Speed-up benefits of Convex Separable Formulations in MOSEK ?

[Tường Nguyễn]

  When formulating a convex optimization problem for MOSEK, does using a separable formulation offer any inherent computational speed-up?

Particularly, I was wondering if  this convex separable structure allows MOSEK to:

• Exploit sparsity more effectively in the KKT matrix?

• Benefit from parallelization or decomposition techniques more readily?

I understand that most convex problems can be reformulated into a conic form, but I am curious if preserving the separable structure in the input model offers any advantages before the final internal transformation. 

[Erling Andersen]

Since, you can only only input convex problems on conic form, then I am not sure which choice you are thinking about.

Maybe, presenting a concrete example would help. Could be

https://docs.mosek.com/whitepapers/qmodel.pdf

answer your question.

Also I suggest you read 

https://link.springer.com/article/10.1007/s10107-002-0349-3

and

https://link.springer.com/article/10.1007/s10107-021-01631-4

if you want to know how Mosek exploits the conic structure.

[Tường Nguyễn]

Thank you, what I  mean is that, if I can find a formulation such that all decision variables are separable like this, then can mosek take advantage of the fact that everything is cleanly separated to speed up the solving process. 


Thank you and best regard!

[Erling D. Andersen]

Since you have to input the problem using cones, then the formulation on the right is closer to what is possible as long as you use Mosek.

Stating a problem on conic form has many advantages e.g.

* The primal-dual algorithm can exploit conic duality.

* The primal-dual algorithm can exploit the power of symmetric cones (=quadratic cones).

* Presolve is easier for conic problems than for general nonlinear problems.

* ...

IMO, stating a convex problem in conic form will, in most cases, be the best you can do.

[Tường Nguyễn]

Thank you, I understand it now !