= Convex perturbation duality
For a <convex perturbation function>, define
$$
\varphi(x)=f(x,0),\qquad\psi(y)=-f^*(0,y),\qquad p(z)=\inf_x f(x,z),\qquad q(v)=\sup_y[-f^*(v,y)].
$$
The primal and dual values are $p(0)$ and $q(0)$; the signed dual marginal $q$ is concave. The identity $p^*(y)=f^*(0,y)$ gives $q(0)=p^{**}(0)$. If $p$ is proper and finite near zero, <subgradients> at zero prove <strong duality> with dual attainment. In finite dimensions the relative-interior condition $0\in\operatorname{ri}(\operatorname{dom}p)$ also suffices, provided $p(0)$ is finite. This does not itself prove primal attainment.
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