[Enhancement] Use proximal operator solution in single-column decomposition

[TioMinho]

Summary

The SLS problem (with $\mathcal{H}_2$-norm) has a closed-form solution in the unconstrained case when a single-column decomposition is used for the separable problems. Specifically, the solution falls into a linear system which can be solved offline. This could be a way to improve dramatically the performance of synthesis in a broad wide of cases.

Proposal

• [ ] Implement a 'fallback' internal function to be called when single-column decomposition is used. Initially, we can try simply solving the associated (sparse) linear system.
• [ ] Compute and store a QR/Cholesky sparse decomposition of the linear operator that solves the optimization. This should be also be done only once in the case of output-feedback SLS.

Notes

• In case the performance increase is too dramatic, maybe we should evaluate the need for Distributing the columns. Still, there is a semantic advantage of distribution (=distributed synthesis where different $\mu$-Controllers compute different portions of the closed-loop map)

Implemented in: TBD

[TioMinho]

I have partially addressed this issue in commit 418873004cbbbc09a84dde80190641e4f1d86d05.

The solution of the KKT System for the equality constrained quadratic program (ECQP) from the H_2 single-column problem is implemented. However, the code is not memory-efficient, as it creates the Hessian and constraint matrices through a lot of allocations.

The issue remains open until memory usage is made more efficient (and maybe the function is further specialized)