Proposition [lcc-001T]

April 25, 2024
Eigil Fjeldgren Rischel

Given a minmax problem $L = (X,Y,L)$ where $X$ is a finite-dimensional real vector space, let $L|_{0} = (X, Y \oplus X^*, L \oplus - \langle -,- \rangle )$ Note that $(L|_0)^+(x) = \infty$ when $x = 0$ and $L^+(0)$ otherwise. Thus this amounts to adding a constraint that $x = 0$ . Analogously, define $L|^0 = (L^*|_0)^* = (X \oplus Y^*, Y, L \oplus \langle -, - \rangle )$

Then the Legendre transform $f^* = ((f|_0)^*)^+$ (viewing both $f$ and $f^*$ as minmax problems using the inclusion $\mathsf {Conv} \to \mathsf {Minmax}$ )