Theorem 1 — strong duality $I=J$, a dual optimizer $(\lambda^*,\varphi_{\lambda^*})$ and complementary slackness

Let $S$ be a Polish space, $\mu$ a Borel probability measure on $S$, $c : S\times S\to[0,\infty)$ a lower semicontinuous cost with $c(x,y)=0$ if and only if $x=y$ (Assumption (A1)), $f : S\to\mathbb R$ upper semicontinuous and $\mu$-integrable (Assumption (A2)), and $\delta>0$. Let $I=\sup\{\int f(y)\,d\pi(x,y):\pi\in\Phi_{\mu,\delta}\}$ be the worst-case expectation of $f$ over transport plans out of $\mu$ with cost at most $\delta$, and $J=\inf\{\lambda\delta+\int\varphi\,d\mu:(\lambda,\varphi)\in\Lambda_{c,f}\}$ its dual. For $\lambda\ge0$ let $\varphi_\lambda(x)=\sup_{y\in S}\{f(y)-\lambda c(x,y)\}$. Then:

1. (a) Strong duality.

2. (b) Dual attainment. There is $\lambda\ge0$ such that $(\lambda,\varphi_\lambda)\in\Lambda_{c,f}$ and $J(\lambda,\varphi_\lambda)=J$.
3. (b) Complementary slackness, "if". For $\pi^*\in\Phi_{\mu,\delta}$ and $(\lambda^*,\varphi_{\lambda^*})\in\Lambda_{c,f}$, if

then $\pi^*$ is a primal optimizer ($I(\pi^*)=I$), $(\lambda^*,\varphi_{\lambda^*})$ is a dual optimizer ($J(\lambda^*,\varphi_{\lambda^*})=J$), and $I(\pi^*)=J(\lambda^*,\varphi_{\lambda^*})$. 4. (b) Complementary slackness, "only if". Conversely, if $\pi^*$ and $(\lambda^*,\varphi_{\lambda^*})$ are as in 3., are primal and dual optimizers with $I(\pi^*)=J(\lambda^*,\varphi_{\lambda^*})$, and $J(\lambda^*,\varphi_{\lambda^*})<\infty$, then (8a) and (8b) hold.

Theorem 1 says that the worst-case expectation over an optimal-transport ball on a general Polish space, with only lower semicontinuous costs and upper semicontinuous integrable $f$, equals a one-dimensional dual problem over $\lambda$, and it characterizes worst-case transport plans.

Formalization Note The finiteness hypothesis $J(\lambda^*,\varphi_{\lambda^*})<\infty$ in item 4 is implicit in the paper: when $f$ outgrows $c$ on a set of positive $\mu$-measure, $\varphi_{\lambda}\equiv\infty$ there, $I=J=\infty$, and an optimal $\pi^*$ with $I(\pi^*)=\infty$ can exist while (8a) is impossible ($f(y)-\lambda^*c(x,y)$ is finite). The direction "if" holds without it. By weak duality, "primal and dual optimizers satisfying $I(\pi^*)=J(\lambda^*,\varphi_{\lambda^*})$" is equivalent to $I(\pi^*)=J(\lambda^*,\varphi_{\lambda^*})$ alone; the statement spells out all three equalities. In (8b) the cost integral is converted to a real number, which is safe because it is at most $\delta$. The space $S$ is a Polish space with its Borel σ-algebra; the cost $c$ is real-valued and written curried, $c\,x\,y = c(x,y)$; (A1) is the structure AssumptionA1; (A2) is the pair of hypotheses UpperSemicontinuous f and Integrable f μ. Values that can be infinite ($I$, $J$, $I(\pi)$, $J(\lambda,\varphi)$, $\varphi_\lambda$) live in EReal; an integral of an extended-real function is $\int\varphi^+ - \int\varphi^-$ with lower Lebesgue integrals, and $\infty-\infty$ evaluates to $-\infty$, so a coupling with $\int f^-\,d\pi=\infty$ never raises the primal supremum (the paper's footnote 2 reading).