In papers I read sometimes that the convex conjugate of $$f(Lx) : L \mbox{ invertible}$$ is

$$ f^*({L^{-1}}^T y)$$

The convex conjugate is defined as

$$ f^*: {\mathbb{R}^N}^* \rightarrow \mathbb{R}: y \mapsto \sup_{x \in \mathbb{R}^N} \ \langle x, y \rangle - f(x) $$

I tried to comprehend how to come up with the conjugate but I am failing. Can anybody help me up? Thanks


For ease of subsequent notations, replace $L$ by $L^{-1}$ in the discussion, and introduce the auxiliary function $g = f \circ L^{-1}: x \mapsto f(L^{-1}x)$.

Now, for any $y \in \mathbb R^N$, one has

$$ \begin{split} g^*(y) &:= \sup_{x \in \mathbb R^N}\langle x, y\rangle - g(y) = \sup_{x \in \mathbb R^N}\langle x, y\rangle - f(L^{-1}x)\\ &= \sup_{z \in \mathbb R^N}\langle Lz, y\rangle - f(z)\;(\text{change of variable }z = L^{-1}x)\\ &=\sup_{z \in \mathbb R^N}\langle z, L^Ty\rangle - f(z) =: f^*(L^Ty).\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \Box \end{split} $$


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