Quasi-Gap and Gap Functions for Non-Smooth Multi-Objective Semi-Infinite Optimization Problems

Document Type: بنیادی - نظری


1 Department of Mathematics‎, ‎Payame Noor University, P.O‎. ‎Box‎, ‎19395-3697‎, ‎Tehran‎, ‎Iran.

2 Department of Mathematics, Yazd branch, Islamic Azad University, Yazd, Iran



In this paper‎, ‎we introduce and study some new single-valued gap functions for non-differentiable semi-infinite multiobjective optimization problems with locally Lipschitz data‎. ‎Since one of the fundamental properties of gap function for optimization problems is its abilities in characterizing the solutions of the problem in question‎, ‎then the essential properties of the newly introduced gap functions are established‎. ‎All results are given in terms of the Clarke subdifferential.


bibitem{ALTa} L‎. ‎Altangerel‎, ‎R.I‎. ‎Bot‎, ‎G‎. ‎Wanka‎, ‎Conjugate duality in vector optimization and some applications to the vector variational inequality‎, ‎Journal of Mathematical Analysis and Applications 329 (2007) 1010-1035‎.


‎bibitem{CARKAN1} G‎. ‎Caristi‎, ‎N‎. ‎Kanzi‎, ‎Karush-Kuhn-Tuker Type Conditions for Optimality of Non-Smooth Multiobjective Semi-Infinite Programming‎, ‎International Journal of Mathematical Analysis 9 (2015) 1929-1938‎.


‎bibitem{CaKaSo} G‎. ‎Caristi‎, ‎N‎. ‎Kanzi‎, ‎M.‎, ‎Soleymani-Damaneh‎, ‎On gap functions for nonsmooth multiobjective‎

‎optimization problems‎, ‎Optim‎. ‎Lett‎. ‎12 (2018) 273-286‎.


‎bibitem{CHGOYA} C.Y‎. ‎Chen‎, ‎C.J‎. ‎Goh‎, ‎X.Q‎. ‎Yang‎, ‎The gap function of a convex multicriteria optimization problem‎, ‎European Journal of Operational Research 111 (1998) 142-151‎.


‎bibitem{CHUKIM} T.D‎. ‎Chuong‎, ‎D.S‎. ‎Kim‎, ‎Nonsmooth semi-infinite multiobjective optimization problems‎, ‎Journal of Optimization; Theory and Applications 160 (2014) 748-762‎ .



‎bibitem{CL} F.H‎. ‎Clarke‎, ‎Optimization and nonsmooth analysis‎. ‎Wiley‎, ‎Interscience (1983)‎.



‎bibitem{HEA} D.W‎. ‎Hearn‎, ‎The gap function of a convex program‎, ‎Operations Research Letters 1 (1982) 67-71‎.



‎bibitem{KANNOB4} 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) 1537-1544‎.



‎bibitem{KAN1} N‎. ‎Kanzi‎, ‎Karush-Kuhn-Tucker Types Optimality Conditions for Non-Smooth Semi-Infinite Vector Optimization Problems‎, ‎J‎. ‎Mathematical Extension 9 (2015) 45-56‎.


‎bibitem{KanSol} N‎. ‎Kanzi‎, ‎M‎. ‎Soleyman-damaneh‎, ‎Slater CQ‎, ‎optimality and duality for quasiconvex semi-infinite optimization problems‎, ‎J‎. ‎Math.Anal.Appl‎. ‎434 (2016) 638-651‎.


‎bibitem{KanSC} N‎. ‎Kanzi‎, ‎J‎. ‎Shaker Ardekani‎, ‎G‎. ‎Caristi‎, ‎Optimality‎, ‎scalarization and duality in linear vector semi-infinite programming‎, ‎Optimization‎. ‎67 (2018) 523-536‎.



‎bibitem{Mas} G‎. ‎Mastroeni‎, ‎Gap functions for equilibrium problems‎, ‎J‎. ‎Glob‎. ‎Optim‎. ‎27 (2003) 411–426‎.


‎bibitem{MiS} H‎. ‎Mirzaee‎, ‎M‎. ‎Soleimani-damaneh‎, ‎Optimality‎, ‎duality and gap function for quasi variational inequality problems‎, ‎ESAIM Control Optim‎. ‎Calc‎. ‎Var‎. ‎23 (2017) 297–308‎.


‎bibitem{SOLEY} M‎. ‎Soleimani-damaneh‎, ‎The gap function for optimization problems in Banach spaces‎, ‎Nonlinear Analysis 69 (2008) 716-723‎.