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@jemoka / Jemoka Knowledge Base / wiki/concepts/dual_transformation.md
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--- 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 & \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 “norm conjugate”: \begin{align} \max_{v}\quad & b^{T}v \\ \textrm{s.t.} \quad & A^{T}v = 0 \\ & \norm{v} \leq 1 \end{align}