Lagrange Duality in Set Optimization

Based on the complete-lattice approach, a new Lagrangian duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. "Saddle sets" replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand and the existence of a saddle set for the Lagrangian on the other hand

In vitro and in vivo characterization of neural stem cells

Neural stem cells are defined as clonogenic cells with self-renewal capacity and the ability to generate all neural lineages (multipotentiality). Cells with these characteristics have been isolated from the embryonic and adult central nervous system. Under specific conditions, these cells can differentiate into neurons, glia, and non-neural cell types, or proliferate in long-term cultures as cell clusters termed “neurospheres”. These cultures represent a useful model for neurodevelopmental studies and a potential cell source for cell replacement therapy. Because no specific markers are available to unequivocally identify neural stem cells, their functional characteristics (self-renewal and multipotentiality) provide the main features for their identification. Here, we review the experimental and ultrastructural studies aimed at identifying the morphological characteristics and the antigenic markers of neural stem cells for their in vitro and in vivo identification

Subdifferential and optimality conditions for the difference of set-valued mappings

In this paper, an existence theorem of the subgradients for set-valued mappings, which introduced by Borwein (Math Scand 48:189-204, 1981), and relations between this subdifferential and the subdifferential introduced by Baier and Jahn (J Optim Theory Appl 100:233-240, 1999), are obtained. By using the concept of this subdifferential, the sufficient optimality conditions for generalized D. C. multiobjective optimization problems are established. And the necessary optimality conditions, which are the generalizations of that in Gadhi (Positivity 9:687-703, 2005), are also established. Moreover, by using a special scalarization function, a real set-valued optimization problem is introduced and the equivalent relations between the solutions are proved for the real set-valued optimization problem and a generalized D. C. multiobjective optimization problem