Interesting paper: solving SOCPs using a NLP solver

[Transurgeon]

I had a discussion with @kevin-tracy yesterday about the IPOPT project and he pointed me to an interesting paper: https://vanderbei.princeton.edu/ps/socp.pdf.

Notably the author discusses the issue of non-differentiability at optimality of the second-order cone (when the norm is equal to zero). There are a few perturbations methods that are proposed, some convex and others not. The paper then concludes with some discussion of certain applications (which we could try also!) and the reformulations that worked best.

This is making me appreciate a lot the conic solvers that support SOCPs and how they are able to deal with this non-differentiability issue. @dance858 , I think it could be a nice read.

Additional note: Kevin worked on a solver called CALIPSO for non-convex problems with SOC and complementarity constraints. It could be a TODO for us to support conic constraints in NLPs too (but maybe that's too much work).

[dance858]

Thank you for sharing this reference! We have at least one test case for IPOPT and SOCPs, right?

I just read the CALIPSO paper; that's really impressive work! Congratulations @kevin-tracy! Kevin, if we wanted to implement some of the problems from section 5 of the paper in Python, what would be the best way to proceed? Specifically, how can we get realistic problem instances and the exact mathematical formulations?