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 problem
(1)
subject to
(2)
Large strongly penalizes violations and approaches the hard-margin solution when separation is possible. Small tolerates more violations in exchange for a wider, more strongly regularized margin.

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