Two-Level Optimization Problems with Infinite Number of Convex Lower Level Constraints

Document Type: بنیادی - نظری

Author

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

Abstract

‎This paper proposes a new form of optimization problem which is a two-level programming problem with infinitely many lower level constraints‎. ‎Firstly‎, ‎we consider some lower level constraint qualifications (CQs) for this problem‎. ‎Then‎, ‎under these CQs‎, ‎we derive formula for estimating the subdifferential of its valued function‎. ‎Finally‎, ‎we present some necessary optimality conditions as Fritz-John type for the problem.

Keywords

Main Subjects


‎bibitem{1} Burachik R.‎, ‎Jeyakumar V‎. ‎(2005)‎. ‎``Dual condition for the convex subdifferential sum formula with applications"‎, ‎Journal of Convex Analysis‎, ‎15‎, ‎540-554‎.

‎bibitem{2} Clarke F‎. ‎H‎. ‎(1983)‎. ‎``Optimization and non-smooth analysis"‎, ‎Wiley-Interscience‎.

‎bibitem{3} Dinh N.‎, ‎Goberna M‎. ‎A.‎, ‎Lopez M‎. ‎A‎. ‎(2006)‎. ‎``From linear to convex system‎: ‎consistency‎, ‎Farkas' lemma and applications"‎, ‎Journal of Convex Analysis‎, ‎13‎, ‎279-290‎.

‎bibitem{4} Dinh N.‎, ‎Goberna M‎. ‎A.‎, ‎Lopez M‎. ‎A.‎, ‎Son T‎. ‎Q‎. ‎(2007)‎. ‎``New Farkas-type results with applications to convex infinite programmings"‎, ‎ESIAM‎: ‎Optimization Calculus Variation‎, ‎13‎, ‎580-597‎.

‎bibitem{5} Dinh N.‎, ‎Mordukhovich B‎. ‎S.‎, ‎Nghia T‎. ‎T‎. ‎A‎. ‎(2010)‎. ‎``Sub-differentials of value functions and optimality conditions for some class of DC and bilevel infinite and semi-infinite programs"‎, ‎Mathematical Programming‎, ‎123‎, ‎101-138‎.

‎bibitem{6} Dinh N.‎, ‎Nghia T‎. ‎T‎. ‎A.‎, ‎Vallet G‎. ‎(2006)‎. ‎``A closedness condition and its applications to DC programs with convex constraints"‎, ‎Preprint of the Laboratory of Applied Mathematics of Pau 0622‎.

‎bibitem{7} Dinh N.‎, ‎Vallet G.‎, ‎Nghia T‎. ‎T‎. ‎A‎. ‎(2008)‎. ‎``Farkas-type results and duality for DC programming with convex constraints"‎, ‎Journal of Convex Analysis‎, ‎15‎, ‎235-262‎.

‎‎‎

‎bibitem{8} Hiriart‎- ‎Urruty J‎. ‎B.‎, ‎Lemarechal C‎. ‎(1991)‎. ‎``Convex Analysis and Minimization Algorithms‎, ‎I & II"‎, ‎Springer‎, ‎Berlin‎, ‎Heidelberg‎.

‎bibitem{9} Jeyakumar V.‎, ‎Dinh N.‎, ‎Lee G‎. ‎M‎. ‎(2004)‎. ‎``A new closed cone constraint qualification for convex programs"‎, ‎Applied Mathematics Research Report AMR04/8‎, ‎School of Mathematics‎, ‎University of new South Wales‎, ‎Australia‎.

‎bibitem{10} Kanzi N‎. ‎``Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems"‎, ‎Journal of Global Optimization‎, ‎(DOI 10.1007/s10898-001-9828-5)‎.‎

 

‎bibitem{11} Kanzi N.‎, ‎Nobakhtian S‎. ‎(2010)‎. ‎``Optimality conditions for nonsmooth semi-infinite programming"‎, ‎Optimization‎, ‎59‎, ‎717-727‎.

‎bibitem{12} Kanzi N.‎, ‎Nobakhtian S‎. ‎(2010)‎. ‎``Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems"‎, ‎European Journal of Operational Research‎, ‎205‎, ‎253-263‎.

‎bibitem{13} Li W.‎, ‎Nahak C.‎, ‎Singer I‎. ‎(2000)‎. ‎``Constraint qualifications in semi-infinite systems of convex inequalities"‎, ‎SIAM Journal of Optimization‎, ‎11‎, ‎31-52‎.

‎bibitem{14} Stein O‎. ‎(2003)‎. ‎``Bi-level strategies in semi-infinite programming"‎, ‎Kluwer‎, ‎Boston‎.