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

@misc{bao2026secondorder,
  title = {{Second-Order Sensitivity of Efficient Solution Maps in Parametric Vector Optimization with Set Constraints}},
  author = {N. X. D. Bao and Tan H. Cao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.28965},
  eprint = {2606.28965},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.28965},
}

TY  - UNPB
TI  - Second-Order Sensitivity of Efficient Solution Maps in Parametric Vector Optimization with Set Constraints
AU  - N. X. D. Bao
AU  - Tan H. Cao
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2606.28965
UR  - https://arxiv.org/abs/2606.28965
AB  - We study second-order fixed-ray Dini sensitivity of the efficient solution map S for the parametric vector problem min_C f(p,x) subject to x∈ H(p). Under a value-to-decision error bound (VDB) -- an inverse stability estimate -- the second-order semi-derivative of the marginal map Φ transfers to an exact second-order Dini formula for S. The result separates value sensitivity from the inverse recovery of efficient decisions. This mechanism allows a nonsingleton efficient-value fiber and requires neither injectivity nor strict monotonicity of ∇_x f. For structured feasible maps H(p)={x∈Ω: g(p,x)∈ D}, we derive primitive-data formulas for D^2H, D^2Φ, and D^2S under Robinson metric regularity, second-order regularity of Ω and D, objective-aware recession conditions, and directional second-order semi-derivability of the data. Polyhedral inequality/equality systems give explicit specializations. A parametric multi-objective portfolio family verifies (VDB) with an explicit constant, obtained directly from the model's feasible directions, while a two-branch example shows how the formula operates on a nonsingleton decision fiber.
ER  -