Convex perturbation function ($f(x,z)$)

A jointly proper convex function $f(x,z)$ represents an optimization problem at perturbation $z=0$ and a family of nearby problems at other $z$. Its primal value function is $p(z)={\inf }_{x}f(x,z)$. Relaxing an inequality by $z$ gives the example $f(x,z)=J(x)+{\delta }_{(-\infty ,0]}(r(x)-\sigma -z)$, using an indicator functional. The resulting convex perturbation duality expresses multipliers as supporting subgradients of the value function.

For a convex perturbation function, defineThe primal and dual values are $p(0)$ and $q(0)$; the signed dual marginal $q$ is concave. The identity $p^{\ast }(y)=f^{\ast }(0,y)$ gives $q(0)=p^{\ast \ast }(0)$. If $p$ is proper and finite near zero, subgradients at zero prove strong duality with dual attainment. In finite dimensions the relative-interior condition $0\in \mathrm{ri}(\mathrm{dom}p)$ also suffices, provided $p(0)$ is finite. This does not itself prove primal attainment.

An optimal dual variable $y^{\ast }\in \partial p(0)$ bounds the effect of perturbing a convex value function:If $p$ is finite convex near zero, $p^{\prime }(0;d)={\max }_{y\in \partial p(0)}⟨y,d⟩$. 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.