With hard margin support vector machines (SVMs), it suffices to find the critical points of the Lagrangian $L = \frac{1}{2}||\theta||^2 - \sum_{n=1}^{N} \alpha_n (y^{(n)} (\vec{\theta}^T\vec{x}^{(n)} + \theta_0) - 1)$ over all $ \alpha_n$, where $\vec{\theta}^T\vec{x}^{(n)} + \theta_0 = 0$ is the seperating hyperplane, $y^{(n)}$ is the label (1 or -1), and there are $N$ data points.
I don't have much of a background in optimization, however, so I wanted to ask for methods that are commonly used to solve this optimization problem. Since $\alpha_n$ are unknown, for example, I wasn't sure how we could use gradient descent here, since we cannot compute the derivative numerically given unknown $\alpha_n$.