How do I calculate the optimal hyperplane of SVMs by hand?

I am trying to get a better understanding of SVMs and their optimization process. I understand that we have a constrained optimization problem that we have to solve with Lagrange multipliers.

The formula that we need to maximize is

where the alphas are the Lagrange multipliers, the x-vectors the vectors of the training data and the scalar y is the label for the positive/negative class {-1, 1}. N is the total number of training vectors.

In my mind, the alphas are optimized in such a way that non-relevant vectors (non-support vectors) are equal to zero and the real support vectors get a value greater than zero. In most tutorials, however, the lecturer stops and leaves the optimization to an algorithm of sorts. I want to understand how the machine calculates these parameters with only a few, say four or five vectors.

Is it correct that the first thing that I have to do is to take the derivative of L w.r.t. every alpha, so that I get a system of linear equations that I can solve for every alpha?

To make my thoughts clear, consider the case with only 2 support vectors, where the first vector has class 1 and the second vector class -1. The system would look like this:

I set all equations equal to zero because I want to find the maximum. This leaves me with a system of linear equations, when reordering the alphas:

I would be grateful for some advice, if this is correct or not. I have stumbled across calculations similar to this (e.g. https://axon.cs.byu.edu/Dan/478/misc/SVM.example.pdf), but I cannot reproduce the equations there with my approach.

Also, I would be interested in how I am going to continue from here. I know that I can calculate the normal vector of the plane with all the alphas, but I do not know how to calculate the offset b of the plane.

$$\def\o{{\tt1}}\def\D{{\rm Diag}}\def\p#1#2{\frac{\partial #1}{\partial #2}}$$Let $$X$$ be the matrix whose colums are the $$x_k$$ vectors. Let's also define the vectors $$(a,y)$$ whose components are $$(\alpha_k,y_k)$$, and their corresponding diagonal matrices \eqalign{ A = \D(a) \quad\quad Y=\D(y) \\ } It will also be convenient to define the symmetric matrix $$S=YX^TXY$$ Write the Lagrangian in terms of these new variables and the all-ones vector $$\o$$.
Then calculate the differential and the gradient. \eqalign{ L &= \o:a - \tfrac 12S:aa^T \\ dL &= \o:da - \tfrac 12S:\left(da\,a^T+a\,da^T\right) \\ &= \o:da - \tfrac 12\left(S+S^T\right):da\,a^T \\ &= \o:da - S:da\,a^T \\ &= \left(\o - Sa\right):da \\ \p{L}{a} &= \left(\o - Sa\right) \\ } Set the gradient to zero and solve for the vector of Lagrange multipliers \eqalign{ Sa = \o \quad\implies\quad a = S^{-1}\o \\\\ }
In the above, a colon is used as a convenient product notation for the trace function, i.e. $$A:B = {\rm Tr}(A^TB)$$ Note that when $$(A,B)$$ are vectors, this reduces to the ordinary dot product.
The transpose and cyclic properties of the trace allow the terms in such a product to be rearranged in many equivalent ways, e.g. \eqalign{ A:B &= B:A = B^T:A^T \\ BC:A &= B:AC^T = C:B^TA \\ }