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On second-order composed proto-differentiability of proper perturbation maps in parametric vector optimization problems. (English) Zbl 1487.90596

Let \(P\), \(X\) and \(Y\) be Euclidean spaces, \(K\) be a pointed closed convex cone in \(Y\) with nonempty interior, \(f:P\times X\rightarrow Y\), and \( C:P\rightrightarrows X\). The parametric vector optimization problem under consideration consists in finding the set of proper minimal points, with respect to the ordering induced by \(K,\) of \(F\left( p\right) :=f\left( p,C\left( p\right) \right) \). A set-valued map is said to be second-order composed proto-differentiable when its second-order contingent derivative (in the sense of G. Isac and A. A. Khan [Acta Math. Vietnam. 34, No. 1, 81–90 (2009; Zbl 1180.90246)]) coincides with its second-order composed adjacent derivative (in the sense of Nguyen Le Hoang Anh [Comput. Appl. Math. 38, No. 3, Paper No. 145, 22 p. (2019; Zbl 1438.46050)]). The author gives conditions for the second-order composed proto-differentiability of \(F\), of the proper perturbation map \(\mathcal{P} :P\rightrightarrows Y\), which assigns to \(p\in \mathcal{P}\) the set \( \mathcal{P}\left( p\right) \) of proper minimal points of \(F\left( p\right) \), and of the proper efficient solution map \(\mathcal{S}:P\rightrightarrows X\), defined by \(\mathcal{S}\left( p\right) :=\left\{ x\in C\left( p\right) \mid f\left( p,x\right) \in \mathcal{P}\left( p\right) \right\} \). He also presents some illustrative examples.

MSC:

90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
49J52 Nonsmooth analysis
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