A Kriging model-based expensive multiobjective optimization algorithm using R2 indicator of expectation improvement.

*(English)*Zbl 1459.90193Summary: Most of the multiobjective optimization problems in engineering involve the evaluation of expensive objectives and constraint functions, for which an approximate model-based multiobjective optimization algorithm is usually employed, but requires a large amount of function evaluation. Aiming at effectively reducing the computation cost, a novel infilling point criterion EIR2 is proposed, whose basic idea is mapping a point in objective space into a set in expectation improvement space and utilizing the R2 indicator of the set to quantify the fitness of the point being selected as an infilling point. This criterion has an analytic form regardless of the number of objectives and demands lower calculation resources. Combining the Kriging model, optimal Latin hypercube sampling, and particle swarm optimization, an algorithm, EIR2-MOEA, is developed for solving expensive multiobjective optimization problems and applied to three sets of standard test functions of varying difficulty and comparing with two other competitive infill point criteria. Results show that EIR2 has higher resource utilization efficiency, and the resulting nondominated solution set possesses good convergence and diversity. By coupling with the average probability of feasibility, the EIR2 criterion is capable of dealing with expensive constrained multiobjective optimization problems and its efficiency is successfully validated in the optimal design of energy storage flywheel.

##### MSC:

90C29 | Multi-objective and goal programming |

62K05 | Optimal statistical designs |

90C30 | Nonlinear programming |

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\textit{D. Han} and \textit{J. Zheng}, Math. Probl. Eng. 2020, Article ID 9474580, 16 p. (2020; Zbl 1459.90193)

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##### References:

[1] | Deb, K., Multi-Objective Optimization Using Evolutionary Algorithms (2001), Hoboken, NJ, USA: John Wiley & Sons, Hoboken, NJ, USA · Zbl 0970.90091 |

[2] | Abraham, A.; Jain, L.; Abraham, A.; Jain, L.; Goldberg, R., Evolutionary multiobjective optimization, Advanced Information and Knowledge Processing, 1-6 (2005), London, UK: Springer, London, UK · Zbl 1079.90593 |

[3] | Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T., A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6, 2, 182-197 (2002) |

[4] | Zitzler, E.; Laumanns, M.; Thiele, L., SPEA2: improving the strength pareto evolutionary algorithm, Proceedings of the EUROGEN’2001 |

[5] | Zhang, Q.; Li, H., MOEA/D: a multiobjective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11, 712-731 (2007) |

[6] | Coello Coello, C. A.; Lechuga, M. S., MOPSO: a proposal for multiple objective particle swarm optimization, Proceedings of the 2002 Congress on Evolutionary Computation, IEEE Computer Society |

[7] | Ong, Y. S.; Nair, P. B.; Keane, A. J., Evolutionary optimization of computationally expensive problems via surrogate modeling, AIAA Journal, 41, 4, 687-696 (2003) |

[8] | Jones, D. R., Taxonomy of global optimization methods based on response surfaces, Journal of Global Optimization, 21, 4, 345-383 (2001) · Zbl 1172.90492 |

[9] | Clarke, S. M.; Griebsch, J. H.; Simpson, T. W., Analysis of support vector regression for approximation of complex engineering analyses, Journal of Mechanical Design, 127, 6, 1077-1087 (2005) |

[10] | Gaspar-Cunha, A.; Vieira, A., A multi-objective evolutionary algorithm using neural networks to approximate fitness, Evaluations, 6, 20 (2005) |

[11] | Kleijnen, J. P. C., Kriging metamodeling in simulation: a review, European Journal of Operational Research, 192, 3, 707-716 (2009) · Zbl 1157.90544 |

[12] | Antony, J., Design of Experiments for Engineers and Scientists (2014), Amsterdam, Netherlands: Elsevier, Amsterdam, Netherlands |

[13] | Parr, J.; Holden, C. M.; Forrester, A. I.; Keane, A. J., Review of efficient surrogate infill sampling criteria with constraint handling, Proceedings of the 2nd International Conference on Engineering Optimization |

[14] | Emmerich, M. T. M.; Yang, K.; Deutz, A. H.; Bartz-Beielstein, T.; Filipi, B.; Koros̆ec, P., Infill criteria for multiobjective bayesian optimization, High-Performance Simulation-Based Optimization, 3-16 (2020), Cham, Switzerland: Springer International Publishing, Cham, Switzerland |

[15] | Jones, D. R.; Schonlau, M.; Welch, W. J., Efficient global optimization of expensive black-box functions, Journal of Global Optimization, 13, 4, 455-492 (1998) · Zbl 0917.90270 |

[16] | Forrester, A.; Sobester, A.; Keane, A., Engineering Design via Surrogate Modelling: A Practical Guide (2008), Hoboken, NJ, USA: John Wiley & Sons, Hoboken, NJ, USA |

[17] | Viana, F.; Haftka, R., Surrogate-based optimization with parallel simulations using the probability of improvement, Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, American Institute of Aeronautics and Astronautics |

[18] | Knowles, J., ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems, IEEE Transactions on Evolutionary Computation, 10, 1, 50-66 (2006) |

[19] | Davins-Valldaura, J.; Moussaoui, S.; Pita-Gil, G.; Plestan, F., ParEGO extensions for multi-objective optimization of expensive evaluation functions, Journal of Global Optimization, 67, 1-2, 79-96 (2017) · Zbl 1359.90127 |

[20] | Shinkyu, J.; Shigeru, O., Efficient global optimization (EGO) for multi-objective problem and data mining, Proceedings of the 2005 IEEE Congress on Evolutionary Computation |

[21] | Murata, T.; Ishibuchi, H., MOGA: multi-objective genetic algorithms, Proceedings of the 1995 IEEE International Conference on Evolutionary Computation |

[22] | Zhang, Q.; Liu, W.; Tsang, E.; Virginas, B., Expensive multiobjective optimization by MOEA/D with Gaussian process model, IEEE Transactions on Evolutionary Computation, 14, 456-474 (2010) |

[23] | Svenson, J.; Santner, T., Multiobjective optimization of expensive-to-evaluate deterministic computer simulator models, Computational Statistics & Data Analysis, 94, 250-264 (2016) · Zbl 06918665 |

[24] | Shimoyama, K.; Jeong, S.; Obayashi, S., Kriging-surrogate-based optimization considering expected hypervolume improvement in non-constrained many-objective test problems, Proceedings of the 2013 IEEE Congress on Evolutionary Computation, IEEE |

[25] | Decker, K. M., The Monte Carlo method in science and engineering: theory and application, Computer Methods in Applied Mechanics and Engineering, 89, 1-3, 463-483 (1991) |

[26] | Li, M.; Yao, X., Quality evaluation of solution sets in multiobjective optimisation: a survey, ACM Computing Surveys, 52, 2, 1-38 (2019) |

[27] | Ishibuchi, H.; Imada, R.; Setoguchi, Y.; Nojima, Y., Reference point specification in inverted generational distance for triangular linear Pareto front, IEEE Transactions on Evolutionary Computation, 22, 6, 961-975 (2018) |

[28] | Sierra, M. R.; Coello Coello, C. A.; Coello Coello, C. A.; Hernández Aguirre, A.; Zitzler, E., Improving PSO-based multi-objective optimization using crowding, mutation and epsilon-dominance. evolutionary multi-criterion optimization, Lecture Notes in Computer Science, 505-519 (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1109.68631 |

[29] | Zitzler, E.; Thiele, L.; Laumanns, M.; Fonseca, C.; da Fonseca, V., Performance assessment of multiobjective optimizers: an analysis and review, IEEE Transactions on Evolutionary Computation, 7, 2, 117-132 (2003) |

[30] | Li, B.; Tang, K.; Li, J.; Yao, X., Stochastic ranking algorithm for many-objective optimization based on multiple indicators, IEEE Transactions on Evolutionary Computation, 20, 6, 924-938 (2016) |

[31] | Bader, J.; Zitzler, E., HypE: an algorithm for fast hypervolume-based many-objective optimization, Evolutionary Computation, 19, 45-76 (2010) |

[32] | Ishibuchi, H.; Imada, R.; Setoguchi, Y.; Nojima, Y., How to specify a reference point in hypervolume calculation for fair performance comparison, Evolutionary Computation, 26, 3, 411-440 (2018) |

[33] | Pamulapati, T.; Mallipeddi, R.; Suganthan, P. N., I_SDE+—an indicator for multi and many-objective optimization, IEEE Transactions on Evolutionary Computation, 23, 346-352 (2019) |

[34] | Wang, R.; Chen, S.; Ma, L.; Cheng, S.; Shi, Y., Multi-indicator bacterial foraging algorithm with kriging model for many-objective optimization, Advances in Swarm Intelligence, 10941, 530-539 (2018), Cham, Switzerland: Springer International Publishing, Cham, Switzerland |

[35] | Phan, D. H.; Suzuki, J., R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization, Proceedings of the IEEE Congress on Evolutionary Computation |

[36] | Gómez, R. H.; Coello, C. A. C., MOMBI: a new metaheuristic for many-objective optimization based on the R2 indicator, Proceedings of the IEEE Congress on Evolutionary Computation |

[37] | Brockhoff, D.; Wagner, T.; Trautmann, H., On the properties of the R2 indicator, Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation, Association for Computing Machinery |

[38] | Sobester, A.; Leary, S. J.; Keane, A. J., On the design of optimization strategies based on global response surface approximation models, Journal of Global Optimization, 33, 1, 31-59 (2005) · Zbl 1137.90743 |

[39] | Pan, I.; Babaei, M.; Korre, A.; Durucan, S., Artificial neural network based surrogate modelling for multi-objective optimisation of geological CO_2 storage operations, Energy Procedia, 63, 3483-3491 (2014) |

[40] | Parr, J. M.; Forrester, A. I. J.; Keane, A. J.; Holden, C. M. E., Enhancing intill sampling criteria for surrogate-based constrained optimization, Journal of Computational Methods in Sciences and Engineering, 12, 1-2, 25-45 (2012) · Zbl 1250.90089 |

[41] | Li, F.; Liu, J.; Tan, S.; Yu, X., R2-MOPSO: a multi-objective particle swarm optimizer based on R2-indicator and decomposition, Proceedings of the IEEE Congress on Evolutionary Computation (CEC) |

[42] | Sato, H., Inverted PBI in MOEA/D and its impact on the search performance on multi and many-objective optimization, Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, Association for Computing Machinery |

[43] | Deutz, A.; Emmerich, M.; Yang, K.; Deb, K.; Goodman, E.; Coello Coello, C. A., The Expected R2-Indicator Improvement for Multi-Objective Bayesian Optimization. Evolutionary Multi-Criterion Optimization, 359-370 (2019), Cham, Switzerland: Springer International Publishing, Cham, Switzerland |

[44] | Park, J.-S., Optimal Latin-hypercube designs for computer experiments, Journal of Statistical Planning and Inference, 39, 1, 95-111 (1994) · Zbl 0803.62067 |

[45] | Kennedy, J.; Eberhart, R., Particle swarm optimization, Proceedings of ICNN’95-International Conference on Neural Networks |

[46] | Rehman, S. u.; Langelaar, M., Expected improvement based infill sampling for global robust optimization of constrained problems, Optimization and Engineering, 18, 723-753 (2017) · Zbl 1390.90457 |

[47] | Han, Z.; Liu, F.; Xu, C.; Zhang, K.; Zhang, Q., Efficient multi-objective evolutionary algorithm for constrained global optimization of expensive functions, Proceedings of the IEEE Congress on Evolutionary Computation (CEC) |

[48] | Jiang, L.; Zhang, W.; Ma, G. J.; Wu, C. W., Shape optimization of energy storage flywheel rotor, Structural and Multidisciplinary Optimization, 55, 2, 739-750 (2017) |

[49] | Mohan, S. C.; Maiti, D. K., Structural optimization of rotating disk using response surface equation and genetic algorithm, International Journal for Computational Methods in Engineering Science and Mechanics, 14, 2, 124-132 (2013) |

[50] | Sun, G.; Tian, Y.; Wang, R.; Fang, J.; Li, Q., Parallelized multiobjective efficient global optimization algorithm and its applications, Structural and Multidisciplinary Optimization, 61, 2, 763-786 (2020) |

[51] | Han, D.; Zheng, J. R., Solving expensive multi-objective optimization problems by kriging model with multi-point updating strategy, Applied Mechanics and Materials, 685, 667-670 (2014) |

[52] | Yang, K.; Palar, P. S.; Emmerich, M.; Shimoyama, K.; Bäck, T., A multi-point mechanism of expected hypervolume improvement for parallel multi-objective bayesian global optimization, Proceedings of the Genetic and Evolutionary Computation Conference, Association for Computing Machinery |

[53] | He, Y.; Sun, J.; Song, P.; Wang, X.; Usmani, A. S., Preference-driven Kriging-based multiobjective optimization method with a novel multipoint infill criterion and application to airfoil shape design, Aerospace Science and Technology, 96 (2020) |

[54] | Liu, J.; Song, W.-P.; Han, Z.-H.; Zhang, Y., Efficient aerodynamic shape optimization of transonic wings using a parallel infilling strategy and surrogate models, Structural and Multidisciplinary Optimization, 55, 3, 925-943 (2017) |

[55] | Ishibuchi, H.; Setoguchi, Y.; Masuda, H.; Nojima, Y., Performance of decomposition-based many-objective algorithms strongly depends on Pareto front shapes, IEEE Transactions on Evolutionary Computation, 21, 2, 169-190 (2017) |

[56] | Xu, S.; Chen, H.; Liang, X.; He, M., A modified MOEAD with an adaptive weight adjustment strategy, Proceedings of the International Conference on Intelligent Computing, Automation and Systems (ICICAS) |

[57] | Dai, C.; Lei, X.; He, X., A decomposition-based evolutionary algorithm with adaptive weight adjustment for many-objective problems, Soft Computing (2019) |

[58] | Li, M.; Yao, X., What weights work for you? Adapting weights for any Pareto front shape in decomposition-based evolutionary multiobjective optimisation, Evolutionary Computation, 28, 2, 227-253 (2020) |

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