---
title: "dual transformation"
type: concept
source: https://www.jemoka.com/posts/kbhdual_transformation/
confidence: high
status: active
---

introducing new variables Consider the original problem:
\begin{align} \min_{x}\quad &amp; \norm{Ax - b} \end{align}
Now, we can reformulate and write \(y = Ax - b\). Now, remember the conjugate of the norm is:
\begin{equation} \norm{z} = \begin{cases} 0, \norm{z} \leq 1\\ \infty \end{cases} \end{equation}
And remember the dual is some massaging of the conjugate. And thus we can formulate a &ldquo;norm conjugate&rdquo;:
\begin{align} \max_{v}\quad &amp; b^{T}v \\ \textrm{s.t.} \quad &amp; A^{T}v = 0 \\ &amp; \norm{v} \leq 1 \end{align}
