International Journal of Nonlinear Analysis and Applications

On generalization of optimality conditions for multiobjective semi-infinite problems

Articles in Press

Document Type : Research Paper

Author

Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran

Abstract

In this paper, we study the multiobjective semi-infinite programming problem with inequality constraints, in which the objective and the constraint functions are not necessarily continuous. If Ω is a local cone approximation, we consider the notion of Ω-subdifferential for functions. Then, we present the Karush-Kuhn-Tucker type necessary and sufficient optimality conditions under an Abadie type qualification for the considered problems via Ω-subdifferential.

Keywords

[1] G. Caristi and M. Ferrara, Semi-infinite multiobjective programming with grneralized invexity, Math. Rep. 62 (2010), 217–233.
[2] G. Caristi and N. Kanzi, Karush-Kuhn-Tucker type conditions for optimality of non-smooth multiobjective semi-infinite programming, Int. J. Math. Anal. 39 (2015), 1929–1938.
[3] M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.
[4] G. Giorgi, A. Gwirraggio, and J. Thierselder, Mathematics of Optimization; Smooth and Nonsmooth cases, Elsevier, 2004.
[5] M.A. Goberna and N. Kanzi, Optimality conditions in convex multiobjective SIP, Math. Program.. 164, (2017), 167–191.
[6] M.A. Goberna and M.A. L´opez, Linear Semi-Infinite Optimazation, Wiley, Chichester, 1998.
[7] B.D. Graven, Invex functions and constrained local minima, Bul. Aust. Math. Soc. 24 (1981), 357–366.
[8] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545–550.
[9] S. Habibi, N. Kanzi, and A. Ebadian, Weak Slater qualification for nonconvex multiobjective semi-infinite pro[1] programming, Iran J. Sci. Technol. Trans. Sci. 44 (2020), 1417–424.
[10] R. Hettich and O. Kortanek, Semi-infinite programming: theory, methods, and applications, Siam Riv. 35 (1993), 380–429.
[11] J.B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, I & II, Springer, Berlin, Heidelberg, 1991.
[12] N. Kanzi, Necessary optimality conditions for nonsmooth semi-infinite programming problems, J. Glob. Optim. 49 (2011), no. 4, 713–725.
[13] N. Kanzi, Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems, J. Glob. Optim. 56 (2013), 417–430.
[14] N. Kanzi, Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints, SIAM J. Optim. 24 (2014), 55–572.
[15] N. Kanzi, Non-Lipschitz semi-infinite optimization problems involving local cone approximation, Iran. J. Oper. Res. 5 (2014), no. 2, 1–11.
[16] N. Kanzi, Necessary and sufficient conditions for (weakly) efficient of non-differentiable multi-objective semi-infinite programming problems, Iran. J. Sci. Technol. Trans. Sci. 42 (2018), no. 3, 1537–1544.
[17] N. Kanzi and S. Nobakhtian, Nonsmooth semi-infinite programming problems with mixed constraints, J. Math. Anal. Appl. 351 (2008), 170–181.
[18] N. Kanzi and S. Nobakhtian, Optimality conditions for nonsmooth semi-infinite programming, Optimization 59 (2008), 717–727.
[19] N. Kanzi and S. Nobakhtian, Nonsmooth multiobjective semi-infinite programming problems with mixed constraints, Pacific J. Optim. 17 (2017), 43–53.
[20] N. Kanzi, J. Shaker Ardekani, and G. Caristi, Optimality, scalarization and duality in linear vector semi-infinite programming, Optimization 67 (2018), 523–536.
[21] N. Kanzi and M. Soleimani-damaneh, Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems, J. Math. Anal. Appl. 434 (2016), 638–651.
[22] W. Li, C. Nahak, and I. Singer, Constraint qualifications in semi-infinite systems of convex inequalities, SIAM J. Optim. 11 (2000), 31–52.
[23] M.A. Lopez, and G. Still, Semi-infinite programming, Eur. J. Oper. Res. 180 (2007), 491–518.
[24] S.S. Mishra, S.Y. Wang, and K.K. Lai, Generalied Convexity and Vector Optimization, Springer, Verlag, Berlin, 2009.
[25] B.S. Mordukhovich and T.T.A. Nghia, Constraint qualification and optimality conditions in semi-infinite and infinite programming, Math. Program. 139 (2012), 271–300.
[26] X.Y. Zheng and X. Yang, Lagrange multipliers in nonsmooth semi-infinite optimization problems, J. Oper. Res. 32 (2007), 168–181.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 25 April 2025
  • Receive Date: 01 January 2024
  • Revise Date: 04 June 2024
  • Accept Date: 15 June 2024