Structural and Multidisciplinary Optimization

, Volume 58, Issue 4, pp 1505–1522 | Cite as

A parallel boundary search particle swarm optimization algorithm for constrained optimization problems

  • Zhao Liu
  • Zeyang Li
  • Ping Zhu
  • Wei Chen


During the past decade, considerable research has been conducted on constrained optimization problems (COPs) which are frequently encountered in practical engineering applications. By introducing resource limitations as constraints, the optimal solutions in COPs are generally located on boundaries of feasible design space, which leads to search difficulties when applying conventional optimization algorithms, especially for complex constraint problems. Even though penalty function method has been frequently used for handling the constraints, the adjustment of control parameters is often complicated and involves a trial-and-error approach. To overcome these difficulties, a modified particle swarm optimization (PSO) algorithm named parallel boundary search particle swarm optimization (PBSPSO) algorithm is proposed in this paper. Modified constrained PSO algorithm is adopted to conduct global search in one branch while Subset Constrained Boundary Narrower (SCBN) function and sequential quadratic programming (SQP) are applied to perform local boundary search in another branch. A cooperative mechanism of the two branches has been built in which locations of the particles near boundaries of constraints are selected as initial positions of local boundary search and the solutions of local boundary search will lead the global search direction to boundaries of active constraints. The cooperation behavior of the two branches effectively reinforces the optimization capability of the PSO algorithm. The optimization performance of PBSPSO algorithm is illustrated through 13 CEC06 test functions and 5 common engineering problems. The results are compared with other state-of-the-art algorithms and it is shown that the proposed algorithm possesses a competitive global search capability and is effective for constrained optimization problems in engineering applications.


Particle swarm optimization Constrained optimization problems Subset constraints boundary narrower function Parallel boundary search Diversity enhancement 



The research leading to the above results was supported by National Natural Science Foundation of China (Grant No. 11772191), National Science Foundation for Young Scientists of China (Grant No. 51705312) and National Postdoctoral Foundation of China (Grant No. 17Z102060055). The authors also acknowledge the support from the Adjunct Professor position provided by the Shanghai Jiao Tong University to Prof. Wei Chen.


  1. Bonyadi MR, Michalewicz Z (2014) On the edge of feasibility: a case study of the particle swarm optimizer. Ieee Congress on Evolutionary Computation. p. 3059–66Google Scholar
  2. Kennedy J, Eberhart R (1995) Particle swarm optimization, in: 1995 I.E. international conference on neural networks proceedings, pp. 1942–1948(vol 48)Google Scholar
  3. Parsopoulos KE, Vrahatis MN (2002) Particle Swarm Optimization Method for Constrained Optimization Problem: CiteSeer,Google Scholar
  4. Pulido GT, Coello CAC (2004) A constraint-handling mechanism for particle swarm optimization. Cec2004: Proceedings of the 2004 Congress on Evolutionary Computation. p. 1396–403 Vol.2Google Scholar
  5. Hu XH, Eberhart R (2002) Solving constrained nonlinear optimization problems with particle swarm optimization, in: 6th world multi-conference on Systemics, cybernetics and informatics (SCI 2002)/8th international conference on information systems analysis and synthesis (ISAS 2002)Google Scholar
  6. Guo CX, Hu JS, Ye B, Cao YJ (2004) Swarm intelligence for mixed-variable design optimization. J Zhejiang Univ (Sci) 5:851–860CrossRefzbMATHGoogle Scholar
  7. He Q, Wang L (2007a) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20:89–99CrossRefGoogle Scholar
  8. He Q, Wang L (2007b) A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Appl Math Comput 186:1407–1422MathSciNetzbMATHGoogle Scholar
  9. Paquet U, Engelbrecht AP (2007) Particle swarms for linearly constrained optimisation. Fundamenta Informaticae 76:147–170MathSciNetzbMATHGoogle Scholar
  10. Liang JJ, Shang Z, Li Z (2010) Coevolutionary comprehensive learning particle swarm optimizer. IEEE Congress on Evolutionary Computation:1–8Google Scholar
  11. C-l S, J-c Z, J-s P (2011) An improved vector particle swarm optimization for constrained optimization problems. Inf Sci 181:1153–1163CrossRefGoogle Scholar
  12. Bonyadi M, Li X, Michalewicz Z (2013) A hybrid particle swarm with velocity mutation for constraint optimization problems. Proceeding of the Fifteenth Conference on Genetic and Evolutionary Computation Conference. p. 1–8Google Scholar
  13. Mezura-Montes E, Coello Coello CA (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm and Evolutionary Computation 1:173–194CrossRefGoogle Scholar
  14. Shi Y, Eberhart R (1999) Modified particle swarm optimizer. IEEE international conference on evolutionary computation proceedings, 1998 I.E. world congress on Computational Intelligence 1999. p. 69–73Google Scholar
  15. Kennedy J (1997) The particle swarm: social adaptation of knowledge. IEEE International Conference on Evolutionary Computation 1997. p. 303–8Google Scholar
  16. Eberhart RC, Shi YH (2001) Particle swarm optimization: Developments, applications and resources,Google Scholar
  17. Yang JM, Chen YP, Horng JT, Kao CY (1997) Applying family competition to evolution strategies for constrained optimization. Evolutionary Programming Vi, International Conference, Ep97, Indianapolis, Indiana, Usa, April 13-16, , Proceedings1997. P. 201–11Google Scholar
  18. Liu Z, Lu J, Zhu P (2016) Lightweight design of automotive composite bumper system using modified particle swarm optimizer. Compos Struct 140:630–643CrossRefGoogle Scholar
  19. Clerc M (2006) Confinements and biases in particle swarm optimisationGoogle Scholar
  20. Liang JJ (2006) PNS. Problem Definitions and Evaluation criteria for the CEC 2006 special session on constrained real-parameter OptimizationGoogle Scholar
  21. Mezura-Montes E, Coello CACA (2005) Simple multimembered evolution strategy to solve constrained optimization problems. IEEE Trans Evol Comput 9:1–17CrossRefzbMATHGoogle Scholar
  22. Wang Y, Cai Z, Guo G, Zhou Y (2007) Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems. Ieee Transactions on Systems Man and Cybernetics Part B-Cybernetics 37:560–575CrossRefGoogle Scholar
  23. Wang Y, Cai Z, Zhou Y, Zeng W (2008) An adaptive tradeoff model for constrained evolutionary optimization. IEEE Trans Evol Comput 12:80–92CrossRefGoogle Scholar
  24. Mezura-Montes E, Cetina-Dominguez O (2012) Empirical analysis of a modified artificial bee Colony for constrained numerical optimization. Appl Math Comput 218:10943–10973MathSciNetzbMATHGoogle Scholar
  25. Dhadwal MK, Jung SN, Kim CJ (2014) Advanced particle swarm assisted genetic algorithm for constrained optimization problems. Comput Optim Appl 58:781–806MathSciNetCrossRefzbMATHGoogle Scholar
  26. Venter G, Haftka RT (2010) Constrained particle swarm optimization using a bi-objective formulation. Struct Multidiscip Optim 40:65–76MathSciNetCrossRefzbMATHGoogle Scholar
  27. Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic selection for constrained optimization. Inf Sci 178:3043–3074CrossRefGoogle Scholar
  28. Garcia S, Molina D, Lozano M, Herrera F (2009) A study on the use of non-parametric tests for analyzing the evolutionary algorithms' behaviour: a case study on the CEC'2005 special session on real parameter optimization. J Heuristics 15:617–644CrossRefzbMATHGoogle Scholar
  29. Wang Y, Cai Z, Zhou Y, Fan Z (2009) Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique. Struct Multidiscip Optim 37:395–413CrossRefGoogle Scholar
  30. Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012a) Water cycle algorithm - a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110:151–166CrossRefGoogle Scholar
  31. Wang L, Li LP (2010) An effective differential evolution with level comparison for constrained engineering design. Struct Multidiscip Optim 41:947–963CrossRefGoogle Scholar
  32. Coelho LDS (2010) Gaussian quantum-behaved particle swarm optimization approaches for constrained engineering design problems. Expert Syst Appl 37:1676–1683CrossRefGoogle Scholar
  33. Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl Soft Comput 13:2592–2612CrossRefGoogle Scholar
  34. Zahara E, Kao YT (2009) Hybrid Nelder–mead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 36:3880–3886CrossRefGoogle Scholar
  35. Guedria NB (2016) Improved accelerated PSO algorithm for mechanical engineering optimization problems. Appl Soft Comput 2016:455–467CrossRefGoogle Scholar
  36. Coello CAC, Montes EM (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16:193–203CrossRefGoogle Scholar
  37. Coello CAC, Becerra RL (2004) Efficient evolutionary optimization through the use of a cultural algorithm. Eng Optim 36:219–236CrossRefGoogle Scholar
  38. Lampinen J (2002) A constraint handling approach for the differential evolution algorithm. Cec'02: proceedings of the 2002 congress on evolutionary computation, Vols 1 and 2, . p. 1468–73Google Scholar
  39. Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl Soft Comput 10:629–640CrossRefGoogle Scholar
  40. Kashan AH (2011) An efficient algorithm for constrained global optimization and application to mechanical engineering design: league championship algorithm (LCA): Butterworth-HeinemannGoogle Scholar
  41. Yang XS (2010) Engineering optimization—an introduction with metaheuristic applications. New York. John Wiley & Sons, Wiley, pp 291–298CrossRefGoogle Scholar
  42. Cuevas E, Cienfuegos M (2014) A new algorithm inspired in the behavior of the social-spider for constrained optimization. Expert Syst Appl 41:412–425CrossRefGoogle Scholar
  43. Ray T, Liew KM (2003) Society and civilization: an optimization algorithm based on the simulation of social behavior. IEEE Trans Evol Comput 7:386–396CrossRefGoogle Scholar
  44. Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012b) Water cycle algorithm – a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct:151–166Google Scholar
  45. Gandomi AH, Yang X-S, Alavi AH (2011) Mixed variable structural optimization using firefly algorithm. Comput Struct 89:2325–2336CrossRefGoogle Scholar
  46. He Q, Wang L (2007c) A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Appl Math Comput 186:1407–1422MathSciNetzbMATHGoogle Scholar
  47. Fz H, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186:340–356MathSciNetzbMATHGoogle Scholar
  48. Akay B, Karaboga D (2012) Artificial bee colony algorithm for large-scale problems and engineering design optimization. J Intell Manuf 23:1001–1014CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The State key laboratory of Mechanical System and Vibration, School of Mechanical EngineeringShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

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