Tadeusz ANTCZAK, Gabriel RUİZ GARZON

On semi-G-V -type I concepts for directionally differentiable multiobjective programming problems

In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with directionally differentiable functions is considered. The so-called G-V -type I objective andconstraint functions and their generalizations are introduced for such nonsmooth vector optimizationproblems. Based upon these generalized invex functions, necessary and sufficient optimality conditionsare established for directionally differentiable multiobjective programming problems. Thus, new FritzJohn type and Karush-Kuhn-Tucker type necessary optimality conditions are proved for the considereddirectionally differentiable multiobjective programming problem. Further, weak, strong and converseduality theorems are also derived for Mond-Weir type vector dual programs.

PDF

___

[1] Agarwal, R P, Ahmad, I, Al-Homidan, S, Optimality and duality for nonsmooth multiobjective programming problems involving generalized d-?-(?, ?)-Type I invex functions, Journal of Nonlinear and Convex Analysis 13 733-744 (2012).
[2] Aghezzaf, B, Hachimi, M, Generalized invexity and duality in multiobjective programming problems, Journal of Global Optimization, 18, 91-101 (2000).
[3] Ahmad, I, Efficiency and duality in nondifferentiable multiobjective programming involving directional derivative, Applied Mathematics, 2, 452-460 (2011).
[4] Antczak, T, Multiobjective programming under dinvexity, European Journal of Operational Research, 137, 28-36 (2002).
[5] Antczak, T, Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d-r-type I functions, Journal of Computational and Applied Mathematics, 225, 236-250 (2009).
[6] Antczak, T, On G-invex multiobjective programming. Part I. Optimality, Journal of Global Optimization, 43, 97-109 (2009).
[7] Antczak, T, On G-invex multiobjective programming. Part II. Duality, Journal of Global Optimization, 43, 111-140 (2009).
[8] Bazaraa, M S, Sherali, H D, Shetty, C M, Nonlinear programming: theory and algorithms, John Wiley and Sons, New York, (1991).
[9] Ben-Israel, A, Mond, B, What is invexity? Journal of Australian Mathematical Society Ser.B 28, 1-9 (1986).
[10] Craven, B D, Invex functions and constrained local minima, Bull. Austral. Math. Soc. 24, 357-366, (1981).
[11] Hanson, M A, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80, 545-550 (1981).
[12] Hanson, M A, Mond, B, Necessary and sufficient conditions in constrained optimization, Mathematical Programming, 37, 51-58 (1987).
[13] Hanson, M A, R.Pini, C.Singh, Multiobjective programming under generalized type I invexity, Journal of Mathematical Analysis and Applications, 261, 562- 577 (2001).
[14] Hachimi, M, Aghezzaf, B, Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions, Journal of Mathematical Analysis and Applications, 296, 382-392 (2004).
[15] Jeyakumar, V, Mond, B, On generalized convex mathematical programming, Journal of Australian Mathematical Society Ser.B, 34, 43-53 (1992).
[16] Kaul, R N, Suneja, S K, Srivastava, M K, Optimality criteria and duality in multiple objective optimization involving generalized invexity, Journal of Optimization Theory and Applications, 80, 465-482 (1994).
[17] Kuk, H, Tanino, T, Optimality and duality in nonsmooth multi objective optimization involving generalized Type I functions, Computers and Mathematics with Applications, 45, 1497-1506 (2003).
[18] Mishra, S K, Wang, S Y, Lai, K K, Optimality and duality in nondifferentiable and multi objective programming under generalized d-invexity, Journal of Global Optimization 29 425-438 (2004).
[19] Mishra, S K, Wang, S Y, Lai, K K, Nondifferentiable multiobjective programming under generalized d-univexity, European Journal of Operational Research, 160, 218-226 (2005).
[20] Mishra, S K, Noor, M A, Some nondifferentiable multiobjective programming problems. Journal of Mathematical Analysis and Applications, 316, 472-482 (2006).
[21] Mukherjee, R N, Singh, A K, Multi-objective optimization involving non-convex semi-differentiable functions, Indian Journal Pure and Application Mathematics, 21, 326-329 (1990).
[22] Preda, V, Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions, Optimization, 36, 219-230 (1996).
[23] Slimani, H, Radjef, M S, Nondifferentiable multiobjective programming under generalized dI -invexity, European Journal of Operational Research, 202, 32-41 (2010).
[24] Suneja, S K, Srivastava, M K, Optimality and duality in non differentiable multi objective optimization involving d-Type I and related functions, Journal of Mathematical Analysis and Applications, 206, 465-479 (1997).
[25] Suneja, S K, Gupta, S, Duality in multiobjective nonlinear programming involving semilocally convex and related functions, European Journal of Operational Research, 107, 675-685 (1998).
[26] Weir, T, Mond, B, Pre-invex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136, 29-38 (1988).
[27] Ye, Y L, d-invexity and optimality conditions, Journal of Mathematical Analysis and Applications, 162, 242-249 (1991).