K-epiderivatives for set-valued functions and optimization

About contingent epiderivatives

“…On the one hand, using different kinds of minimal elements, see [11,13]; on the other hand, considering different types of tangent cones, see [3,6,12,15]. However, all definitions have the same mathematical structure.…”

Higher order weak epiderivatives and applications to duality and optimality conditions

“…In general, since the epigraph of a set-valued map has nicer properties than its graph, it is useful to employ the epiderivatives in set-valued optimization. As to other concepts of epiderivatives for set-valued maps and applications to optimality conditions, one can refer to [6][7][8][9] and the references therein. Recently, Jahn et al [10] introduced second-order contingent epiderivative and generalized contingent epiderivative for set-valued maps and obtained some second-order optimality conditions based on these concepts.…”

A multiplier rule in set-valued optimisation

“…In recent years a great deal of attention has been given to the characterisation of the weak-minimality for (Po) and (Pj) by employing various notions of derivatives for set-valued maps, see [2,3,4,7,8,11,12,13] and the references therein. A common strategy adopted in these works is to use direct arguments, based on the derivatives chosen for the set-valued maps involved, to verify a claim that the images of the derivatives do not intersect with certain open cones.…”

“…A common strategy adopted in these works is to use direct arguments, based on the derivatives chosen for the set-valued maps involved, to verify a claim that the images of the derivatives do not intersect with certain open cones. Here it should be noted that such a disjunction is given 94 A.A. Khan and F. Raciti [2] in the image spaces. Further, this disjunction is then combined with some separation arguments and multiplier rules are obtained.…”

“…It suffices to show that there exists m 6 N such that x + \ n x n £ F [y -int(K)] f or every n ^ m. By the definition of DiF(x,y){-), there exist (y n ) C K with y n -> y and ni € N such that y + A n y n € F(x + A n i n ) for every n ^ n\. Since y £ -int(A') and y n -> y, there exists n 2 G N such that X n y n £ -int(i^) for every n ^ ra 2 . This implies that y + X n y n £ F(x + A n x n ) D (y -int(A')) for n ^ m := max{ni,n 2 }.…”