Second-Order Sensitivity of Efficient Solution and Marginal Maps in Parametric Vector Optimization with Set Constraints

We develop a second-order sensitivity theory for the efficient solution map $S$ of a parametric vector optimization problem $\min_C f(p,x)$ subject to $x\in H(p)$. The main point is the passage from efficient values to efficient decisions. Under a value-to-decision error bound (VDB), second-order information for the marginal map $\Phi$ lifts to a second-order Dini formula for $S$. We first work in the abstract inclusion model $x\in H(p)$, where outer and inner estimates yield second-order semi-derivability of $S$. We then specialize to structured feasible maps $H(p)=\{x\in\Omega:g(p,x)\in D\}$. Under Robinson metric regularity along $\Omega$, second-order regularity of $\Omega$ and $D$, and directional second-order semi-derivability of the data, we obtain explicit formulas for $\DD H$, $\DD\Phi$, and $\DD S$. The framework is specialized to polyhedral inequality/equality systems and illustrated by a robust multi-objective portfolio model and a DC-dispatch model for electricity markets, with a brief discussion of complementarity-based extensions.