I still think that utilizing ideas from ROCK-type methods (extensions to Runge-Kutta Chebyshev for better stability in the complex plane) and other methods utilized for stabilizing stiff ODEs could be helpful, since I haven’t really found methods that are very effective for highly ill-conditioned local optimization problems where a nonlinear preconditioner is hard to specify. That said, since you don’t care about “accuracy”, you might as well use a low order method, which then means that implicit Euler or Rosenbrock Euler methods might be the things to look at. And indeed, if you search the literature, you find recent research in proximal optimization methods have a lot of links to implicit Euler:
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