Regularity Conditions for Non-Differentiable Infinite Programming Problems using Michel-Penot Subdifferential

Document Type: بنیادی - نظری


payame Noor university of Yazd


In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz‎. ‎Necessary optimality conditions and regularity conditions are given‎. ‎Our approach are based on the Michel-Penot subdifferential.


Main Subjects

[1] Borwein‎ ‎J‎. ‎M.‎, ‎Zhu‎ ‎Q‎. ‎J‎. ‎(1999) " ‎A Survey of subdifferential calculus with applications ''‎, ‎Nonlinear Analysis‎, ‎38‎, ‎687-773‎.

[2] Hiriart-Urruty‎ ‎J‎. ‎B.‎, ‎Lemarechal‎ ‎C‎. ‎(1991)‎ " ‎Convex analysis and minimization algorithms‎, ‎I & II‎ ", ‎Springer‎, ‎Berlin‎, ‎Heidelberg‎.

[3] Giorgi‎ ‎J.‎, ‎Gwirraggio‎ ‎A.‎, ‎Thierselder‎ ‎J‎. ‎(2004)‎ " ‎Mathematics of optimization; smooth and non-smooth cases‎ ", ‎Elsivier‎.

[4] Michel‎ ‎P.‎, ‎Penot J‎. ‎P‎. ‎(1984)‎ " ‎Calculsous differentiel pour des fonctions lipschitziennes et non-lipschitziennes ''‎, ‎Academic Sciences Paris (I); Mathematics‎, ‎12‎, ‎269-272‎.

[5] Michel‎ ‎P.‎, ‎Penot‎ ‎J‎. ‎P‎. ‎(1992)‎ " ‎A generalized derivative for calm and stable functions ''‎, ‎Differential and Integral Equations‎, ‎5‎, ‎433‎- ‎454‎.

[6] Mordukhovich‎ ‎B‎. ‎S.‎, ‎Nghia‎ ‎T‎. ‎T‎. ‎A‎. ‎(2013)‎ " ‎Constraint qualification and optimality conditions in semi-infinite and infinite programming ''‎, ‎Mathematical Programming‎, ‎139‎, ‎271-300‎.

[7] Mordukhovich‎ ‎B‎. ‎S.‎, ‎Nghia‎ ‎T‎. ‎T‎. ‎A‎. ‎(2012)‎ " ‎Nonsmooth cone-constrained optimization with applications to semi-infinite programming ''‎, ‎Optimization‎, ‎online‎, ‎3/3396‎.

[8] Mordukhovich‎ ‎B‎. ‎S.‎, ‎Nghia‎ ‎T‎. ‎T‎. ‎A‎. ‎(2011)‎ " ‎Sub-differentials of non-convex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data ''‎, ‎Optimization‎, ‎online‎, ‎12/3261‎.