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
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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