I am studying Support Vector Machines from ISLR and I am stuck on the following optimization problem (page 346) -

\begin{align} \underset{w}{\text{maximize }} & \frac{M}{\|w\|} \\[6pt] \text{subject to } &y^{(i)}(w^Tx^{(i)} +b) \geq M(1 - \epsilon_i), \; i = 1, \ldots, m.\\[6pt] &\epsilon_i \geq 0, \; i = 1, \ldots, m.\\[6pt] &\sum_{1}^m \epsilon_i\ \leq C \end{align}

Here $M$ is the functional margin.

How is this optimization problem equivalent to the more conventional optimization problem (ESL page 439) - \begin{equation*} \begin{aligned} & \underset{w}{\text{minimize}} & & \|w\|^2 + C \sum_{1}^{m} \epsilon_i\\ & \text{subject to} & & y^{(i)}(w^Tx^{(i)} +b) \geq 1 - \epsilon_i, \; i = 1, \ldots, m. & & \epsilon_i \geq 0, \; i = 1, \ldots, m. \end{aligned} \end{equation*}

I understand how maximization became minimization and how M became 1. However, I am unable to understand how did $C$ went up into the objective function (along with the $\epsilon_i$)

I apologize if this is the wrong place to post this question. Kindly migrate it to a more suitable forum, if needed.


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