MDS minimization with gradient descent
Just concatenate all of the variables into a single, long vector.  In your case, you'll have a $2n$-dimensional vector:

$$v = (v_{1,x},v_{1,y},v_{2,x},v_{2,y},\dots,v_{n,x},v_{n,y})$$

where $v_{i,x}$ is the x-coordinate of $v_i$ and $v_{i,y}$ is the y-coordinate of $v_i$.  Now treat this as a function of $v$, i.e.,

$$f(v) = \sum_{i,j} (||v_i - v_j|| - d_{i,j})^2$$

This gives you an objective function $f: \mathbb{R}^{2n} \to \mathbb{R}$, which you can try to maximize.  For instance, you can apply the gradient descent method to $f$ (after computing the partial derivatives of $f$).



That said: for your particular problem, gradient descent might not work very well, unless you happen to have a starting point (initial value for $v_1,\dots,v_n$) that is close to the optimum value.  There might be many local optima that gradient descent could get stuck in.  The only way to know for sure is to experiment with a bit and see what happens.