Given functions $f$ and $g$, P1 and P2 below are said to be two equivalent formulations of the support vector machine. How does one show the equivalence?
P1: \begin{align} \beta^*(\lambda)=\text{argmin}_\beta &\left(\sum_{i=1}^n\big(1-f(\beta;x_i)\big)_++\lambda g(\beta)\right) \\ & g(\beta^*(\lambda))=0 \end{align} where $\lambda\ge0$ is a constant and $u_+:=\max(0,u)$.
P2: \begin{align} \max_\beta\ &M \\ &g(\beta) = 0 \\ &f(\beta;x_i) \ge M(1-\epsilon_i) \\ &\epsilon_i\ge0,\ \sum_{i=1}^n\epsilon_i\le C \\ & \forall i \end{align} where $C\ge0$ is a parameter.
Obviously $\lambda g(\beta)$ in P1 is the Lagrange multiplier. I do not see immediately how $M$ in P2 emerges from P1.