= Sensitivity analysis in convex perturbation duality
An optimal dual variable $y^*\in\partial p(0)$ bounds the effect of perturbing a convex value function:
$$
p(z)\geq p(0)+\langle y^*,z\rangle.
$$
If $p$ is finite convex near zero, $p'(0;d)=\max_{y\in\partial p(0)}\langle y,d\rangle$. A singleton <subdifferential> gives differentiability and a first-order expansion. For an upper-bound constraint relaxed by $z$, the derivative equals the negative of the nonnegative <Lagrange multiplier>; relaxing the bound can decrease the minimum value.
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