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A stopping criterion for decomposition-based multi-objective evolutionary algorithms

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Abstract

This paper proposes a new stopping criterion for decomposition-based multi-objective evolutionary algorithms (MOEA/Ds) to reduce the unnecessary usage of computational resource. In MOEA/D, a multi-objective problem is decomposed into a number of single-objective subproblems using a Tchebycheff decomposition approach. Then, optimal Pareto front (PF) is obtained by optimizing the Tchebycheff objective of all the subproblems. The proposed stopping criterion monitors the variations of Tchebycheff objective at every generation using maximum Tchebycheff objective error (MTOE) of all the subproblems and stops the algorithm, when there is no significant improvement in MTOE. \(\chi ^{2}\) test is used for statistically verifying the significant changes of MTOE for every \(\gamma \) generations. The proposed stopping criterion is implemented in a recently constrained MOEA/D variant, namely CMOEA/D-CDP, and a simulation study is conducted with the constrained test instances for choosing a suitable tolerance value for the MTOE stopping criterion. A comparison with the recent stopping methods demonstrates that the proposed MTOE stopping criterion is simple and has minimum computational complexity. Moreover, the MTOE stopping criterion is tested on real-world application, namely multi-objective \(\hbox {H}_{\infty }\) loop shaping PID controller design. Simulation results revealed that the MTOE stopping criterion reduces the unnecessary usage of computational resource significantly when solving the constrained test instances and multi-objective \(\hbox {H}_{\infty }\) loop shaping PID controller design problems.

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Abbreviations

CDP:

Constraint-domination principle

DM:

Decision maker

EAs:

Evolutionary algorithms

FE:

Function evaluation

GD:

Generational distance

HV:

Hypervolume

MFE:

Maximum number of function evaluation

MOEA:

Multi-objective evolutionary algorithm

MOP:

Multi-objective problems

PF:

Pareto front

MIMO:

Multi-input multi-output

LSSC:

Least square stopping criterion

maxCD:

Maximum crowding distance

IAE:

Integral absolute error

IGD:

Inverted generational distance

MTOE:

Maximum Tchebycheff objective error

SISO:

Single-input single-output

PIs:

Performance indicators

MDR:

Mutual domination rate

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Acknowledgments

The authors gratefully thank the Principal and Management of Thiagarajar College of Engineering, Madurai, India, for their support and continuous encouragement in this research. This research was supported by Indian Government via University Grants Commission’s Maulana Azad National Fellowship scheme (File No.: MANF-MUS-TAM-3045). Also, authors thank to Tobias Wagner, Heike Trautmann and Luis Marti for sharing their MATLAB code of taxonomy of stopping methods for MOEA algorithms.

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Correspondence to K. Mohaideen Abdul Kadhar.

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Funding

This research is funded by University Grants Commission of India via Maulana Azad National Fellowship scheme (File No.: MANF-MUS-TAM-3045).

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Authors declared no conflict of interest.

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Communicated by V. Loia.

Appendix

Appendix

1.1 Calculation of Euclidean distance

In MOEA/D, the subproblems are represented as weight vectors. The Euclidean distance between the two subproblems or weight vectors (for example \(\lambda ^{1}\) and \(\lambda ^{2}\)) is calculated as follows:

$$\begin{aligned} d(\lambda ^{1},\lambda ^{2})=\sqrt{\sum _{k=1}^m {(\lambda _k^1 -\lambda _k^2 )^{2}}} \end{aligned}$$
(21)

where m is the number of objectives in the MOP. In this way, the Euclidean distances between a subproblem (\(\lambda ^{\mathrm{j}}\)) to all subproblems are calculated. Then, T numbers of neighborhood subproblems for a jth subproblem (\(\lambda ^{\mathrm{j}}\)) are selected based on their minimum Euclidean distance (see Tables 8, 9).

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Abdul Kadhar, K.M., Baskar, S. A stopping criterion for decomposition-based multi-objective evolutionary algorithms. Soft Comput 22, 253–272 (2018). https://doi.org/10.1007/s00500-016-2331-7

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