Ideas to make ProximalProjection more robust

[f0uriest]

Right now we still have cases where the proximal optimizer keeps trying smaller and smaller steps until eventually it quits saying Warning: A bad approximation caused failure to predict improvement.

This seems to be due to inconsistent linearizations or a possibly instability, where small changes in the boundary (the x variable the optimizer sees) lead to large changes in the volume (the f values the optimizer sees).

Some possible ideas to make this better:

• Have some "patience" criteria, where if the perturb/solve stuff doesn't find an acceptable step in N steps, we go back and do a full continuation solve.
• Have some check on the step size in both boundary and volume and reject steps if the ratio is too large.
• Use the 2nd order perturbations to do the above, where we reject a step if the 2nd order correction is larger than some fraction of the first order step (similar to geodesic acceleration).
• If several attempts don't find an acceptable step, return a smaller function value to force it to accept a "bad" step in the hopes that linearizing around the new point will be better.

[dpanici]

Case for a precise-QA-like equilibrium in free boundary, each step in the proximal just gives a bad approximation, files are here: https://drive.google.com/drive/folders/19-F7cVb1yxks42eCI1Nx7DlxjCkyxaaf?usp=drive_link

[dpanici]

@dpanici check thru notebooks and link here along with current commit that it occurs at

[dpanici]

• https://desc-docs--1189.org.readthedocs.build/en/1189/notebooks/tutorials/free_boundary_equilibrium.html#Solovev-Tokamak SOLOVEV free bdry (on PR Add easier to read print option #1189 )
• https://desc-docs--1189.org.readthedocs.build/en/1189/notebooks/tutorials/advanced_optimization.html#Multigrid-method-with-proximal-optimizer QH proximal optimization (again on Add easier to read print option #1189 )