LCPs and NCPs for differential inclusions

Hi, thanks for adding me to the project :)

Essentially the problem types LCP and NCP from this repository seem fitting for applications within DI solvers.
Actually, it would be quite cool to differentiate the DI solvers in the end, using the sensitivity tools from here!

The typical requirements are:

• the ability to update the problem fast with new data (similar to SciML's LinearProblem)
• ideally: control over stopping criteria/number of iterations (often a low accuracy solution is sufficient)

Do you think that I could start here by implementing a few classical LCP solvers? For example splitting methods like

• Projected Jacobi method
• Projected Successive Over-Relaxation and Gauss-Seidel

(Down the road, some methods based on the interior point method would be interesting as well.)

With respect to differential inclusions, a typical reference is this book:
Vincent Acary , Bernard Brogliato - Numerical Methods for Nonsmooth Dynamical Systems which is from the authors of SICONSO. Chapter 12 contains a quick view on many methods.

Start by reviewing the repository's existing LCP and NCP problem types and how SciML's LinearProblem handles fast data updates. Define the scope and API for classical LCP solvers, including configurable stopping criteria and iteration limits; done should include agreed solver behavior and tests for the proposed methods.