Skip to main content
Springer Nature Link
Log in

New Hybrid Perturbed Projected Gradient and Simulated Annealing Algorithms for Global Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The main objective of this works is to present an efficient hybrid optimization approach using a new coupling technique for solving constrained engineering design problems. This hybrid is based on the simulated annealing algorithm with the projected gradient and its stochastic perturbation. The proposed hybrid is combined with corrected techniques in order to correct the solutions out of domain and send them to the domain’s border. The proposed algorithm is tested and evaluated on several benchmark functions, as well as on the basis of some engineering design problems. The obtained results are well compared with typical approaches existing in the literature. The solutions obtained by the proposed hybrid are more accurate than those given by other known methods and the performance and efficiency of the proposed algorithm are demonstrated.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data Availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Code Availability

The code source data used to support the findings of this study are available from the corresponding author upon request.

References

  1. Belkourchia, Y., Azrar, L., Zeriab, E.S.M.: A hybrid optimization algorithm for solving constrained engineering design problems. In: 2019 5th International Conference on Optimization and Applications (ICOA), pp. 1–7. IEEE (2019)

  2. Birattari, M., Pellegrini, P., Dorigo, M.: On the invariance of ant colony optimization. IEEE Trans. Evol. Comput. 11(6), 732–742 (2007)

    Google Scholar 

  3. Bouhadi, M.: Contribution à l’optimisation globale avec contraintes. Approche stochastique. Ph.D. thesis, Thesis, Université Mohammed V, Faculté des Sciences, Rabat, Maroc, 1997. 106 (1997)

  4. Bouleau, N., Bensoussan, A.: Probabilités de l’Ingénieur: Variables Aléatoires et Simulation. Hermann Paris (1986)

  5. Boussaïd, I., Lepagnot, J., Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Cagnina, L.C., Esquivel, S.C., Coello, C.A.C.: Solving engineering optimization problems with the simple constrained particle swarm optimizer. Informatica 32(3), 319–326 (2008)

    MATH  Google Scholar 

  7. Coello, C.A.C.: Use of a self-adaptive penalty approach for engineering optimization problems. Comput. Ind. 41(2), 113–127 (2000)

    Google Scholar 

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

    Google Scholar 

  9. De Cursi, J.S., Ellaia, R., Bouhadi, M.: Global optimization under nonlinear restrictions by using stochastic perturbations of the projected gradient. In: Frontiers in Global Optimization, pp. 541–561. Springer (2004)

  10. Deb, K.: Geneas: a robust optimal design technique for mechanical component design. In: Evolutionary Algorithms in Engineering Applications, pp. 497–514 (1997)

  11. Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2–4), 311–338 (2000)

    MATH  Google Scholar 

  12. Dimopoulos, G.G.: Mixed-variable engineering optimization based on evolutionary and social metaphors. Comput. Methods Appl. Mech. Eng. 196(4–6), 803–817 (2007)

    MATH  Google Scholar 

  13. Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp. 39–43. IEEE (1995)

  14. Eskandar, H., Sadollah, A., Bahreininejad, A., Hamdi, M.: Water cycle algorithm—a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput. Struct. 110, 151–166 (2012)

    Google Scholar 

  15. Fesanghary, M., Mahdavi, M., Minary-Jolandan, M., Alizadeh, Y.: Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems. Comput. Methods Appl. Mech. Eng. 197(33–40), 3080–3091 (2008)

    MATH  Google Scholar 

  16. Gandomi, A.H., Alavi, A.H.: Krill herd: a new bio-inspired optimization algorithm. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4831–4845 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Gandomi, A.H., Yang, X.S., Alavi, A.H.: Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng. Comput. 29(1), 17–35 (2013)

    Google Scholar 

  18. Garg, H.: Solving structural engineering design optimization problems using an artificial bee colony algorithm. J. Ind. Manag. Optim. 10(3), 777–794 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Garg, H.: A hybrid PSO-GA algorithm for constrained optimization problems. Appl. Math. Comput. 274, 292–305 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Goldberg, D.E., Richardson, J.: Genetic algorithms with sharing for multimodal function optimization. In: Genetic Algorithms and Their Applications: Proceedings of the Second International Conference on Genetic Algorithms, vol. 4149. Lawrence Erlbaum, Hillsdale (1987)

  21. Golinski, J.: An adaptive optimization system applied to machine synthesis. Mech. Mach. Theory 8(4), 419–436 (1973)

    Google Scholar 

  22. Gu, L., Yang, R., Tho, C.H., Makowskit, M., Faruquet, O., Li, Y.L.: Optimisation and robustness for crashworthiness of side impact. Int. J. Veh. Des. 26(4), 348–360 (2001)

    Google Scholar 

  23. Guedria, N.B.: Improved accelerated PSO algorithm for mechanical engineering optimization problems. Appl. Soft Comput. 40, 455–467 (2016)

    Google Scholar 

  24. Han, Y.Y., Gong, D., Sun, X.: A discrete artificial bee colony algorithm incorporating differential evolution for the flow-shop scheduling problem with blocking. Eng. Optim. 47(7), 927–946 (2015)

    MathSciNet  Google Scholar 

  25. Hashim, F.A., Houssein, E.H., Mabrouk, M.S., Al-Atabany, W., Mirjalili, S.: Henry gas solubility optimization: a novel physics-based algorithm. Future Gener. Comput. Syst. 101, 646–667 (2019)

    Google Scholar 

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

    Google Scholar 

  27. He, S., Prempain, E., Wu, Q.: An improved particle swarm optimizer for mechanical design optimization problems. Eng. Optim. 36(5), 585–605 (2004)

    MathSciNet  Google Scholar 

  28. Hedar, A.R., Fukushima, M.: Derivative-free filter simulated annealing method for constrained continuous global optimization. J. Glob. Optim. 35(4), 521–549 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Heidari, A.A., Mirjalili, S., Faris, H., Aljarah, I., Mafarja, M., Chen, H.: Harris hawks optimization: algorithm and applications. Future Gener. Comput. Syst. 97, 849–872 (2019)

    Google Scholar 

  30. Himmelblau, D.M.: Applied Nonlinear Programming. McGraw-Hill Companies (1972)

    MATH  Google Scholar 

  31. Homaifar, A., Qi, C.X., Lai, S.H.: Constrained optimization via genetic algorithms. Simulation 62(4), 242–253 (1994)

    Google Scholar 

  32. Hu, X., Eberhart, R.C., Shi, Y.: Engineering optimization with particle swarm. In: Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS’03 (Cat. No. 03EX706), pp. 53–57. IEEE (2003)

  33. Karaboga, D., Basturk, B.: A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J. Glob. Optim. 39(3), 459–471 (2007)

    MathSciNet  MATH  Google Scholar 

  34. Kashan, A.H.: An efficient algorithm for constrained global optimization and application to mechanical engineering design: league championship algorithm (LCA). Comput. Aided Des. 43(12), 1769–1792 (2011)

    Google Scholar 

  35. Kaveh, A., Talatahari, S.: Engineering optimization with hybrid particle swarm and ant colony optimization (2009)

  36. Kaveh, A., Talatahari, S.: An improved ant colony optimization for constrained engineering design problems. Eng. Comput. (2010)

  37. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN’95-International Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE (1995)

  38. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)

    MathSciNet  MATH  Google Scholar 

  39. Lee, K.S., Geem, Z.W.: A new structural optimization method based on the harmony search algorithm. Comput. Struct. 82(9–10), 781–798 (2004)

    Google Scholar 

  40. Lee, K.S., Geem, Z.W.: A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput. Methods Appl. Mech. Eng. 194(36–38), 3902–3933 (2005)

    MATH  Google Scholar 

  41. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Trans. Evol. Comput. 10(3), 281–295 (2006)

    Google Scholar 

  42. Liang, J.J., Qu, B., Suganthan, P.N., Hernández-Díaz, A.G.: Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China and Nanyang Technological University, Singapore, Technical Report, vol. 201212, no. 34, pp. 281–295 (2013)

  43. Liang, J.J., Qu, B.Y., Suganthan, P.N.: Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore, vol. 635, p. 490 (2013)

  44. Liu, H., Cai, Z., Wang, Y.: Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl. Soft Comput. 10(2), 629–640 (2010)

    Google Scholar 

  45. Mahdavi, M., Fesanghary, M., Damangir, E.: An improved harmony search algorithm for solving optimization problems. Appl. Math. Comput. 188(2), 1567–1579 (2007)

    MathSciNet  MATH  Google Scholar 

  46. Mehta, V.K., Dasgupta, B.: A constrained optimization algorithm based on the simplex search method. Eng. Optim. 44(5), 537–550 (2012)

    MathSciNet  Google Scholar 

  47. Mezura-Montes, E., Coello, C.A.C.: An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int. J. Gen. Syst. 37(4), 443–473 (2008)

    MathSciNet  MATH  Google Scholar 

  48. Mezura-Montes, E., Velázquez-Reyes, J., Coello, C.C.: Modified differential evolution for constrained optimization. In: 2006 IEEE International Conference on Evolutionary Computation, pp. 25–32. IEEE (2006)

  49. Midilli, Y.E., Parshutin, S.: Review for optimisation of neural networks with genetic algorithms and design of experiments in stock market prediction. In: Information Technology & Management Science (RTU Publishing House), vol. 22 (2019)

  50. Millan-Paramo, C., Abdalla Filho, J.E.: Size and shape optimization of truss structures with natural frequency constraints using modified simulated annealing algorithm. Arab. J. Sci. Eng. 45(5), 3511–3525 (2020)

    Google Scholar 

  51. Millan-Paramo, C., Filho, J.E.A.: Exporting water wave optimization concepts to modified simulated annealing algorithm for size optimization of truss structures with natural frequency constraints. Eng. Comput. 37(1), 763–777 (2021)

    Google Scholar 

  52. Omran, M.G., Al-Sharhan, S.: Improved continuous ant colony optimization algorithms for real-world engineering optimization problems. Eng. Appl. Artif. Intell. 85, 818–829 (2019)

    Google Scholar 

  53. Rajasekaran, S.: On simulated annealing and nested annealing. J. Glob. Optim. 16(1), 43–56 (2000)

    MathSciNet  MATH  Google Scholar 

  54. Rao, S.S.: Engineering Optimization: Theory and Practice. Wiley (2019)

    Google Scholar 

  55. Rizk-Allah, R.M., El-Sehiemy, R.A., Wang, G.G.: A novel parallel hurricane optimization algorithm for secure emission/economic load dispatch solution. Appl. Soft Comput. 63, 206–222 (2018)

    Google Scholar 

  56. Rosen, J.: The gradient projection method for nonlinear programming. Part II. Nonlinear constraints. J. Soc. Ind. Appl. Math. 9(4), 514–532 (1961)

    MATH  Google Scholar 

  57. Sadollah, A., Bahreininejad, A., Eskandar, H., Hamdi, M.: Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl. Soft Comput. 13(5), 2592–2612 (2013)

    Google Scholar 

  58. Sandgren, E.: Nonlinear integer and discrete programming in mechanical design. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 26584, pp. 95–105. American Society of Mechanical Engineers (1988)

  59. dos Santos Coelho, L.: Gaussian quantum-behaved particle swarm optimization approaches for constrained engineering design problems. Expert Syst. Appl. 37(2), 1676–1683 (2010)

    Google Scholar 

  60. Shi, Y., Eberhart, R.: A modified particle swarm optimizer. In: 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98TH8360), pp. 69–73. IEEE (1998)

  61. Wang, G., Guo, L., Wang, H., Duan, H., Liu, L., Li, J.: Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Comput. Appl. 24(3–4), 853–871 (2014)

    Google Scholar 

  62. Wang, G.G., Gandomi, A.H., Alavi, A.H.: An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl. Math. Model. 38(9–10), 2454–2462 (2014)

    MathSciNet  MATH  Google Scholar 

  63. Wang, G.G., Gandomi, A.H., Alavi, A.H., Deb, S.: A hybrid method based on krill herd and quantum-behaved particle swarm optimization. Neural Comput. Appl. 27(4), 989–1006 (2016)

    Google Scholar 

  64. Wang, G.G., Gandomi, A.H., Zhao, X., Chu, H.C.E.: Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft. Comput. 20(1), 273–285 (2016)

    Google Scholar 

  65. Wang, G.G., Guo, L., Gandomi, A.H., Alavi, A.H., Duan, H.: Simulated annealing-based krill herd algorithm for global optimization. In: Abstract and Applied Analysis, vol. 2013. Hindawi (2013)

  66. Wang, H., Yi, J.H.: An improved optimization method based on krill herd and artificial bee colony with information exchange. Memet. Comput. 10(2), 177–198 (2018)

    Google Scholar 

  67. Wu, G., Mallipeddi, R., Suganthan, P.N.: 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 (2017)

  68. Yang, X.S.: Engineering Optimization: An Introduction with Metaheuristic Applications. Wiley (2010)

    Google Scholar 

  69. Yassin, B., Lahcen, A., Zeriab, E.S.M.: Hybrid optimization procedure applied to optimal location finding for piezoelectric actuators and sensors for active vibration control. Appl. Math. Model. 62, 701–716 (2018)

    MathSciNet  MATH  Google Scholar 

  70. Yi, J.H., Deb, S., Dong, J., Alavi, A.H., Wang, G.G.: An improved NSGA-III algorithm with adaptive mutation operator for big data optimization problems. Future Gener. Comput. Syst. 88, 571–585 (2018)

    Google Scholar 

  71. Yi, J.H., Xing, L.N., Wang, G.G., Dong, J., Vasilakos, A.V., Alavi, A.H., Wang, L.: Behavior of crossover operators in NSGA-III for large-scale optimization problems. Inf. Sci. 509, 470–487 (2020)

    MathSciNet  Google Scholar 

  72. Yildiz, B.S., Pholdee, N., Bureerat, S., Yildiz, A.R., Sait, S.M.: Enhanced grasshopper optimization algorithm using elite opposition-based learning for solving real-world engineering problems. Eng. Comput. 1–13 (2021)

  73. Zhang, M., Luo, W., Wang, X.: Differential evolution with dynamic stochastic selection for constrained optimization. Inf. Sci. 178(15), 3043–3074 (2008)

    Google Scholar 

Download references

Acknowledgements

We would like to thank the Moroccan Ministry of Higher Education, Scientific Research and Innovation and the CNRST who funded this work through the Project PPR2/06/2016 as well as the OCP Foundation through the APRD research program.

Author information

Authors and Affiliations

  1. Research Center STIS, M2CS, Department of Applied Mathematics and Informatics, ENSAM, Mohammed V University in Rabat, Rabat, Morocco

    Yassin Belkourchia, Mohamed Zeriab Es-Sadek & Lahcen Azrar

  2. Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    Lahcen Azrar

Authors
  1. Yassin Belkourchia
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mohamed Zeriab Es-Sadek
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Lahcen Azrar
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

YB and MZE performed the programming, analysis, interpretation of the results, also the main contributor in writing the manuscript. LA supervised, validated and reviewed the work, also responsible for the project administration and funding acquisition.

Corresponding author

Correspondence to Lahcen Azrar.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Andrew Ning.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belkourchia, Y., Es-Sadek, M.Z. & Azrar, L. New Hybrid Perturbed Projected Gradient and Simulated Annealing Algorithms for Global Optimization. J Optim Theory Appl 197, 438–475 (2023). https://doi.org/10.1007/s10957-023-02210-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02210-7

Keywords