Perturbation function ($f(x,u)$)

A perturbation function encodes a family of objectives and constraints indexed by a perturbation $u$, with the original problem at $u=0$. Extended-real values encode constraints. Its marginal $p(u)={\inf }_{x}f(x,u)$ measures the optimal value's response to perturbation; jointly convex data give convex perturbation duality.