For any solution \(x^\star\) to the original problem, we have \(E x^\star = 0\), and thus \(P_E x^\star = x^\star\); since \(G x^\star \geq 0\), the reformulated system \(G P_E x^\star \geq 0 \Leftrightarrow G x^\star \geq 0\) is also satisfied. Moreover, for any solution \(\tilde{x}\) to the reformulated system, it suffices to consider \(x^\star = P_E \tilde{x}\) to obtain a solution to the original system.
The second reformulation, based on projectors, is interesting to me because the reformulated system preserves the primal solution space, while eliminating equality constraints, which tend to be hard to handle with Lagrangian methods.
Perceptron as a dual ascent
This insight is copied straight from Soheili's dissertation
(Elementary Algorithms...). The vanilla Perceptron method takes a system (normalised such that each row has unit norm) of the form