Question on the implementation

[pbeste18]

Hi-
I noticed that the implementation of ADMM isn't exactly the same as what's in the paper. It seems you are using something along the lines of OSQP with a slight modification. Is there a reason you chose to do it this way (as opposed to making the objective fully linearly separable...) ? I tried implementing the procedure and have not been able to get it to work. I love the paper and believe it will be incredibly useful for my problem. Thank you in advance.

[moehle]

Hi! Thanks for the question, and sorry for the delayed response.

You're exactly right: We follow something a little closer to OSQP in terms of problem formulation that what we have in our own paper. The motivation is subtle, and pretty minor. It's so that the resulting KKT matrix (in equation (13) in our paper, (24) in the OSQP paper) is quasi-definite. This guarantees that the LDL factorization exists, regardless of how we permute the rows and columns of the KKT matrix.

It's actually pretty similar to adding regularization to the bottom right of the KKT matrix, and doing iterative refinement. The clever trick here is that the iterative refinement is essentially embedded in the ADMM iterations, so there aren't two iteration loops (ADMM and iterative refinement).

I'd love to hear more about your application! Feel free to message me.