Soft Computing

, Volume 19, Issue 6, pp 1461–1475 | Cite as

A two-layer surrogate-assisted particle swarm optimization algorithm

  • Chaoli Sun
  • Yaochu Jin
  • Jianchao Zeng
  • Yang Yu


Like most evolutionary algorithms, particle swarm optimization (PSO) usually requires a large number of fitness evaluations to obtain a sufficiently good solution. This poses an obstacle for applying PSO to computationally expensive problems. This paper proposes a two-layer surrogate-assisted PSO (TLSAPSO) algorithm, in which a global and a number of local surrogate models are employed for fitness approximation. The global surrogate model aims to smooth out the local optima of the original multimodal fitness function and guide the swarm to fly quickly to an optimum or the global optimum. In the meantime, a local surrogate model constructed using the data samples near each particle is built to achieve a fitness estimation as accurate as possible. The contribution of each surrogate in the search is empirically verified by experiments on uni- and multi-modal problems. The performance of the proposed TLSAPSO algorithm is examined on ten widely used benchmark problems, and the experimental results show that the proposed algorithm is effective and highly competitive with the state-of-the-art, especially for multimodal optimization problems.


Particle swarm optimization Surrogate-assisted optimization Computationally expensive optimization problems 



This work was supported in part by Youth Foundation of Shanxi Province of China under Grant No. 2011021019-3, the Doctoral Foundation of Taiyuan University of Science and Technology under Grant No. 20122010, and the State Key Laboratory of Software Engineering, Nanjing University, China, Project No. KFKT2013A05.


  1. Abou El-Ela A, Fetouh T, Bishr M, Saleh R (2008) Power systems operation using particle swarm optimization technique. Electr Power Syst Res 78(11):1906–1913Google Scholar
  2. Acar E, Rais-Rohani M (2009) Ensemble of metamodels with optimized weight factors. Struct Multidiscip Optim 37(3):279–294CrossRefGoogle Scholar
  3. Bird S, Li X (2010) Improving local convergence in particle swarms by fitness approximation using regression. In: Computational intelligence in expensive optimization problems. Adaptation learning and optimization, vol 2. Springer, Berlin, Heidelberg, New York, pp 265–293Google Scholar
  4. Buche D, Schraudolph NN, Koumoutsakos P (2005) Accelerating evolutionary algorithms with gaussian process fitness function models. IEEE Trans Syst Man Cybern Part C Appl Rev 35(2):183–194CrossRefGoogle 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. Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, pp 39–43Google Scholar
  7. Eberhart RC, Shi Y (2000) Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the 2000 congress on evolutionary computation, vol 1, pp 84–88Google Scholar
  8. Farina M (2002) A neural network based generalized response surface multiobjective. In: Proceedings of the 2002 congress on evolutionary computation, vol 1, pp 956–961Google Scholar
  9. Fonseca LG, Lemonge AC, Barbosa HJ (2012) A study on fitness inheritance for enhanced efficiency in real-coded genetic algorithms. In: Proceedings of the 2012 IEEE congress on evolutionary computation (CEC), pp 1–8Google Scholar
  10. Goel T, Haftka RT, Shyy W, Queipo NV (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216Google Scholar
  11. He S, Prempain E, Wu Q (2004) An improved particle swarm optimizer for mechanical design optimization problems. Eng Optim 36(5):585–605CrossRefMathSciNetGoogle Scholar
  12. Hendtlass T (2007) Fitness estimation and the particle swarm optimisation algorithm. In: Proceedings of the IEEE congress on evolutionary computation, pp 4266–4272Google Scholar
  13. Jin Y, Sendhoff B (2004) Reducing fitness evaluations using clustering techniques and neural network ensembles. In: Proceedings of the genetic and evolutionary computation (GECCO 2004). Lecture notes in computer science, vol 3102. Springer, New York, pp 688– 699Google Scholar
  14. Jin Y, Olhofer M, Sendhoff B (2002) A framework for evolutionary optimization with approximate fitness. IEEE Trans Evol Comput 6(5):481–494CrossRefGoogle Scholar
  15. Jin Y (2005) A comprehensive survey of fitness approximation in evolutionary computation. Soft Comput 9(1):3–12CrossRefGoogle Scholar
  16. Joseph VR, Hung Y, Sudjianto A (2008) Blind kriging: a new method for developing metamodels. J Mech Des 130(3):031102.1– 031102.8Google Scholar
  17. Kattan A, Galvan E (2012) Evolving radial basis function networks via gp for estimating fitness values using surrogate models. In: Proceedings of the 2012 IEEE congress on evolutionary computation (CEC), pp 1–7Google Scholar
  18. Lian Y, Liou M-S (2005) Multiobjective optimization using coupled response surface model. AIAA J 43(6):1316–1325CrossRefGoogle Scholar
  19. Lim D, Jin Y, Ong Y-S, Sendhoff B (2010) Generalizing surrogate-assisted evolutionary computation. IEEE Trans Evol Comput 14(3):329–355CrossRefGoogle Scholar
  20. Liu B, Zhang Q, Gielen G (2014) A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems. IEEE Trans Evol Comput 18(2):180–192Google Scholar
  21. Lu J, Li B, Jin Y (2013) An evolution strategy assisted by an ensemble of local gaussian process models. In: Proceedings of the fifteenth annual conference on genetic and evolutionary computation conference, ACM, pp 447–454Google Scholar
  22. Lu X, Tang K, Yao X (2011) Classification-assisted differential evolution for computationally expensive problems. In: Proceedings of the 2011 IEEE congress on evolutionary computation (CEC), pp 1986–1993Google Scholar
  23. Ong YS, Nair PB, Keane AJ, Wong KW (2004) Surrogate-assisted evolutionary optimization frameworks for high-fidelity engineering design problems. In: Jin Y (ed) Knowledge incorporation in evolutionary computation. Studies in fuzziness and soft computing series. Springer, pp 307–331Google Scholar
  24. Ong YS, Nair PB, Keane AJ (2003) Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J 41(4):687–696CrossRefGoogle Scholar
  25. Ong Y-S, Nair PB, Lum KY (2006) Max-min surrogate-assisted evolutionary algorithm for robust design. IEEE Trans Evol Comput 10(4):392–404CrossRefGoogle Scholar
  26. Parno M, Hemker T, Fowler K (2012) Applicability of surrogates to improve efficiency of particle swarm optimization for simulation-based problems. Eng Optim 44(5):521–535CrossRefGoogle Scholar
  27. Praveen C, Duvigneau R (2009) Low cost pso using metamodels and inexact preevaluation: application to aerodynamic shape design. Comput Methods Appl Mech Eng 198(9):1087–1096CrossRefzbMATHGoogle Scholar
  28. Ratle A (2001) Kriging as a surrogate fitness landscape in evolutionary optimization. AI EDAM 15(01):37–49Google Scholar
  29. Regis RG (2014) Particle swarm with radial basis function surrogates for expensive blackbox optimization. J Comput Sci 5(1):12–23CrossRefMathSciNetGoogle Scholar
  30. Reyes-Sierra M, Coello CAC (2005) A study of fitness inheritance and approximation techniques for multi-objective particle swarm optimization. In: Proceedings of the 2005 IEEE congress on evolutionary computation, vol 1, pp 65–72Google Scholar
  31. Sha D, Hsu C-Y (2008) A new particle swarm optimization for the open shop scheduling problem. Comput Oper Res 35(10):3243–3261CrossRefzbMATHGoogle Scholar
  32. Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: Proceedings of the 1998 IEEE international conference on evolutionary computation, IEEE world congress on computational intelligence, pp 69–73Google Scholar
  33. Smith RE, Dike BA, Stegmann SA (1995) Fitness inheritance in genetic algorithms. In: Proceedings of the 1995 ACM symposium on applied computing, pp 345–350Google Scholar
  34. Storn R (1996) On the usage of differential evolution for function optimization. In: Proceedings of the 1996 biennial conference of the North American Fuzzy Information Processing Society, NAFIPS, pp 519–523Google Scholar
  35. Suganthan PN, Hansen N, Liang JJ, Deb K, Chen Y, Auger A, Tiwari S (2005) Problem definitions and evaluation criteria for the cec 2005 special session on realparameter optimization. Technical Report, Nanyang Technological University, Singapore and KanGAL Report #2005005, IIT Kanpur, India Google Scholar
  36. Sun C, Zeng J, Pan J, Jin Y (2013) Similarity-based evolution control for fitness estimation in particle swarm optimization. In: Proceedings of the 2013 IEEE symposium on computational intelligence in dynamic and uncertain environments (CIDUE), pp 1–8Google Scholar
  37. Sun C, Zeng J, Pan J, Xue S, Jin Y (2012) A new fitness estimation strategy for particle swarm optimization. Inf Sci 221:355–370Google Scholar
  38. Sun X, Gong D, Jin Y, Chen S (2013) A new surrogate-assisted interactive genetic algorithm with weighted semisupervised learning. IEEE Trans Cybern 43(2):685–698CrossRefGoogle Scholar
  39. Tang Y, Chen J, Wei J (2013) A surrogate-based particle swarm optimization algorithm for solving optimization problems with expensive black box functions. Eng Optim 45(5):557–576CrossRefMathSciNetGoogle Scholar
  40. Tenne Y, Armfield SW (2009) A framework for memetic optimization using variable global and local surrogate models. Soft Comput 13(8–9):781–793CrossRefGoogle Scholar
  41. Ulmer H, Streichert F, Zell A (2003) Evolution strategies assisted by Gaussian processes with improved preselection criterion. In: Proceedings of the 2003 congress on evolutionary computation (CEC’03), vol 1, pp 692–699Google Scholar
  42. Zhou Z, Ong YS, Nguyen MH, Lim D (2005) A study on polynomial regression and gaussian process global surrogate model in hierarchical surrogate-assisted evolutionary algorithm. In: Proceedings of the 2005 IEEE congress on evolutionary computation, vol 3, pp 2832–2839Google Scholar
  43. Zhou Z, Ong YS, Nair PB, Keane AJ, Lum KY (2007) Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans Syst Man Cybern Part C Appl Rev 37(1):66–76CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Chaoli Sun
    • 1
  • Yaochu Jin
    • 2
  • Jianchao Zeng
    • 1
  • Yang Yu
    • 3
  1. 1.Complex System and Computational Intelligence LaboratoryTaiyuan University of Science and TechnologyTaiyuanChina
  2. 2.Department of ComputingUniversity of SurreyGuildfordUK
  3. 3.National Key Laboratory for Novel Software TechnologyNanjing UniversityNanjingChina

Personalised recommendations