with a unique optimal solution at \(x = \frac{y^{\top}A}{\|y^{\top}A\|_2}\), which is exactly the (rescaled) solution found by the Perceptron. If \(y^{\top}A = 0\), Farkas's lemma tells us the system is infeasible.
The Lagrangian perspective is interesting because it lets us optimise over different domains than the unit ball, e.g., the unix box of 0/1 solutions, or any other domain with a linear optimisation oracle. It also helps us understand why the Perceptron's implicit dual ascent method works so well over the unit ball: the Lagrangian subproblem always has exactly one optimal solution, so the dual is always differentiable! That doesn't meant we can "just" use accelerated smooth optimisation methods: I expect the dual's Lipschitz constant is pretty awful, so we might still want to add a smoothing term.
Chubanov's method as a dual ascent over a projection reformulation
The reformulation for