Skip to main content
SpringerLink
Log in

A modified augmented Lagrangian with improved grey wolf optimization to constrained optimization problems

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

This paper presents a novel constrained optimization algorithm named MAL-IGWO, which integrates the benefit of the improved grey wolf optimization (IGWO) capability for discovering the global optimum with the modified augmented Lagrangian (MAL) multiplier method to handle constraints. In the proposed MAL-IGWO algorithm, the MAL method effectively converts a constrained problem into an unconstrained problem and the IGWO algorithm is applied to deal with the unconstrained problem. This algorithm is tested on 24 well-known benchmark problems and 3 engineering applications, and compared with other state-of-the-art algorithms. Experimental results demonstrate that the proposed algorithm shows better performance in comparison to other approaches.

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
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Ali MM, Zhu WX (2013) A penalty function-based differential evolution algorithm for constrained global optimization. Comput Optim Appl 54(3):707–739

    Article  MathSciNet  MATH  Google Scholar 

  2. Rao RV, Savsani VJ, Vakharia DP (2011) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315

    Article  Google Scholar 

  3. Deb K, Srivastava S (2012) A genetic algorithm based augmented Lagrangian method for constrained optimization. Comput Optim Appl 51(3):869–902

    Article  MathSciNet  MATH  Google Scholar 

  4. Long W, Liang XM, Huang YF, Chen YX (2013) A hybrid differential evolution augmented Lagrangian method for constrained numerical and engineering optimization. Comput Aided Des 45(12):1562–1574

    Article  MathSciNet  Google Scholar 

  5. Wang Y, Cai Z, Zhou Y, Zeng W (2008) An adaptive tradeoff model for constrained evolutionary optimization. IEEE Trans Evol Comput 12(1):80–92

    Article  Google Scholar 

  6. Tuba M, Bacanin N (2014) Improved seeker optimization algorithm hybridized with firefly algorithm for constrained optimization problems. Neurocomputing 143:197–207

    Article  Google Scholar 

  7. Gandomi AH, Yang XS, Talatahari S, Deb S (2012) Couple eagle strategy and differential evolution for unconstrained and constrained global optimization. Comput Math Appl 63(1):191–200

    Article  MathSciNet  MATH  Google Scholar 

  8. Long W, Liang X, Huang Y, Chen Y (2014) An effective hybrid cuckoo search algorithm for constrained global optimi- zation. Neural Comput Appl 25(3–4):911–926

    Article  Google Scholar 

  9. Brajevic I (2015) Crossover-based artificial bee colony algorithm for constrained optimization problems. Neural Comput Appl 26(7):1587–1601

    Article  Google Scholar 

  10. Sadollah A, Eskandar H, Bahreininejad A, Kim JH (2015) Water cycle algorithm with evaporation rate for solving constrained and unconstrained optimization problems. Appl Soft Comput 30:58–71

    Article  Google Scholar 

  11. Wang Y, Cai Z (2012) Combining multiobjective optimization with differential evolution to solve constrained optimization problems. IEEE Trans Evol Comput 16(1):117–134

    Article  Google Scholar 

  12. Niu B, Wang JW, Wang H (2014) Bacterial-inspired algorithm for solving constrained optimization problems. Neurocomputing 148:54–62

    Article  Google Scholar 

  13. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69(3):46–61

    Article  Google Scholar 

  14. Zhu A, Xu C, Li Z, Wu J, Liu Z (2015) Hybridizing grey wolf optimization with differential evolution for global optimization and test scheduling for 3D stacked SoC. J Syst Eng Electron 26(2):317–328

    Article  Google Scholar 

  15. Saremi S, Mirjalili SZ, Mirjalili SM (2015) Evolutionary population dynamics and grey wolf optimizer. Neural Comput Appl 26(5):1257–1263

    Article  Google Scholar 

  16. Emary E, Zawbaa HM, Hassanien AE (2016) Binary grey wolf optimization approaches for feature selection. Neurocomputing 172:371–381

    Article  Google Scholar 

  17. Kamboj VK (2015) A novel hybrid PSO-GWO approach for unit commitment problem. Neural Comput Appl. doi:10.1007/s00521-015-1962-4

    Google Scholar 

  18. Jayakumar N, Subramanian S, Ganesan S, Elanchezhian EB (2016) Grey wolf optimization for combined heat and power dispatch with cogeneration systems. Electr Power Energy Syst 74:252–264

    Article  Google Scholar 

  19. El-Gaafary AAM, Mohamed YS, Hemeida AM, Mohamed AA (2015) Grey wolf optimization for multi input multi output system. Univers J Commun Netw 3(1):1–6

    Article  Google Scholar 

  20. Komaki GM, Kayvanfar V (2015) Grey wolf optimizer algorithm for the two-stage assembly flow shop scheduling problem with release time. J Comput Sci 8:109–120

    Article  Google Scholar 

  21. Song X, Tang L, Zhao S, Zhang X, Li L, Huang J, Cai W (2015) Grey wolf optimizer for parameter estimation in surface waves. Soil Dyn Earthq Eng 75:147–157

    Article  Google Scholar 

  22. Mirjalili S (2015) How effective is the grey wolf optimizer in training multi-layer perceptrons. Appl Intell 43(1):150–161

    Article  Google Scholar 

  23. Madadi A, Motlagh MM (2014) Optimal control of DC motor using grey wolf optimizer algorithm. Tech J Eng Appl Sci 4(4):373–379

    Google Scholar 

  24. El-Fergany AA, Hasanien HM (2015) Single and multi-objective optimal power flow using grey wolf optimizer and differential evolution algorithms. Electr Power Compon Syst 43(13):1548–1559

    Article  Google Scholar 

  25. Kamboj VK, Bath SK, Dhillon JS (2015) Solution of non-convex economic load dispatch problem using grey wolf optimizer. Neural Comput Appl. doi:10.1007/s00521-015-1934-8

    Google Scholar 

  26. Sulaiman MH, Mustaffa Z, Mohamed MR, Aliman O (2015) Using the gray wolf optimizer for solving optimal reactive power dispatch problem. Appl Soft Comput 32:286–292

    Article  Google Scholar 

  27. Metz MC, Vucetich JA, Smith DW, Stahler DR, Peterson RO (2011) Effect of sociality and season on gray wolf (Canis lupus) foraging behavior: implications for estimating summer kill rate. PLoS ONE 6(3):1–10

    Article  Google Scholar 

  28. Muro C, Escobedo R, Spector L, Coppinger RP (2011) Wolf-pack (Canis lupus) hunting strategies emerge from simple rules in computational simulations. Behav Process 88(3):192–197

    Article  Google Scholar 

  29. Michalewicz Z, Schoenauer M (1996) Evolutionary algorithm for constrained parameter optimization problems. Evol Comput 4(1):1–32

    Article  Google Scholar 

  30. Mezura-Montes E, Coello CAC (2005) Self-adaptive fitness formulation for constrained optimization. IEEE Trans Evol Comput 9(1):1–17

    Article  MATH  Google Scholar 

  31. Costa L, Santo IACPE, Fernandes EMGP (2012) A hybrid genetic pattern search augmented Lagrangian method for constrained global optimization. Appl Math Comput 218(18):9415–9426

    MathSciNet  MATH  Google Scholar 

  32. Liang XM, Hu JB, Zhong WT, Qian JX (2001) A modified augmented Lagrange multiplier methods for large-scale optimization. Chin J Chem Eng 9(2):167–172

    Google Scholar 

  33. 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 41:1–8

    Google Scholar 

  34. Rocha AMA, Martins TF, Fernandes EM (2011) An augmented Lagrangian fish swarm based method for global optimization. J Comput Appl Math 235(16):4611–4620

    Article  MathSciNet  MATH  Google Scholar 

  35. Mahdavi A, Shiri ME (2015) An augmented Lagrangian ant colony based method for constrained optimization. Comput Optim Appl 60(1):263–276

    Article  MathSciNet  MATH  Google Scholar 

  36. Mezura-Montes E, Cetina-Dominguez O (2012) Empirical analysis of a modified artificial bee colony for constrained numerical optimization. Appl Math Comput 218(22):10943–10973

    MathSciNet  MATH  Google Scholar 

  37. Wang Y, Cai ZX, Guo GQ, Zhou YR (2007) Multiobjective optimization and hybrid evolutionary algorithm to solve constrained optimization problems. IEEE Trans Syst Man Cybern 37(3):560–575

    Article  Google Scholar 

  38. Lin CH (2013) A rough penalty genetic algorithm for constrained optimization. Inf Sci 241:119–137

    Article  Google Scholar 

  39. Deb K (2000) A efficient constraint handling method for genetic algorithms. Comput Meth Appl Mech Eng 186(2–4):311–338

    Article  MATH  Google Scholar 

  40. Belegundu AD (1982) A study of mathematical programming methods for structural optimization. Ph.D. Thesis, Deparment of Civil and Environmental Engineering, University of Iowa, Iowa

  41. Mezura-Montes E, Coello CAC (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16(3):193–203

    Article  Google Scholar 

  42. He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problem. Eng Appl Artif Intell 20(1):89–99

    Article  MathSciNet  Google Scholar 

  43. Shen H, Zhu Y, Niu B, Wu QH (2009) An improved group search optimizer for mechanical design optimization problems. Progress Nat Sci 19(1):91–97

    Article  Google Scholar 

  44. Wang Y, Cai ZX, Zhou YR (2009) Accelerating adaptive trade-off model using shrinking space technique for constrained evolutionary optimization. Int J Numer Meth Eng 77(11):1501–1534

    Article  MathSciNet  MATH  Google Scholar 

  45. Huang FZ, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356

    MathSciNet  MATH  Google Scholar 

  46. Sandgren E (1990) Nonlinear integer and discrete programming in mechanical design optimization. ASME J Mech Des 112(2):223–229

    Article  Google Scholar 

  47. Mezura-Montes E, Coello CAC (2005) Useful infeasible solutions in engineering optimization with evolutionary algorithm. MICAI’2005 Lect Notes Artif Int 3789:652–662

    Google Scholar 

  48. Akhtar S, Tai K, Ray T (2002) A socio-behavioral simulation model for engineering design optimization. Eng Optim 34(4):341–354

    Article  Google Scholar 

  49. Gandomi AH, Yang XS, Alavi AH (2013) Cuckoo search algorithm: a meta-heuristic approach to solve structural optimization problem. Eng Comput 29(1):17–35

    Article  Google Scholar 

  50. Mezura-Montes E, Coello CAC, Ricardo L (2003) Engineering optimization using a simple evolutionary algorithm. In: Proceedings of International Conference on Tools Artificial Intelligence, pp 149–156

  51. Ray T, Saini P (2001) Engineering design optimization using a swarm with an intelligent information sharing among individuals. Eng Optim 33(6):735–748

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant No. 61463009, the Humanities and Social Sciences Planning Foundation of Ministry of Education of China under Grant No. 12XJA910001, the Beijing Natural Science Foundation under Grant No. 4122022, the 125 Special Major Science and Technology of Department of Education of Guizhou Province under Grant No. [2012]011, and the Science and Technology Foundation of Guizhou Province under Grant No. [2016]2082.

Author information

Authors and Affiliations

  1. Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guizhou, China

    Wen Long, Shaohong Cai, Jianjun Jiao & Wenzhuan Zhang

  2. School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China

    Ximing Liang

Authors
  1. Wen Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ximing Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shaohong Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianjun Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wenzhuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ximing Liang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Long, W., Liang, X., Cai, S. et al. A modified augmented Lagrangian with improved grey wolf optimization to constrained optimization problems. Neural Comput & Applic 28 (Suppl 1), 421–438 (2017). https://doi.org/10.1007/s00521-016-2357-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-016-2357-x

Keywords

Search

Navigation