TAILIEUCHUNG - Optimality conditions via weak subdifferentials in reflexive Banach spaces
In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. | Turk J Math (2017) 41: 1 – 8 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Optimality conditions via weak subdifferentials in reflexive Banach spaces Sara HASSANI1,2,∗, Musa MAMMADOV1,2 , Mina JAMSHIDI3 1 Federation University of Australia, Ballarat, Victoria, Australia 2 National Information and Communications Technology Australia, Sydney, Australia 3 Faculty of Science and New Technologies, Kerman Graduate University of Advanced Technology, Kerman, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. Key words: Supporting cone, weak subdifferential and nonconvex optimization 1. Introduction The generalization of concepts of ordinary derivatives and normal cones plays an important role in the study of necessary and sufficient conditions of optimality for nonsmooth and nonconvex optimization problems. The notion of subdifferentials was introduced by Rockafellar [17] to deal with optimization problems involving convex and nonsmooth functions. Since then, different notions of subdifferentials and normal cones have been introduced, which are applicable for different classes of optimization problems. We mention here the concepts of the Fr´echet subdifferential [3, 15], Clarke’s subdifferential [4], and limiting Fr´echet subdifferentials [15, 16]. In [1, 2], the notion of a supporting cone was introduced and led to so-called weak subdifferentials. To eliminate the duality gap in nonconvex programming, an augmented Lagrangian is used that is constructed by supporting cones [2, 5, 6]. Later in [12], the concept of an augmented dual cone was introduced in
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