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

© 2016 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )

Download PDF

Cite

XML

Abstract

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 and constraint functions and their generalizations are introduced for such nonsmooth vector optimization problems. Based upon these generalized invex functions, necessary and sufficient optimality conditions are established for directionally differentiable multiobjective programming problems. Thus, new Fritz John type and Karush-Kuhn-Tucker type necessary optimality conditions are proved for the considered directionally differentiable multiobjective programming problem. Further, weak, strong and converse duality theorems are also derived for Mond-Weir type vector dual programs.

Keywords

multiobjective programming

(weak) Pareto optimal solution

G-V-invex function

G-Fritz John necessary optimality conditions

G-Karush-Kuhn-Tucker necessary optimality conditions

duality

Conflict of interest

The authors declare they have no competing interests.

References

[1] Agarwal, R P, Ahmad, I, Al-Homidan, S, Optimality and duality for nonsmooth multiobjective program- ming problems involving generalized d-ρ-(η,θ)-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, E伍ciency and duality in nondiferentiable multiobjective programming involving directional de- rivative, Applied Mathematics, 2, 452-460 (2011).

[4] Antczak, T, Multiobjective programming under d- invexity, European Journal of Operational Research, 137, 28-36 (2002).

[5] Antczak, T, Optimality conditions and duality for nondiferentiable multiobjective programming prob- lems involving d-r-type I functions, Journal of Com- putational and Applied Mathematics, 225, 236-250 (2009).

[6] Antczak, T, On G-invex multiobjective program- ming. PartI. Optimality, Journal of Global Optimiza- tion, 43, 97-109 (2009).

[7] Antczak, T, On G-invex multiobjective program- ming. Part II. Duality, Journal of Global Optimiza- tion, 43, 111-140 (2009).

[8] Bazaraa, M S, Sherali, H D, Shetty, C M, Nonlin- ear programming: theory and algorithms, John Wi- ley 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 lo- cal minima, Bull. Austral. Math. Soc. 24, 357-366, (1981).

[11] Hanson, M A, On su伍ciency of the Kuhn-Tucker con- ditions, Journal of Mathematical Analysis and Appli- cations, 80, 545-550 (1981).

[12] Hanson, M A, Mond, B, Necessary and su伍cient conditions in constrained optimization, Mathematical Programming, 37, 51-58 (1987).

[13] Hanson, M A, R.Pini, C.Singh, Multiobjective pro- gramming under generalized type I invexity, Journal of Mathematical Analysis and Applications, 261, 562- 577 (2001).

[14] Hachimi, M, Aghezzaf, B, Su伍ciency and duality in diferentiable multiobjective programming involving generalized type I functions, Journal of Mathemati- cal 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 Optimiza- tion Theory and Applications, 80, 465-482 (1994).

[17] Kuk, H, Tanino, T, Optimality and duality in non- smooth multi objective optimization involving gener- alized 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 nondiferentiable and multi objective pro- gramming under generalized d-invexity, Journal of Global Optimization 29 425-438 (2004).

[19] Mishra, S K, Wang, S Y, Lai, K K, Nondifer- entiable multiobjective programming under general- izedd-univexity, European Journal of Operational Re- search, 160, 218-226 (2005).

[20] Mishra, S K, Noor, M A, Some nondiferentiable mul- tiobjective programming problems. Journal of Math- ematical Analysis and Applications, 316, 472-482 (2006).

[21] Mukherjee, R N, Singh, A K, Multi-objective op- timization involving non-convex semi-diferentiable functions, Indian Journal Pure and Application Mathematics, 21, 326-329 (1990).

[22] Preda, V, Optimality conditions and duality in multi- ple objective programming involving semilocally con- vex and related functions, Optimization, 36, 219-230 (1996).

[23] Slimani, H, Radjef, M S, Nondiferentiable multiob- jective programming under generalized dI-invexity, European Journal of Operational Research, 202, 32-41 (2010).

[24] Suneja, S K, Srivastava, M K, Optimality and du- ality in non diferentiable multi objective optimiza- tion involving d-Type I and related functions, Jour- nal of Mathematical Analysis and Applications, 206, 465-479 (1997).

[25] Suneja, S K, Gupta, S, Duality in multiobjective non- linear 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 ob- jective optimization, Journal of Mathematical Analy- sis and Applications, 136, 29-38 (1988).

[27] Ye, Y L, d-invexity and optimality conditions, Jour- nal of Mathematical Analysis and Applications, 162, 242-249 (1991).

Previous article in this issue

Next article in this issue

An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing