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Minimax Theory and its Applications 01 (2016), No. 2, 163--195
Copyright Heldermann Verlag 2016

On the Cone Minima and Maxima of Directed Convex Free Disposal Subsets and Applications

Mohamed Ait Mansour
Dép. de Mathématiques et Informatiques, Faculté Poly-Disciplinaire, Université Cadi Ayyad, Route Sidi Bouzid, 4600 Safi, Morocco

Hassan Riahi
Dép. de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, 40000 Marrakech, Morocco

We first present new existence theorems of cone-supremum/infimum for directed convex and/or free disposal subsets in their closure. Then, we provide various conditions through which this kind of subsets admits a cone-maximum/minimum point, the so-called strongly maximal/minimal or ideal efficient points with respect to a cone. Next, we present a unifying result on the existence of these remarkable points, which we apply to extend, improve and unify the existence of an ideal efficient point for hypo/epi-graphical level sets of a given vector-valued function recently considered in joint papers of M. Ait Mansour, C. Malivert, A. Metrane, H. Riahi and M.Théra. A global set-valued analysis on the hypo/epi-profile mappings for general vector-valued maps is also presented. As a consequence, we extend the regularizations and radial epi-derivatives of M. Ait Mansour and H. Riahi [Extended radial epiderivatives of nonconvex vector-valued maps, Optimization, to appear] and F. F. Bazán [Radial epiderivative and asymptotic functions in non-convex vector optimization, SIAM J. Optimization 14 (2003) 284--305] and, henceforth, obtain optimality conditions for global strong Pareto optimums of non-convex nondifferentiable extended vector-valued maps under different assumptions on the ordering cone and the topology of the target space, improving and generalizing the classic global optimality conditions of quasi-convex differentiable extended real-valued functions.

Keywords: Closed convex upward/downward directed sets, downward/upward free disposal sets, cone-supremum/infimum, cone-maximal/minimal points, strongly maximal/minimal points, vector-valued maps, hypo/epi-graphical level sets, semi-continuity, regularizations, exte

MSC: 06A06, 58C07, 49J53,49J52, 90C46, 90C29, 90C26

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