Annals of Operations Research

, Volume 251, Issue 1–2, pp 73–87 | Cite as

Nondifferentiable minimax programming problems with applications



This paper is devoted to the study of optimality conditions and duality in nondifferentiable minimax programming problems and applications. Employing some advanced tools of variational analysis and generalized differentiation, we establish new necessary conditions for optimal solutions of a minimax programming problem involving inequality and equality constraints. Sufficient conditions for the existence of such solutions to the considered problem are also obtained by way of \(L\)-invex-infine functions. We state a dual problem to the primal one and explore weak, strong and converse duality relations between them. In addition, some of these results are applied to a nondifferentiable multiobjective optimization problem.


Minimax programming problem Optimality condition Duality Limiting subdifferential \(L\)-invex-infine function 

Mathematics Subject Classification

49K99 65K10 90C29 90C46 



The authors would like to thank the editor and the referees for valuable comments and suggestions.

Conflict of interest

The authors declare that they have no potential conflict of interest.


  1. Ahmad, I., Husain, Z., & Sharma, S. (2008). Second-order duality in nondifferentiable minmax programming involving type-I functions. Journal of Computational and Applied Mathematics, 215(1), 91–102.CrossRefGoogle Scholar
  2. Antczak, T. (2008). Generalized fractional minimax programming with \(B\)-\((p, r)\)-invexity. Computers & Mathematics with Applications, 56(6), 1505–1525.CrossRefGoogle Scholar
  3. Antczak, T. (2011). Nonsmooth minimax programming under locally Lipschitz \((\Phi,\rho )\)-invexity. Applied Mathematics and Computation, 217(23), 9606–9624.CrossRefGoogle Scholar
  4. Bector, C. R. (1996). Wolfe-type duality involving \((B,\eta )\)-invex functions for a minmax programming problem. Journal of Mathematical Analysis and Applications, 201(1), 114–127.CrossRefGoogle Scholar
  5. Bector, C. R., Chandra, S., & Kumar, V. (1994). Duality for a class of minmax and inexact programming problem. Journal of Mathematical Analysis and Applications, 186(3), 735–746.CrossRefGoogle Scholar
  6. Bram, J. (1966). The Lagrange multiplier theorem for max–min with several constraints. SIAM Journal on Applied Mathematics, 14, 665–667.CrossRefGoogle Scholar
  7. Chinchuluun, A., Yuan, D., & Pardalos, P. M. (2007). Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Annals of Operations Research, 154, 133–147.CrossRefGoogle Scholar
  8. Chuong, T. D. (2012). \(L\)-invex-infine functions and applications. Nonlinear Analysis, 75, 5044–5052.CrossRefGoogle Scholar
  9. Chuong, T. D., & Kim, D. S. (2014). Optimality conditions and duality in nonsmooth multiobjective optimization problems. Annals of Operations Research, 217, 117–136.CrossRefGoogle Scholar
  10. Chuong, T. D., Huy, N. Q., & Yao, J.-C. (2009). Subdifferentials of marginal functions in semi-infinite programming. SIAM Journal on Optimization, 20, 1462–1477.CrossRefGoogle Scholar
  11. Golestani, M., & Nobakhtian, S. (2012). Convexificators and strong Kuhn–Tucker conditions. Computers & Mathematics with Applications, 64(4), 550–557.CrossRefGoogle Scholar
  12. Husain, Z., Jayswal, A., & Ahmad, I. (2009). Second order duality for nondifferentiable minimax programming problems with generalized convexity. Journal of Global Optimization, 44(4), 593–608.CrossRefGoogle Scholar
  13. Jayswal, A. (2008). Non-differentiable minimax fractional programming with generalized \(\alpha \)-univexity. Journal of Computational and Applied Mathematics, 214(1), 121–135.CrossRefGoogle Scholar
  14. Jayswal, A., & Stancu-Minasian, I. (2011). Higher-order duality for nondifferentiable minimax programming problem with generalized convexity. Nonlinear Analysis, 74(2), 616–625.CrossRefGoogle Scholar
  15. Lai, H. C., & Huang, T. Y. (2009). Optimality conditions for a nondifferentiable minimax programming in complex spaces. Nonlinear Analysis, 71(3–4), 1205–1212.CrossRefGoogle Scholar
  16. Lai, H. C., & Huang, T. Y. (2012). Nondifferentiable minimax fractional programming in complex spaces with parametric duality. Journal of Global Optimization, 53(2), 243–254.CrossRefGoogle Scholar
  17. Lee, J.-C., & Lai, H.-C. (2005). Parameter-free dual models for fractional programming with generalized invexity. Annals of Operations Research, 133, 47–61.CrossRefGoogle Scholar
  18. Liu, J. C., & Wu, C. S. (1998). On minimax fractional optimality conditions with invexity. Journal of Mathematical Analysis and Applications, 219(1), 21–35.CrossRefGoogle Scholar
  19. Liu, J. C., Wu, C. S., & Sheu, R. L. (1997). Duality for fractional minimax programming. Optimization, 41(2), 117–133.CrossRefGoogle Scholar
  20. Mishra, S. K., & Rueda, N. G. (2006). Second-order duality for nondifferentiable minimax programming involving generalized type I functions. Journal of Optimization Theory and Applications, 130(3), 477–486.CrossRefGoogle Scholar
  21. Mond, B., & Weir, T. (1981). Generalized concavity and duality. In S. Schaible & W. T. Ziemba (Eds.), Generalized concavity in optimization and economics (pp. 263–279). New York: Academic Press.Google Scholar
  22. Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation. I: basic theory. Berlin: Springer.Google Scholar
  23. Rockafellar, R. T. (1970). Convex analysis. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
  24. Sach, P. H., Lee, G. M., & Kim, D. S. (2003). Infine functions, nonsmooth alternative theorems and vector optimization problems. Journal of Global Optimization, 27, 51–81.CrossRefGoogle Scholar
  25. Tanimoto, S. (1980). Nondifferentiable mathematical programming and convex-concave functions. Journal of Optimization Theory and Applications, 31(3), 331–342.CrossRefGoogle Scholar
  26. Wolfe, P. (1961). A duality theorem for nonlinear programming. Quarterly of Applied Mathematics, 19, 239–244.CrossRefGoogle Scholar
  27. Zalmai, G. J. (1989). Optimality conditions and duality for constrained measurable subset selection problems with minmax objective functions. Optimization, 20(4), 377–395.CrossRefGoogle Scholar
  28. Zalmai, G. J. (2003). Parameter-free sufficient optimality conditions and duality models for minmax fractional subset programming problems with generalized \((F,\rho,\theta )\)-convex functions. Computers & Mathematics with Applications, 45(10–11), 1507–1535.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsSaigon UniversityHo Chi Minh CityVietnam
  2. 2.Department of Applied MathematicsPukyong National UniversityBusanKorea

Personalised recommendations