 Solution

The hard-margin constraints are feasible exactly when the classes are separated by a separating hyperplane. Failure after the corrections in part a therefore means the classes overlap.

Introduce slack variables of a support vector machine and solve the soft-margin problemsubject toLarge $C$ strongly penalizes violations and approaches the hard-margin solution when separation is possible. Small $C$ tolerates more violations in exchange for a wider, more strongly regularized margin.