How does SVM handle non-separable data mathematically?

SVM introduces slack variables to allow constraint violations in non-separable cases.

For non-separable data, constraints y_i(w·x_i + b) ≥ 1 are relaxed using ξ_i ≥ 0. The optimization becomes minimizing ||w||² + C Σ ξ_i. This allows controlled misclassification while still maximizing margin.