Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

Main Article Content

Izhar Ahmad
Krishna Kummari
S. Al-Homidan

Abstract

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.

Article Details

References

  1. J. M. Abadie, On the Kuhn-Tucker theorem, Nonlinear Programming, J. Abadie ed., John Wiley, New York, 1967, pp. 21-36.
  2. W. Achtziger and C. Kanzow, Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications, Math. Program. 114 (2008), 69-99.
  3. Y. An, G. Ye, D. Zhao and W. Liu, Hermite-Hadamard type inequalities for interval (h1, h2)-convex functions, Mathematics, 7 (2019), Art. ID 436.
  4. M. S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming. Theory and Algorithms. 2nd edition, John Wiley & Sons, Hoboken, 1993.
  5. M. P. Bendsøe and O. Sigmund, Topology Optimization-Theory, Methods and Applications, 2nd ed., Springer, Heidelberg, Germany, 2003.
  6. A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Method. Oper. Res. 76 (2012), 273-288.
  7. R. W. Cottle, A theorem of Fritz John in mathematical programming, RAND Memorandum RM-3858-PR, RAND Corporation, 1963.
  8. M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control, 7 (1969), 232-241.
  9. A. Khare and T. Nath, Enhanced Fritz John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints, J. Math. Anal. Appl. 472 (2019), 1042-1077.
  10. Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1997.
  11. O. L. Mangasarian and S. Fromovitz, The Fritz John necessary optimality condition in the presence of equality and inequality constraints, J. Math. Anal. Appl. 17 (1967), 37-47.
  12. S. K. Mishra, V. Singh, V. Laha and R. N. Mohapatra, On constraint qualifications for multiobjective optimization problems with vanishing constraints, Optimization Methods, Theory and Applications, Springer Berlin Heidelberg (2015), 95-135.
  13. R. Osuna-G ´omez, B. Hern ´andez-Jim ´enez, Y. Chalco-Cano and G. Ruiz-Garz ´on, New efficiency conditions for multiobjective interval-valued programming problems, Inform. Sci. 420 (2017), 235-248.
  14. J. V. Outrata, M. Koˇcvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 1998.
  15. D. W. Peterson, A review of constraint qualifications in finite-dimensional spaces, SIAM Rev. 15 (1973), 639-654.
  16. A. Sadeghi, M. Saraj and N. M. Amiri, Efficient solutions of interval programming problems with inexact parameters and second order cone constraints, Mathematics, 6 (2018), Art ID 270.
  17. D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. Nonlinear Anal. Optim. 5 (2014), 91-103.
  18. D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval-valued multiobjective programming with generalized differentiable functions, European J. Oper. Res. 254 (2016), 29-39.
  19. M. Slater, Lagrange multipliers revisited: a contribution to nonlinear programming, Cowles Commission Discussion Paper, Mathematics, 403, 1950.
  20. I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D Reidei Publishing Company, Bordrecht, 1984.
  21. Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval- valued programming, J. Ind. Manage. Optim. 9 (2013), 131-142.
  22. Z. Wang and S. C. Fang, On constraint qualifications: motivation, design and inter-relations, J. Ind. Manage. Optim. 9 (2013), 983-1001.
  23. L. Wang, G. Yang, H. Xiao, Q. Sun, J. Ge, Interval optimization for structural dynamic responses of an artillery system under uncertainty, Eng. Optim. 52 (2020), 343-366.
  24. H. C. Wu, On interval valued nonlinear programming problems, J. Math. Anal. Appl. 338 (2008), 299-316.
  25. H. C. Wu, The Karush Kuhn Tuker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. Oper. Res. 196 (2009), 49-60.
  26. H. C. Wu, Solving the interval-valued optimization problems based on the concept of null set, J. Ind. Manage. Optim. 14 (3) (2018), 1157-1178.