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Suppose $K\in \mathbb{R}^{m\times n}$, $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$.

Let a function $F: \mathbb{R}^m \rightarrow \mathbb{R}$, $F(y)$.

Let $y=Kx$, $G(x)=F(Kx)$.

Suppose $G(x)=F(Kx)$, and $G$ and $F$ are two convex functions.

Let the conjugate of $G$ be $G^*$ and the conjugate of $F$ be $F^*$.

Thus,

$F^*(y)=\max_z\{<z,y>-F(z)\}$.

$G^*(x)=\max_u\{<u,x>-G(u)\}$

Is there any relation between $F^*(y)$ and $G^*(x)$.

Vivian
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This is Theorem 11.23b in Rockafellar & Wets. Let $A \in \mathbb{R}^{m \times n}$. Define $(Af)(u) = \inf \{f(x) | Ax = u\}$ and $ (f^* A^T)(y) = f^*(A^T y)$. If $f = gA$ for a proper, lsc, convex function $g$ such that the range of $A$ meets the effective domain of $g$, then $f^* = \text{cl}(A^T g^*)$. So in your case, $G = FK$, therefore $G^* = \text{cl}(K^T F^*)$.

Ronaldo Carpio
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