Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces (Q1029465)

Let \(X\) denote a normed space and \(Y,Z\) Banach spaces partially ordered by closed, convex, pointed cones \(C\) and \(K\), respectively. The authors consider the following general vector optimization problem: \[ \text{minimize }f(x)\text{ subject to }g(x)\in-K,\;x\in S, \] where \(S\) is a nonempty set, \(f:S\to Y\), \(g:S\to Z\). A scalarization technique for the computation of the contingent epi/hypoderivatives of set-valued maps is proposed. The technique makes possible to prove new computational formulas for epi/hypoderivatives of set valued maps taking values in partially ordered Banach spaces with a shrinking Schauder basis. The formulas are applied to give necessary optimality conditions for weak and proper minimizers of the generalized optimization problem mentioned above. The conditions extend the classical results from smooth multiobjective optimization problems.