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

Abstract

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 \(Φ\) 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 \(H(p)=\{x\inΩ:g(p,x)\in D\}\). Under Robinson metric regularity along \(Ω\), second-order regularity of \(Ω\) and \(D\), and directional second-order semi-derivability of the data, we obtain explicit formulas for \(\DD H\), \(\DDΦ\), 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.