø Three typical MOEAs have their own shortcomings for MOPs and MaOPs
Ø An ensemble framework of combining the advantages of three typical MOEAs can effectively handle MOPs
Ø Embedding clustering methods into evolutionary algorithm is an effective way for MaOPs
2019-4-16 45
Future Work
Ø An effective ensemble framework for MaOPs
Ø Using clustering method to self-guide weight vectors in decomposition-based MOEAs for improving diversity
Ø Effective variation operators to produce new solutions of MaOPs
Ø Embedding other machine learning methods into evolutionary algorithm for solving MaOPs
2019-4-16 46
个人主页: http://csse.szu.edu.cn/en/people1bbd.html?30298[1] Qiuzhen Lin, Songbai Liu, et al., A Clustering-based Evolutionary Algorithm for Many-objective Optimization Problems, IEEE Transactions on Evolutionary Computation, in press, online: DOI: 10.1109/TEVC.2018.2866927[2] Qiuzhen Lin, Songbai Liu, et al., Particle Swarm Optimization with A Balanceable Fitness Estimation for Many-objective Optimization Problems, IEEE Transactions on Evolutionary Computation, 2018, 22(1), 32-46.[3] Qiuzhen Lin, Jianyong Chen*, et al., A Hybrid Evolutionary Immune Algorithm for Multiobjective Optimization Problems, IEEE Transactions on Evolutionary Computation, Oct. 2016, 20(5), 711-729.[4]Wenjun Wang, Shaoqiang Yang, Qiuzhen Lin*, et al., An Effective Ensemble Framework for Multiobjective Optimization, IEEE Transactions on Evolutionary Computation, in press, online: DOI: 10.1109/TEVC.2018.2879078.[5] Qiuzhen Lin, Genmiao Jin, et al., A Diversity-Enhanced Resource Allocation Strategy for Decomposition-based Multiobjective Evolutionary Algorithm, IEEE Transactions on Cybernetics, 2018, 48(8), 2388-2401.[6] Qingling Zhu(研究生), Qiuzhen Lin *, et al., An External Archive-Guided Multi-objective Particle Swarm Optimization Algorithm, IEEE Transactions on Cybernetics, Sep 2017, 47(9), 2794 – 2808.[7]Fei Chen, Donghong Wang, Qiuzhen Lin *, et al., Towards Dynamic Verifiable Pattern Matching, IEEE Transactions on Big Data, 2018, in press, DOI: 10.1109/TBDATA.2018.2868657.[8]Lijia Ma, Jianqiang Li*, Qiuzhen Lin, et al., Reliable Link Inference for Network Data With Community Structures, IEEE Transactions on Cybernetics, 2018, in press, DOI: 10.1109/TCYB.2018.2860284.
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Thank you!
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