 Squared-slack support vector machine

The squared-slack support vector machine uses constraints $y_i(w\cdot x_i+b)\ge 1-{\xi }_i$. With an unpenalized intercept, its Lagrangian dual problem maximizes ${\sum }_i{\alpha }_i-\frac{1}{2}{\sum }_{i,j}{\alpha }_i{\alpha }_j{y}_i{y}_{j}K(x_i,x_j)-\frac{1}{4C}{\sum }_i{\alpha }_i^2$ subject to ${\alpha }_i\ge 0$ and ${\sum }_i{\alpha }_i{y}_i=0$. The square adds a diagonal penalty in the dual rather than a finite upper box constraint. The Karush-Kuhn-Tucker conditions give $w={\sum }_i{\alpha }_i{y}_i\varphi (x_i)$ and ${\xi }_i={\alpha }_i/(2C)$ for a feature map realizing the positive-semidefinite kernel $K$.