Advertisement

A dynamic surrogate-assisted evolutionary algorithm framework for expensive structural optimization

  • Mingyuan Yu
  • Xia Li
  • Jing LiangEmail author
Research Paper
  • 93 Downloads

Abstract

In the expensive structural optimization, the data-driven surrogate model has been proven to be an effective alternative to physical simulation (or experiment). However, the static surrogate-assisted evolutionary algorithm (SAEA) often becomes powerless and inefficient when dealing with different types of expensive optimization problems. Therefore, how to select high-reliability surrogates to assist an evolutionary algorithm (EA) has always been a challenging task. This study aimed to dynamically provide an optimal surrogate for EA by developing a brand-new SAEA framework. Firstly, an adaptive surrogate model (ASM) selection technology was proposed. In ASM, according to different integration criteria from the strategy pool, elite meta-models were recombined into multiple ensemble surrogates in each iteration. Afterward, a promising model was adaptively picked out from the model pool based on the minimum root of mean square error (RMSE). Secondly, we investigated a novel ASM-based EA framework, namely ASMEA, where the reliability of all models was updated in real-time by generating new samples online. Thirdly, to verify the performance of the ASMEA framework, two instantiation algorithms are widely compared with several state-of-the-art algorithms on a commonly used benchmark test set. Finally, a real-world antenna structural optimization problem was solved by the proposed algorithms. The results demonstrate that the proposed framework is able to provide a high-reliability surrogate to assist EA in solving expensive optimization problems.

Keywords

Evolutionary algorithm Adaptive surrogate model Expensive optimization Reliability 

Nomenclature

Abbreviation

FEA

finite element analysis

CFD

computational fluid dynamics

EA

evolutionary algorithm

SAEA

surrogate-assisted evolutionary algorithm

ASM

adaptive surrogate model

ASMEA

ASM-based evolutionary algorithm

PSO

particle swarm optimization

DE

differential evolution

ASMPSO

particle swarm optimization based on ASM

ASMDE

differential evolution based on ASM

GP

Gauss process model

KRG

Kriging model

PRS

polynomial response surface model

RBF

radial basis function model

SHEP

Shepard model

ANN

artificial neural network

SVM

support vector machine

RBNN

radial basis neural network

ELM

extreme learning machine

HFSS

high-frequency simulation software

WAS

weighted average surrogate

OWS

optimum weight surrogate

DOE

design of experiment

LHS

Latin hypercube sampling

RMSE

root of mean square error

AE

absolute error

PRESS

prediction residual error sum of square

SR

successful run

VTR

value to reach

Symbols

n

the number of meta-models

k

the number of elite meta-models

D

the number of variables

NP

the size of the population

St

the size of the strategy pool

T1

the train set in the database

T2

the test set in the database

G or Gm

the evolutional generation or maximum G

FES or FESmax

the function evaluations or maximum FES

c1 and c2

learn rates in PSO

F

scale factor in DE

Cr

crossover rate in DE

xmin and xmax

variable space

funASM

evaluation function based on ASM

funreal

evaluation function based on simulation

G-opt

global optimum

L-opt

local optimum

#G

the number of G-opt

#L

the number of L-opt

ASM1

the ASMPSO algorithm

ASM2

the ASMDE algorithm

N

the number of sampling points

S11

return less

L, W, h,

the length, width, height of the patch

L1

edge distance

Greek symbols

τ

scale factor in T1 and T2

ω

inertia weight in PSO

λ

wavelength

Notes

Funding information

This research was supported by the National Natural Science Foundation of China (Grant Nos. 61803054 and 61876169) and the State Education Ministry and Fundamental Research Funds for the Central Universities (2019 CDJSK 04 XK 23).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Acar E (2015) Effect of error metrics on optimum weight factor selection for ensemble of metamodels. Expert Syst Appl 42(5):2703–2709CrossRefGoogle Scholar
  2. Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294CrossRefGoogle Scholar
  3. Allmendinger R, Emmerich MTM, Hakanen J, Jin Y, Rigoni E (2017) Surrogate-assisted multicriteria optimization: complexities, prospective solutions, and business case. J Multi-Criteria Decis Anal 24(1–2)CrossRefGoogle Scholar
  4. Cheng R, Rodemann T, Fischer M, Olhofer M and Jin Y (2017) Evolutionary many-objective optimization of hybrid electric vehicle control: from general optimization to preference articulation. IEEE Transactions on Emerging Topics in Computational Intelligence, PP(99) pp 1–1Google Scholar
  5. Clerc M, Kennedy J (2002) The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 6(1):58–73CrossRefGoogle Scholar
  6. Deb A, Roy JS, Gupta B (2014) Performance comparison of differential evolution, particle swarm optimization and genetic algorithm in the design of circularly polarized microstrip antennas. IEEE Trans Antennas Propag 62(8):3920–3928zbMATHCrossRefGoogle Scholar
  7. Dong J, Li Q, Deng L (2018) Design of fragment-type antenna structure using an improved BPSO. IEEE Trans Antennas Propag 66(2):564–571CrossRefGoogle Scholar
  8. Elsayed SM, Ray T and Sarker RA (2014) A surrogate-assisted differential evolution algorithm with dynamic parameters selection for solving expensive optimization problems. Evolutionary Computation, pp 1062–1068Google Scholar
  9. Feng Z, Zhang Q, Zhang Q, Tang Q, Yang T, Ma Y (2015) A multiobjective optimization based framework to balance the global exploration and local exploitation in expensive optimization. J Glob Optim 61(4):677–694MathSciNetzbMATHCrossRefGoogle Scholar
  10. Garbo A, German BJ (2019) Performance assessment of a cross-validation sampling strategy with active surrogate model selection. Struct Multidiscip Optim 59(6):2257–2272CrossRefGoogle Scholar
  11. Gaspar B, Teixeira AP and Guedes Soares C (2017) Adaptive surrogate model with active refinement combining Kriging and a trust region method. Reliability Engineering & System Safety, PP(165) pp 277–291Google Scholar
  12. Goel T, Haftka RT, Wei S, Queipo NV (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216CrossRefGoogle Scholar
  13. Goudos SK, Gotsis KA, Siakavara K, Vafiadis EE, Sahalos JN (2013) A multi-objective approach to subarrayed linear antenna arrays design based on memetic differential evolution. IEEE Trans Antennas Propag 61(6):3042–3052MathSciNetzbMATHCrossRefGoogle Scholar
  14. Gunst R (1997) Response surface methodology: process and product optimization using designed experiments. J Stat Plan Inference 38(3):284–286Google Scholar
  15. Guo D, Jin Y, Ding J and Chai T (2018) Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems. IEEE Transactions on Cybernetics, PP(99) pp 1–14Google Scholar
  16. Herrera M, Guglielmetti A, Xiao M, Coelho RF (2014) Metamodel-assisted optimization based on multiple kernel regression for mixed variables. Struct Multidiscip Optim 49(6):979–991CrossRefGoogle Scholar
  17. Jin Y (2011) Surrogate-assisted evolutionary computation: recent advances and future challenges. Swarm Evol Comput 1(2):61–70CrossRefGoogle Scholar
  18. Jin Y and Sendhoff B (2002) Fitness approximation in evolutionary computation-a survey. GECCO: Genetic & Evolutionary Computation ConferenceGoogle Scholar
  19. Juan AA, Faulin J, Grasman SE, Rabe M, Figueira G (2015) A review of simheuristics: extending metaheuristics to deal with stochastic combinatorial optimization problems. Oper Res Persp 2(C):62–72MathSciNetGoogle Scholar
  20. Lian Y, Oyama A, Liou MS (2013) Progress in design optimization using evolutionary algorithms for aerodynamic problems. Prog Aerosp Sci 46(5):199–223Google Scholar
  21. Lim D and Sendhoff B (2007) A study on metamodeling techniques, ensembles, and multi-surrogates in evolutionary computation. Conference on Genetic and Evolutionary Computation, pp 1288–1295Google Scholar
  22. Lim D, Jin Y, Ong YS, Sendhoff B (2010) Generalizing surrogate-assisted evolutionary computation. IEEE Trans Evol Comput 14(3):329–355CrossRefGoogle Scholar
  23. Liu, B., Koziel S and Zhang Q (2016a) A multi-fidelity surrogate-model-assisted evolutionary algorithm for computationally expensive optimization problems. J Comput Sc PP(12):28–37MathSciNetCrossRefGoogle Scholar
  24. Liu B, Zhang QF and Gielen G (2016b) A surrogate-model-assisted evolutionary algorithm for computationally expensive design optimization problems with inequality constraints. Simulation-Driven Modeling and Optimization, pp 347–370Google Scholar
  25. Liu B, Koziel S, Ali N (2017) SADEA-II: a generalized method for efficient global optimization of antenna design. J Comput Des Eng 4(2):86–97Google Scholar
  26. Mallipeddi R and Lee M (2012) Surrogate model assisted ensemble differential evolution algorithm. Evolutionary Computation, pp 1–8Google Scholar
  27. Martin JD, Simpson TW (2004) Use of kriging models to approximate deterministic computer models. AIAA J 43(4):853–863CrossRefGoogle Scholar
  28. Miruna JAS, Baskar S (2015) Surrogate assisted-hybrid differential evolution algorithm using diversity control. Expert Syst 32(4):531–545CrossRefGoogle Scholar
  29. Müller J, Piché R (2010) Mixture surrogate models based on Dempster-Shafer theory for global optimization problems. J Glob Optim 51(1):79–104MathSciNetzbMATHCrossRefGoogle Scholar
  30. Müller J, Shoemaker CA (2014) Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems. J Glob Optim 60(2):123–144MathSciNetzbMATHCrossRefGoogle Scholar
  31. Nicosia G, Rinaudo S, Sciacca E (2008) An evolutionary algorithm-based approach to robust analog circuit design using constrained multi-objective optimization. Knowl-Based Syst 21(3):175–183CrossRefGoogle Scholar
  32. Ohno M, Yoshimatsu A, Kobayashi M, Watanabe S (2002) A framework for evolutionary optimization with approximate fitness functions. IEEE Trans Evol Comput 6(5):481–494CrossRefGoogle Scholar
  33. Ponweiser W, Wagner T and Vincze M (2008) Clustered multiple generalized expected improvement: a novel infill sampling criterion for surrogate models. Evolutionary Computation, pp 3515–3522Google Scholar
  34. Qin AK, Huang VL, Suganthan PN (2009) Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans Evol Comput 13(2):398–417CrossRefGoogle Scholar
  35. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Kevin Tucker P (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28zbMATHCrossRefGoogle Scholar
  36. Regis RG, Shoemaker CA (2007) Improved strategies for radial basis function methods for global optimization. J Glob Optim 37(1):113–135MathSciNetzbMATHCrossRefGoogle Scholar
  37. Sun C, Jin Y, Zeng J, Yu Y (2015) A two-layer surrogate-assisted particle swarm optimization algorithm. Soft Comput 19(6):1461–1475CrossRefGoogle Scholar
  38. Sun C, Jin Y, Cheng R, Ding J, Zeng J (2017) Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems. IEEE Trans Evol Comput 21(4):644–660CrossRefGoogle Scholar
  39. Tian J, Tan Y, Zeng JC, Sun CL, Jin YC (2019) Multiobjective infill criterion driven gaussian process-assisted particle swarm optimization of high-dimensional expensive problems. IEEE Trans Evol Comput 23(3):459–472CrossRefGoogle Scholar
  40. Varghese V, Ramu P, Krishnan V and Saravana Kumar G (2016) Pull out strength calculator for pedicle screws using a surrogate ensemble approach. Comput Methods Programs Biomed, PP(137):11–22CrossRefGoogle Scholar
  41. Viana FAC, Haftka RT, Steffen V (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscip Optim 39(4):439–457CrossRefGoogle Scholar
  42. Wang C, Duan Q, Gong W, Ye A, Di Z, Miao C (2014) An evaluation of adaptive surrogate modeling based optimization with two benchmark problems. Environ Model Softw 60(76):167–179CrossRefGoogle Scholar
  43. Wang H, Jin Y, Janson JO (2016) Data-driven surrogate-assisted multi-objective evolutionary optimization of a trauma system. IEEE Trans Evol Comput 20(6):939–952CrossRefGoogle Scholar
  44. Wang H, Jin Y and Doherty J (2017) Committee-based active learning for surrogate-assisted particle swarm optimization of expensive problems. IEEE Transection on Cybernetics, PP(99), pp 1–14Google Scholar
  45. Wang H, Jin Y, Sun C and Doherty J (2018) Offline data-driven evolutionary optimization using selective surrogate ensembles. IEEE Transactions on Evolutionary Computation, PP(99), pp 1–1Google Scholar
  46. Ye P, Pan G (2016) Global optimization method using adaptive and parallel ensemble of surrogates for engineering design optimization. Optimization 66(7):1135–1155MathSciNetzbMATHCrossRefGoogle Scholar
  47. Ye P, Pan G (2017) Global optimization method using ensemble of metamodels based on fuzzy clustering for design space reduction. Eng Comput 33(3):573–585CrossRefGoogle Scholar
  48. Yu H, Tan Y, Zeng J, Sun C and Jin Y (2018) Surrogate-assisted hierarchical particle swarm optimization. Information Sciences, PP(454), pp 59–72MathSciNetCrossRefGoogle Scholar
  49. Zerpa LE, Queipo NV, Pintos S, Salager JL (2005) An optimization methodology of alkaline–surfactant–polymer flooding processes using field scale numerical simulation and multiple surrogates. J Pet Sci Eng 47(3–4):197–208CrossRefGoogle Scholar
  50. Zhang Q, Liu W, Tsang E, Virginas B (2010) Expensive multiobjective optimization by MOEA/D with gaussian process model. IEEE Trans Evol Comput 14(3):456–474CrossRefGoogle Scholar
  51. Zhang J, Chowdhury S, Zhang J, Messac A, Castillo L (2013) Adaptive hybrid surrogate modeling for complex systems. AIAA J 51(3):643–656CrossRefGoogle Scholar
  52. Zhou XJ, Ma YZ, Li XF (2011) Ensemble of surrogates with recursive arithmetic average. Struct Multidiscip Optim 44(5):651–671CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The School of AutomationChongqing UniversityChongqingChina
  2. 2.The School of the Third Affiliated HospitalZhengzhou UniversityZhengzhouChina
  3. 3.The School of Electrical EngineeringZhengzhou UniversityZhengzhouChina

Personalised recommendations