= Solution
The <kernel trick> replaces inner products $\phi(x_i)^T\phi(x_j)$ by evaluations of a <positive-semidefinite kernel> $k(x_i,x_j)$ without explicitly constructing the feature vectors. In the corresponding reproducing-kernel Hilbert space, the hard-margin problem is
$$
\min_{b,f\in\mathcal H}\frac12\|f\|_{\mathcal H}^2
\quad\text{subject to}\quad
y_i\{b+f(X_i)\}\geq1.
$$
Equivalently, its dual is
$$
\max_{\alpha_i\geq0}
\left\{
\sum_i\alpha_i
-\frac12\sum_{i,j}\alpha_i\alpha_jy_iy_jk(X_i,X_j)
\right\},
\qquad
\sum_i\alpha_iy_i=0.
$$
There is no upper bound $\alpha_i\leq C$ because this is the hard-margin, rather than soft-margin, problem.
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