Editing Support vector machine (section)

=== Primal ===
Minimizing {{EquationNote|2|(2)}} can be rewritten as a constrained optimization problem with a differentiable objective function in the following way.

For each <math>i \in \{1,\,\ldots,\,n\}</math> we introduce a variable <math> \zeta_i = \max\left(0, 1 - y_i(\mathbf{w}^\mathsf{T} \mathbf{x}_i - b)\right)</math>. Note that <math> \zeta_i</math> is the smallest nonnegative number satisfying <math> y_i(\mathbf{w}^\mathsf{T} \mathbf{x}_i - b) \geq 1 - \zeta_i.</math>

Thus we can rewrite the optimization problem as follows

<math display="block"> \begin{align}
&\text{minimize } \frac 1 n \sum_{i=1}^n \zeta_i + \lambda \|\mathbf{w}\|^2 \\[0.5ex]
&\text{subject to } y_i\left(\mathbf{w}^\mathsf{T} \mathbf{x}_i - b\right) \geq 1 - \zeta_i \, \text{ and } \, \zeta_i \geq 0,\, \text{for all } i.
\end{align} </math>

This is called the ''primal'' problem.