A Bi-objective Based Hybrid Evolutionary-Classical Algorithm for Handling Equality Constraints

  • Rituparna Datta
  • Kalyanmoy Deb
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)


Equality constraints are difficult to handle by any optimization algorithm, including evolutionary methods. Much of the existing studies have concentrated on handling inequality constraints. Such methods may or may not work well in handling equality constraints. The presence of equality constraints in an optimization problem decreases the feasible region significantly. In this paper, we borrow our existing hybrid evolutionary-cum-classical approach developed for inequality constraints and modify it to be suitable for handling equality constraints. This modified hybrid approach uses an evolutionary multi-objective optimization (EMO) algorithm to find a trade-off frontier in terms of minimizing the objective function and the constraint violation. A suitable penalty parameter is obtained from the frontier and then used to form a penalized objective function. The procedure is repeated after a few generations for the hybrid procedure to adaptively find the constrained minimum. Unlike other equality constraint handling methods, our proposed procedure does not require the equality constraints to be transformed into an inequality constraint. We validate the efficiency of our method on six problems with only equality constraints and two problems with mixed equality and inequality constraints.


Evolutionary multi-objective optimization Constraint handling Equality constraint Penalty Function Bi-Objective optimization Hybrid methodology 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Deb, K.: Optimization for Engineering Design: Algorithms and Examples. Prentice-Hall, New Delhi (1995)Google Scholar
  2. 2.
    Reklaitis, G.V., Ravindran, A., Ragsdell, K.M.: Engineering Optimization Methods and Applications. Wiley, New York (1983)Google Scholar
  3. 3.
    Richardson, J.T., Palmer, M.R., Liepins, G.E., Hilliard, M.R.: Some guidelines for genetic algorithms with penalty functions. In: Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 191–197. Morgan Kaufmann Publishers Inc., San Francisco (1989)Google Scholar
  4. 4.
    Gen, M., Cheng, R.: A survey of penalty techniques in genetic algorithms. In: Proceedings of IEEE International Conference on Evolutionary Computation. IEEE Press, Los Alamitos (1996)Google Scholar
  5. 5.
    Deb, K.: An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering 186(2-4), 311–338 (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Coello, C., Carlos, A.: Use of a self-adaptive penalty approach for engineering optimization problems. Computers in Industry 41(2), 113–127 (2000)CrossRefGoogle Scholar
  7. 7.
    Surry, P.D., Radcliffe, N.J., Boyd, I.D.: A multi-objective approach to constrained optimisation of gas supply networks: The COMOGA method. In: Fogarty, T.C. (ed.) AISB-WS 1995. LNCS, vol. 993, pp. 166–180. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  8. 8.
    Camponogara, E., Talukdar, S.N.: A genetic algorithm for constrained and multiobjective optimization. In: 3rd Nordic Workshop on Genetic Algorithms and Their Applications (3NWGA), pp. 49–62 (1997)Google Scholar
  9. 9.
    Angantyr, A., Andersson, J., Aidanpaa, J.-O.: Constrained optimization based on a multiobjective evolutionary algorithm. In: Proceedings of Congress on Evolutionary Computation, pp. 1560–1567 (2003)Google Scholar
  10. 10.
    Coello, C.A.C.: Treating objectives as constraints for single objective optimization. Engineering Optimization 32(3), 275–308 (2000)CrossRefGoogle Scholar
  11. 11.
    Deb, K., Lele, S., Datta, R.: A hybrid evolutionary multi-objective and SQP based procedure for constrained optimization. In: Kang, L., Liu, Y., Zeng, S. (eds.) ISICA 2007. LNCS, vol. 4683, pp. 36–45. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Echeverri, M.G., Lezama, J.M.L., Romero, R.: An efficient constraint handling methodology for multi-objective evolutionary algorithms. Revista Facultad de Ingenieria-Universidad de Antioquia 49, 141–150 (2009)Google Scholar
  13. 13.
    Burke, E.K., Smith, A.J.: Hybrid evolutionary techniques for the maintenance schedulingproblem. IEEE Transactions on Power Systems 15(1), 122–128 (2000)CrossRefGoogle Scholar
  14. 14.
    Fatourechi, M., Bashashati, A., Ward, R.K., Birch, G.E.: A hybrid genetic algorithm approach for improving the performance of the LF-ASD brain computer interface. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2005, vol. 5 (2005)Google Scholar
  15. 15.
    Victoire, T., Jeyakumar, A.E.: A modified hybrid EP-SQP approach for dynamic dispatch with valve-point effect. International Journal of Electrical Power & Energy Systems 27(8), 594–601 (2005)CrossRefGoogle Scholar
  16. 16.
    Hinterding, R.: Constrained parameter optimisation: equality constraints. In: Proceedings of the 2001 Congress on Evolutionary Computation, vol. 1, pp. 687–692. IEEE, Los Alamitos (2002)Google Scholar
  17. 17.
    Peconick, G., Wanner, E.F., Takahashi, R.H.C.: Projection-based local search operator for multiple equality constraints within genetic algorithms. In: IEEE Congress on Evolutionary Computation, CEC 2007, pp. 3043–3049. IEEE, Los Alamitos (2008)Google Scholar
  18. 18.
    Lin, C.Y., Wu, W.H.: Adaptive penalty strategies in genetic search for problems with inequality and equality constraints. In: IUTAM Symposium on Evolutionary Methods in Mechanics, pp. 241–250. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Shang, W., Zhao, S., Shen, Y.: A flexible tolerance genetic algorithm for optimal problems with nonlinear equality constraints. Advanced Engineering Informatics 23(3), 253–264 (2009)CrossRefGoogle Scholar
  20. 20.
    Deb, K., Datta, R.: A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach. In: Proceedings of the Congress on Evolutionary Computation (CEC 2010), pp. 1–8 (2010)Google Scholar
  21. 21.
    Deb, K.: Multi-objective optimization using evolutionary algorithms. Wiley, Chichester (2001)zbMATHGoogle Scholar
  22. 22.
    Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello Coello, C.A., Deb, K.: Problem definitions and evaluation criteria for the CEC 2006: Special session on constrained real-parameter optimization. Technical report, Nanyang Technological University, Singapore (2006)Google Scholar
  23. 23.
    Zavala, A.E.M., Aguirre, A.H., Diharce, E.R.V.: Continuous Constrained Optimization with Dynamic Tolerance Using the COPSO Algorithm, pp. 1–24. Springer, Heidelberg (2009)Google Scholar
  24. 24.
    Takahama, T., Sakai, S.: Solving Difficult Constrained Optimization Problems by the ε Constrained Differential Evolution with Gradient-Based Mutation, pp. 51–72. Springer, Heidelberg (2009)Google Scholar
  25. 25.
    Brest, J.: Constrained Real-Parameter Optimization with ε Self-Adaptive Differential Evolution, pp. 73–94. Springer, Heidelberg (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Rituparna Datta
    • 1
  • Kalyanmoy Deb
    • 1
  1. 1.Kanpur Genetic Algorithms Laboratory (KanGAL), Department of Mechanical EngineeringIndian Institute of Technology KanpurIndia

Personalised recommendations