Characterization of Properly Efficient Solutions for Convex Multiobjective Programming with Nondifferentiable vanishing constraints

Document Type: بنیادی - نظری


Department of Mathematics‎, ‎Payame Noor University (PNU)‎, ‎P.OBox‎, ‎19395-4679‎, ‎Tehran‎, ‎Iran.



This paper studies the convex multiobjective optimization problem with vanishing constraints‎. ‎We introduce a new constraint qualification for these problems‎, ‎and then a necessary optimality condition for properly efficient solutions is presented‎. ‎Finally by imposing some assumptions‎, ‎we show that our necessary condition is also sufficient for proper efficiency‎. ‎Our results are formulated in terms of convex subdifferential.


bibitem{AcKa} {Achtziger W.‎, ‎Kanzow C.} (2007)‎. ‎`` Mathematical programs with vanishing constraints‎: ‎optimality conditions and constraint qualifications''‎, ‎Math‎. ‎Program‎, ‎114‎, ‎69--99.‎



‎bibitem {ehr} {Ehrgott M.} (2005)‎. ‎`` Multicriteria Optimization''‎ , ‎Springer‎, ‎Berlin‎.


‎bibitem{geo} Geoffrion A‎. ‎(1968)‎. ‎`` Proper efficiency and the theory of vector maximization''‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl‎, ‎22‎, ‎618--630‎.


‎bibitem{GoRiTaZa} Gopfert A.‎, ‎Riahi H.‎, ‎Tammer C.‎, ‎Zalinescu C‎. ‎(2003)‎. ‎` Variational methods in partioal ordered spaces''‎, ‎Springer‎, ‎New York‎.


‎bibitem{HJO} Henrion R.‎, ‎Jourani A.‎, ‎Outrata J‎. ‎(2002)‎. ‎`` On the calmness of a class of multifunctions''‎, ‎SIAM J‎. ‎Optim‎, ‎13‎, ‎603--618‎.



‎bibitem{Hiriart} Hiriart-Urruty J.B.‎, ‎Lemarechal C‎. ‎(1991)‎. ‎`` Convex analysis and minimization algorithms''‎, ‎I‎. ‎Berlin‎: ‎Springer‎.



‎bibitem{HoKa2} Hoheisel T.‎, ‎Kanzow C‎. ‎(2008)‎. ‎`` Stationarity conditions for mathematical programs with vanishing constraints using weak constraint qualifications''‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl‎, ‎337‎, ‎292--310‎.



‎Hoheisel T.‎, ‎Kanzow C.‎, ‎Outrata J‎. ‎(2010)‎. ‎`` Exact penalty results for mathematical programs with vanishing constraints''‎, ‎Nonlinear Anal‎, ‎72‎, ‎2514--2526‎.


‎bibitem{HoKa1}Hoheisel T.‎, ‎Kanzow C‎. ‎(2007)‎. ‎`` First-‎- ‎and second--order optimality conditions for mathematical programs with vanishing constraints''‎, ‎Appl Math‎, ‎52‎, ‎495--514‎.



‎ Kazemi S.‎, ‎Kanzi N‎. ‎(2018)‎. ‎`` Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints''‎, ‎Journal of Optimization Theory and Applications‎, ‎DOI 10.1007/s10957--018--1373--7‎.


‎ bibitem{KazKanEba}‎

‎ Kazemi S.‎, ‎Kanzi N.‎, ‎Ebadian A‎. ‎(2019)‎. ‎`` Estimating the Frèchet Normal Cone in Optimization Problems with Nonsmooth Vanishing Constraints''‎, ‎Iranian Journal of Science and Technology‎, ‎Transactions A‎: ‎Science‎, ‎DOI 10.1007/s40995--019--00683--8‎.


‎bibitem{MSLM} Mishra S.K.‎, ‎Singh V.‎, ‎Laha V.‎, ‎Mohapatra R‎. ‎N‎. ‎(2015)‎. ‎`` On Constraint qualifications for multiobjective optimization problems with vanishing constraints''‎, ‎Optimization methods‎, ‎theory and applications‎, ‎Springer‎, ‎Heidelberg‎, ‎95--135‎.


‎bibitem{MSL} Mishra S.K.‎, ‎Singh V.‎, ‎Laha V‎. ‎(2016)‎. ‎`` On duality for mathematical programs with vanishing constraints''‎, ‎Annals of Operations Research‎. ‎243‎, ‎249--272‎.


‎bibitem{MOVNOB} Movahedian N.‎, ‎Nobakhtian S‎. ‎(2010)‎. ‎`` Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints ''‎, ‎Nonlinear Anal‎, ‎72‎, ‎2694--2705‎.


‎bibitem{ROCW} Rockafellar R.T.‎, ‎Wets J.B‎. ‎(1998)‎. ‎`` Variational analysis ''‎, ‎Springer--Verlag‎.


‎bibitem{Ye} Ye J‎. ‎(2005)‎. ‎`` Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints''‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl‎, ‎307‎, ‎350--369‎.