Soft margin nonlinear SVM classification

A soft margin nonlinear SVM classifier is obtained by introducing slack variables and the capacity C. As with Eq. [47], the dual problem is 

Another formulation of support vector machines is the v-SVM in which the parameter C is replaced by a parameter v G [0, 1] that is the lower and upper bound on the number of training patterns that are support vectors and are situated on the wrong side of the hyperplane. v-SVM can be used for both classification and regression, as presented in detail in several reviews, by Scholkopf et al., Chang and Steinwart, and Chen, Lin, and 

With these notations, the primal Lagrangian function of this problem is 

We substitute Eqs. [83] and [84] into Eq. [82], using A, p 8 0, and then we substitute the dot products with kernels, to obtain the following quadratic optimization problem  

From these equations, it follows that the v-SVM classifier is 

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