Skip to main content
SpringerLink
Log in

Adaptive boundary constraint-handling scheme for constrained optimization

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Nature-inspired optimization algorithms are meta-heuristics that mimic nature for solving optimization problems. Many optimization problems are constrained and have a bounded search space from which some solution vectors leave when the variation operators are applied. Therefore, the use of boundary constraint-handling methods (BCHM) is necessary in order to repair the invalid vectors. This paper presents an adaptive scheme to handling boundary constraints in constrained numerical optimization problems. The proposed adaptive scheme operates in two stages: At the first one, when there are still no feasible solutions, a BCHM that benefits the exploration of the search space is employed, and in the second stage, one of several BCHMs, according to their associated probabilities, is selected. The methods’ probabilities are updated every learning period so that the methods that generate the best repaired solutions will have a greater chance of being selected. The proposed scheme has been tested within two nature-inspired optimization algorithms: Particle Swarm Optimization and Differential Evolution employing their canonical version as well as one state-of-the-art version specialized in constrained optimization. A set of sixty single-objective constrained real-parameter optimization problems are solved. The results show that this adaptive scheme has a major impact on the algorithm’s performance, and it is able to promote better final results mainly within high-dimensional problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Alvarez-Benitez JE, Everson RM, Fieldsend JE (2005) A MOPSO algorithm based exclusively on pareto dominance concepts. In: International conference on evolutionary multi-criterion optimization. Springer, Berlin, pp 459–473

  • Arabas J, Szczepankiewicz A, Wroniak T (2010) Experimental comparison of methods to handle boundary constraints in differential evolution. In: International conference on parallel problem solving from nature. Springer, Berlin, pp 411–420

  • Bäck T, Fogel DB, Michalewicz Z (1997) Handbook of evolutionary computation. IOP Publishing Ltd., Bristol

    Book  MATH  Google Scholar 

  • Brest J, Greiner S, Boskovic B, Mernik M, Zumer V (2006) Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 10(6):646–657

    Article  Google Scholar 

  • Chu W, Gao X, Sorooshian S (2011) Handling boundary constraints for particle swarm optimization in high-dimensional search space. Inf Sci 181(20):4569–4581

    Article  Google Scholar 

  • Clerc M (2006) Confinements and biases in particle swarm optimisation. Technical Report hal-00122799. https://hal.archives-ouvertes.fr/hal-00122799

  • Clerc M, Kennedy J (2002) The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 6(1):58–73

    Article  Google Scholar 

  • Deb K (2000) An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng 186(2):311–338

    Article  MATH  Google Scholar 

  • Eberhart RC, Kennedy J, et al (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, vol 1. New York, NY, pp 39–43

  • Elsayed SM, Sarker RA, Mezura-Montes E (2014) Self-adaptive mix of particle swarm methodologies for constrained optimization. Inf Sci 277:216–233

    Article  MathSciNet  MATH  Google Scholar 

  • Gandomi AH, Yang XS (2012) Evolutionary boundary constraint handling scheme. Neural Comput Appl 21(6):1449–1462

    Article  Google Scholar 

  • Helwig S, Wanka R (2007) Particle swarm optimization in high-dimensional bounded search spaces. In: 2007 IEEE swarm intelligence symposium. IEEE, pp 198–205

  • Helwig S, Branke J, Mostaghim S (2013) Experimental analysis of bound handling techniques in particle swarm optimization. IEEE Trans Evol Comput 17(2):259–271

    Article  Google Scholar 

  • Huang T, Mohan A (2005) A hybrid boundary condition for robust particle swarm optimization. IEEE Antennas Wirel Propag Lett 4:112–117. https://doi.org/10.1109/LAWP.2005.846166

    Article  Google Scholar 

  • Jia G, Wang Y, Cai Z, Jin Y (2013) An improved (\(\mu + \lambda \))-constrained differential evolution for constrained optimization. Inf Sci 222:302–322

    Article  MathSciNet  MATH  Google Scholar 

  • Jordehi AR (2015) A review on constraint handling strategies in particle swarm optimisation. Neural Comput Appl 26(6):1265–1275

    Article  Google Scholar 

  • Juárez-Castillo E, Pérez-Castro N, Mezura-Montes E (2015) A novel boundary constraint-handling technique for constrained numerical optimization problems. In: 2015 IEEE congress on evolutionary computation (CEC). IEEE, pp 2034–2041

  • Juárez-Castillo E, Acosta-Mesa HG, Mezura-Montes E (2017a) Empirical study of bound constraint-handling methods in particle swarm optimization for constrained search spaces. In: 2017 IEEE congress on evolutionary computation (CEC). IEEE, pp 604–611

  • Juárez-Castillo E, Pérez-Castro N, Mezura-Montes E (2017b) An improved centroid-based boundary constraint-handling method in differential evolution for constrained optimization. Int J Pattern Recogn Artif Intell 31:1759023

    Article  MathSciNet  Google Scholar 

  • Kennedy J (2006) Swarm intelligence. In: Zomaya AY (ed) Handbook of nature-inspired and innovative computing. Springer, Berlin, pp 187–219

    Chapter  Google Scholar 

  • Kukkonen S, Lampinen J (2005) Gde3: the third evolution step of generalized differential evolution. In: 2005 IEEE congress on evolutionary computation, vol 1. IEEE, pp 443–450

  • Lampinen J (2002) A constraint handling approach for the differential evolution algorithm. In: Proceedings of the congress on evolutionary computation, vol 2. IEEE Computer Society Washington, DC, pp 1468–1473

  • Liang J, Runarsson TP, Mezura-Montes E, Clerc M, Suganthan P, Coello CC, Deb K (2006) Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization. J Appl Mech 41(8):8–31

    Google Scholar 

  • Mallipeddi R, Suganthan PN (2010) Problem definitions and evaluation criteria for the CEC 2010 competition on constrained real-parameter optimization. Nanyang Technological University, Singapore

    Google Scholar 

  • Mezura-Montes E, Coello CAC (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol Comput 1(4):173–194

    Article  Google Scholar 

  • Mezura-Montes E, Miranda-Varela ME, del Carmen Gómez-Ramón R (2010) Differential evolution in constrained numerical optimization: an empirical study. Inf Sci 180(22):4223–4262

    Article  MathSciNet  MATH  Google Scholar 

  • Padhye N, Deb K, Mittal P (2013) Boundary handling approaches in particle swarm optimization. In: Proceedings of seventh international conference on bio-inspired computing: theories and applications (BIC-TA 2012). Springer, Berlin, pp 287–298

  • Padhye N, Mittal P, Deb K (2015) Feasibility preserving constraint-handling strategies for real parameter evolutionary optimization. Comput Optim Appl 62(3):851–890

    Article  MathSciNet  MATH  Google Scholar 

  • Price K, Storn RM, Lampinen JA (2005) Differential evolution: a practical approach to global optimization (natural computing series). Springer, Berlin

    MATH  Google Scholar 

  • Purchla M, Malanowski M, Terlecki P, Arabas J (2004) Experimental comparison of repair methods for box constraints. In: Proceedings of 7th national conference on evolutionary computation and global optimisation, pp 135–142

  • Robinson J, Rahmat-Samii Y (2004) Particle swarm optimization in electromagnetics. IEEE Trans Antennas Propag 52(2):397–407

    Article  MathSciNet  MATH  Google Scholar 

  • Ronkkonen J, Kukkonen S, Price KV (2005) Real-parameter optimization with differential evolution. Proc IEEE CEC 1:506–513

    Google Scholar 

  • Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4(3):284–294

    Article  Google Scholar 

  • Shi Y, Cheng S, Qin Q (2011) Experimental study on boundary constraints handling in particle swarm optimization: from population diversity perspective. Int J Swarm Intell Res 2(3):43–69

    Article  Google Scholar 

  • Simon D (2013) Evolutionary optimization algorithms. Wiley, New York

    Google Scholar 

  • Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Wang BC, Li HX, Yen GG (2016) Incorporating objective function information into the feasibility rule for constrained evolutionary optimization. IEEE Trans Cybern 46(12):2938–2952

    Article  Google Scholar 

  • Wang BC, Li HX, Li JP, Wang Y (2018) Composite differential evolution for constrained evolutionary optimization. IEEE Trans Syst Man Cybern Syst PP:1–14

    Google Scholar 

  • Wei J, Jia L (2013) A novel particle swarm optimization algorithm with local search for dynamic constrained multi-objective optimization problems. In: 2013 IEEE congress on evolutionary computation (CEC). IEEE, pp 2436–2443

  • Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82

    Article  Google Scholar 

  • Wu G, Mallipeddi R, Suganthan P (2016) Problem definitions and evaluation criteria for the CEC 2017 competition on constrained real-parameter optimization. National University of Defense Technology, Changsha, Hunan, PR China and Kyungpook National University, Daegu, South Korea and Nanyang Technological University, Singapore, Technical Report

  • Xu S, Rahmat-Samii Y (2007) Boundary conditions in particle swarm optimization revisited. IEEE Trans Antennas Propag 55(3):760–765

    Article  Google Scholar 

  • Yang XS (2014) Nature-inspired optimization algorithms. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Zhang WJ, Xie XF, Bi DC (2004) Handling boundary constraints for numerical optimization by particle swarm flying in periodic search space. In: CEC2004. Congress on evolutionary computation, 2004, vol 2. IEEE, pp 2307–2311

  • Zhong J, Hu X, Zhang J, Gu M (2005) Comparison of performance between different selection strategies on simple genetic algorithms. In: Computational intelligence for modelling, control and automation, 2005 and international conference on international conference on intelligent agents, web technologies and internet commerce, vol 2. IEEE, pp 1115–1121

Download references

Acknowledgements

The first author acknowledges support from the Mexican National Council for Science and Technology (CONACyT) through a scholarship to pursue graduate studies at the University of Veracruz. Special thanks are due to Dra. Cora Beatriz Excelente Toledo for her support in reviewing this work. This study was funded by the Mexican Council for Science and Technology (CONACyT) (Grant Number 220522).

Author information

Authors and Affiliations

  1. Artificial Intelligence Research Center, University of Veracruz, Sebastián Camacho #5 Centro, 91000, Xalapa, Veracruz, Mexico

    Efrén Juárez-Castillo, Héctor-Gabriel Acosta-Mesa & Efrén Mezura-Montes

  2. Technological University of Huasteca Hidalguense, Huejutla-Chalahuiyapa S/N, Tepoxteco, 43000, Huejutla, Hidalgo, Mexico

    Efrén Juárez-Castillo

Authors
  1. Efrén Juárez-Castillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Héctor-Gabriel Acosta-Mesa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Efrén Mezura-Montes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Efrén Juárez-Castillo.

Ethics declarations

Conflicts of interest

The authors declare that they have no competing interests.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by V. Loia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Juárez-Castillo, E., Acosta-Mesa, HG. & Mezura-Montes, E. Adaptive boundary constraint-handling scheme for constrained optimization. Soft Comput 23, 8247–8280 (2019). https://doi.org/10.1007/s00500-018-3459-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-3459-4

Keywords

Search

Navigation