Higher-order tangent epiderivatives and applications to duality in set-valued optimization
Let \(X,Y\) be real Banach spaces and \(C\) a pointed closed convex cone in \(Y\). For a set-valued mapping \(F:X\to 2^Y\) one defines the graph and the epigraph of \(F\) by \[\mathrm{gr}\, F:=\{(x,y)\in X\times Y:y\in F(x)\}\] and \[\mathrm{epi}_C F=\{(x,y)\in X\times Y: y\in F(x)+C\}\,,\] respectively. Two approaches are known to define contingent derivatives of \(F\) -- using linear approximation of its graph or of its epigraph. In the present paper the authors adopt the second approach -- they consider higher-order tangent epiderivatives, study their properties and give applications to set-valued optimization, namely duality for a mixed constrained optimization problem.
optimization problem with mixed constraints
scientific article; zbMATH DE number 7437469
scientific article