In my personal field, applied optimal transport for PDEs, we often play the following game, so much so that some of my colleagues and I actually call it Brenier's trick (after Yann Brenier): When miniminzing a convex functional $\rho\mapsto F(\rho)$ over some convex subspace (typically the space of probability measures) with linear constraints $L(\rho)=0$, write the constraints as a supremum of linear functionals over auxiliary mutlipliers, and use a convex/concave minmax theorem (Rockafellar, often, or any variant of von Neumann's minmax theorem in infinite dimension) to exchange \begin{multline*} \inf\limits_\rho\Big\{ F(\rho):\quad L(\rho)=0\Big\}=\inf\limits_{\rho} F(\rho)+ \begin{cases} 0 & \text{if }L(\rho)=0\\ +\infty&\text{else} \end{cases} \\ =\inf\limits_\rho\Big\{ F(\rho)+\sup\limits_\phi \langle L(\rho),\phi\rangle\Big\} =\inf\limits_\rho\sup\limits_\phi F(\rho)+\langle L(\rho),\varphi\rangle \\ =\sup\limits_\phi\inf\limits_\rho F(\rho)+\langle L(\rho),\varphi\rangle \end{multline*} and in turn solve the free optimization problem in $\rho$ and then in $\phi$ to retrieve a lot of significant information about the joint optimizer. For example this can be used to retrieve in just a few lines, and at least heuristically, the right equations for Wasserstein geodesics (a forward continuity coupled with a backward Hamilton-Jacobi equation), the so-called Otto's calculus, and many other variants thereof for a whole variety of models. Of course this is nothing but a Lagrange multiplier method for constrained optimization, but it shows up so often in applied optimal transport problems that I thought it would be worth mentioning here.

In my personal field, applied optimal transport for PDEs, we often play the following game, so much so that some of my colleagues and I actually call it Brenier's trick (after Yann Brenier): When miniminzing a convex functional $\rho\mapsto F(\rho)$ over some convex subspace (typically the space of probability measures) with linear constraints $L(\rho)=0$, write the constraints as a supremum of linear functionals over auxiliary mutlipliers, and use a convex/concave minmax theorem (Rockafellar, often, or any variant of von Neumann's minmax theorem in infinite dimension) to swap $\inf\sup=\sup\inf$ as \begin{multline*} \inf\limits_\rho\Big\{ F(\rho):\quad L(\rho)=0\Big\}=\inf\limits_{\rho} F(\rho)+ \begin{cases} 0 & \text{if }L(\rho)=0\\ +\infty&\text{else} \end{cases} \\ =\inf\limits_\rho\Big\{ F(\rho)+\sup\limits_\phi \langle L(\rho),\phi\rangle\Big\} =\inf\limits_\rho\sup\limits_\phi F(\rho)+\langle L(\rho),\varphi\rangle \\ =\sup\limits_\phi\inf\limits_\rho F(\rho)+\langle L(\rho),\varphi\rangle. \end{multline*} One can then solve the free optimization problem in $\rho$ and next in $\phi$ to retrieve a lot of significant information about the joint optimizers. For example this can be used to retrieve in just a few lines, and at least heuristically, the right equations for Wasserstein geodesics (a forward continuity equation coupled with a backward Hamilton-Jacobi equation), the so-called Otto's calculus, and many other variants thereof for a whole variety of models. Of course this is nothing but a Lagrange multiplier method for constrained optimization, but it shows up so often in applied optimal transport problems that I thought it would be worth mentioning here.