ϕ (x_{i})^{T} ϕ (x_{j})

k (x_{i}, x_{j})

without explicitly constructing the feature vectors. In the corresponding reproducing-kernel Hilbert space, the hard-margin problem is

min_{b, f \in H} \frac{1}{2} ∥ f ∥_{H}^{2} subject to y_{i} {b + f (X_{i})} \geq 1.

Equivalently, its dual is

max_{α_{i} \geq 0} {\sum_{i} α_{i} - \frac{1}{2} \sum_{i, j} α_{i} α_{j} y_{i} y_{j} k (X_{i}, X_{j})}, \sum_{i} α_{i} y_{i} = 0.

There is no upper bound

α_{i} \leq C

because this is the hard-margin, rather than soft-margin, problem.

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