An Optimality Theory Based Proximity Measure for Evolutionary Multi-Objective and Many-Objective Optimization

  • Kalyanmoy Deb
  • Mohamed Abouhawwash
  • Joydeep Dutta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9019)


Evolutionary multi- and many-objective optimization (EMO) methods attempt to find a set of Pareto-optimal solutions, instead of a single optimal solution. To evaluate these algorithms, performance metrics either require the knowledge of the true Pareto-optimal solutions or, are ad-hoc and heuristic based. In this paper, we suggest a KKT proximity measure (KKTPM) that can provide an estimate of the proximity of a set of trade-off solutions from the true Pareto-optimal solutions. Besides theoretical results, the proposed KKT proximity measure is computed for iteration-wise trade-off solutions obtained from specific EMO algorithms on two, three, five and 10-objective optimization problems. Results amply indicate the usefulness of the proposed KKTPM as a termination criterion for an EMO algorithm.


Multi-objective optimization Evolutionary optimization Termination criterion KkT optimality conditions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreani, R., Haeser, G., Martinez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60(5), 627–641 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bader, J., Deb, K., Zitzler, E.: Faster hypervolume-based search using Monte Carlo sampling. In: Proceedings of Multiple Criteria Decision Making (MCDM 2008). LNEMS, vol. 634, pp. 313–326. Springer, Heidelberg (2010)Google Scholar
  3. 3.
    Bector, C.R., Chandra, S., Dutta, J.: Principles of Optimization Theory. Narosa, New Delhi (2005)Google Scholar
  4. 4.
    Deb, K.: Multi-objective optimization using evolutionary algorithms. Wiley, Chichester (2001)MATHGoogle Scholar
  5. 5.
    Deb, K.: Scope of stationary multi-objective evolutionary optimization: A case study on a hydro-thermal power dispatch problem. Journal of Global Optimization 41(4), 479–515 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Deb, K., Abouhawwash, M.: An optimality theory based proximity measure for set based multi-objective optimization. COIN Report Number 2014015. Electrical and Computer Engineering, Michigan State University, East Lansing, USA (2014)Google Scholar
  7. 7.
    Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point based non-dominated sorting approach, Part I: Solving problems with box constraints. IEEE Transactions on Evolutionary Computation 18(4), 577–601 (2014)CrossRefGoogle Scholar
  9. 9.
    Deb, K., Tiwari, R., Dixit, M., Dutta, J.: Finding trade-off solutions close to KKT points using evolutionary multi-objective optimization. In: Proceedings of the Congress on Evolutionary Computation (CEC 2007), pp. 2109–2116. IEEE Press, Piscatway (2007)Google Scholar
  10. 10.
    Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. Journal of Global Optimization 56(4), 1463–1499 (2013)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)MATHGoogle Scholar
  12. 12.
    Haeser, G., Schuverdt, M.L.: Approximate KKT conditions for variational inequality problems. Optimization Online (2009)Google Scholar
  13. 13.
    Manoharan, P.S., Kannan, P.S., Baskar, S., Iruthayarajan, M.W.: Evolutionary algorithm solution and kkt based optimality verification to multi-area economic dispatch. Electrical Power and Energy Systems 31, 365–373 (2009)CrossRefGoogle Scholar
  14. 14.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)MATHGoogle Scholar
  15. 15.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1996)Google Scholar
  16. 16.
    Tulshyan, R., Arora, R., Deb, K., Dutta, J.: Investigating ea solutions for approximate KKT conditions for smooth problems. In: Proc. of Genetic and Evolutionary Algorithms Conference (GECCO 2010), pp. 689–696. ACM Press (2010)Google Scholar
  17. 17.
    While, L., Hingston, P., Barone, L., Huband, S.: A faster algorithm for calculating hypervolume. IEEE Trans. on Evolutionary Computation 10(1), 29–38 (2006)CrossRefGoogle Scholar
  18. 18.
    Wierzbicki, A.P.: The use of reference objectives in multiobjective optimization. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making Theory and Applications, pp. 468–486. Springer, Berlin (1980)CrossRefGoogle Scholar
  19. 19.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. Evol. Comput. Journal 8(2), 125–148 (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kalyanmoy Deb
    • 1
  • Mohamed Abouhawwash
    • 1
  • Joydeep Dutta
    • 2
  1. 1.Computational Optimization and Innovation (COIN) LaboratoryMichigan State UniversityEast LansingUSA
  2. 2.Department of Humanities and Social SciencesIndian Institute of Technology KanpurKanpurIndia

Personalised recommendations