Abstract
In this paper, we have considered a nonsmooth multiobjective optimization problem where the objective and constraint functions involved are directionally differentiable. A new class of generalized functions (d − ρ − η − θ)-type I univex is introduced which generalizes many earlier classes cited in literature. Based upon these generalized functions, we have derived weak, strong, converse and strict converse duality theorems for mixed type multiobjective dual program in order to relate the efficient and weak efficient solutions of primal and dual problem.
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References
Hanson, M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Zhao, F.: On sufficiency of the Kuhn-Tucker conditions in nondifferentiable programming. B. Aust. Math. Soc. 46, 385–389 (1992)
Antczak, T.: Multiobjective programming under d-invexity. Eur. J. Oper. Res. 137, 28–36 (2002)
Ye, Y.L.: d-invexity and optimality conditions. J. Math. Anal. Appl. 162, 242–249 (1991)
Bector, C.R., Suneja, S.K., Gupta, S.: Univex functions and univex nonlinear programming. In: Proceeding of the Administrative Sciences Association Of Canada, pp. 115–124 (1992)
Rueda, N.G., Hanson, M.A., Singh, C.: Optimality and duality with generalized convexity. J. Optim. Theory Appl. 86, 491–500 (1995)
Mishra, S.K.: On multiple-objective optimization with generalized univexity. J. Math. Anal. Appl. 224, 131–148 (1998)
Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality for multiple-objective optimization under generalized type I univexity. J. Math. Anal. Appl. 303, 315–326 (2005)
Nahak, C., Mohapatra, R.N.: d − ρ − η − θ invexity in multiobjective optimization. Nonlinear Anal. 70, 2288–2296 (2009)
Mishra, S.K., Wang, S.Y., Lai, K.K.: Nondifferentiable multiobjective programming under generalized d-univexity. Eur. J. Oper. Res. 160, 218–226 (2005)
Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality in nondifferentiable and multiobjective programming under generalized d-invexity. J. Global Optim. 29, 425–438 (2004)
Hanson, M.A., Mond, B.: Necessary and sufficient conditions in constrained optimization. Math. Program. 37, 51–58 (1987)
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Kharbanda, P., Agarwal, D. & Sinha, D. Generalized Univex Functions in Nonsmooth Multiobjective Optimization. J Math Model Algor 12, 393–406 (2013). https://doi.org/10.1007/s10852-013-9218-8
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DOI: https://doi.org/10.1007/s10852-013-9218-8