Support standard conic form in FileFormats.SDPA

[kocvara]

Hello,

I was re-directed here from

jump-dev/Dualization.jl#134

The point is that when read_from_file reads an SDPA input file, it converts the SDP problem into a "dual" version with additional variables and linear qualities. This makes some solvers (Mosek) very inefficient for the generated model, while the original problem, as stored in the SDPA file would be much easier to solve. Example:

model = read_from_file("sdplib/theta1.dat-s")
set_optimizer(model, Mosek.Optimizer)

generates a model with

  Constraints            : 1275                     
  Scalar variables       : 104   
  Matrix variables       : 1     

while the original formulation would just have

  Constraints            : 104             
  Scalar variables       : 0               
  Matrix variables       : 1     

This is contrary to Matlab/Yalmip treatment of SDPA files:

model = read_from_file("sdplib/thetaG11.dat-s")

gives the problem in format A and

dual_model = dualize(model)

in format B, while Matlab/Yalmip

>> [con, obj] = loadsdpafile('../../../sdplib/thetaG11.dat-s');

gives format B and

[cond,objd,X,free] = dualize(con,obj)

gives format A.

[blegat]

It would be possible to add support for reading directly the standard form. The choice to read the SDPA into the geometric form was a bit arbitrary and we thought there is no need to implement both as we have Dualization and the Dualization step should be negligible compared to the solve time.
In any case, the reading would still be geometric form by default and standard form should be manually chosen. We won't automatically adapt the choice between standard or geometric form depending on the solver.
I'd like to check first whether the performance issue in Dualization could be fixed though.

[odow]

@blegat is this needed in light of the updates in Dualization?

[blegat]

I don't think it is now that the performance issue was fixed in Dualization

[kocvara]

Very fast after the update, thanks! Perhaps just one thing - the user could be made aware of the format read by read_from_file(?) .

[blegat]

Indeed, I choose the geometric conic form because it was the one specified as the primal in http://plato.asu.edu/ftp/sdpa_format.txt.
I explicited this choice in the docstring but it seems we forgot to include the docstring in the docs. Should be resolved by #1797.