multiobjective and many-objective optimization using...

Multiobjective and Many-Objective Optimization using Evolutionary Algorithms

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Lin Qiuzhen College of Computer Science and Software Engineering,

Shenzhen University, Shenzhen

qiuzhlin@szu.edu.cn

Outline

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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

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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

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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

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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

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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)

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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)

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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

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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

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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

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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

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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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