[Csdp] Scaling the problem to mitigate the solver "giving up"

[Hans Mittelmann]

Hi Jed,
thanks for sending a nice challenge problem. I am running all the SDP solvers at neos-server.org <http://neos-server.org/>
and have several papers in which I solved ill-conditioned SDPs with increased accuracy.
I ran your problem locally with SDPT3 (fail), SeDuMi(output attached), and am still running it with
SDPA-DD. I am confident the latter will work but it takes time. You can see the accuracy with SeDuMi
and options to obtain higher accuracy. It may be possible to get a little more accuracy or have 
success with another solver I have not tried, yet. Let me know how important it is for you to solve
this and similar problems. Regarding CSDP, Brian can tell you best what do. 
Hans Mittelmann


> On Sep 14, 2017, at 2:47 PM, Jedediah Fry <mfry90 at vt.edu> wrote:
> 
> Hi CSDP users,
> 
> I'm using CSDP to solve large semidefinite programs and have run into some difficulties with the solver.  Typically, the solver either "gives up" with a solution that satisfies my LMIs and has an objective with a somewhat large duality gap (-1e-01 relative real gap), or "gives up" with a solution that does not satisfy my LMIs.   In the latter case, I believe there exists a feasible solution, but the solver gives up with a solution outside the feasible set. In both cases, it typically reports "Stuck at edge of dual feasibility."
> 
> People have talked about similar issues in the CSDP archives, and Dr. Borchers states that scaling is often a good fix (https://list.coin-or.org/pipermail/csdp/2014-January/000125.html <https://urldefense.proofpoint.com/v2/url?u=https-3A__list.coin-2Dor.org_pipermail_csdp_2014-2DJanuary_000125.html&d=DwMFaQ&c=Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=kv9YjHgx2Q3T85jYlx6ozVwGCbdfroU5MhEqr2aByKQ&m=NbDTYgDutbbTwGYD3E_9dY7-6hX7Wu_M825HFuuZexk&s=YCW7GBNNYKnc6k28Pbs2R8oi6oYMNOAmgG7KvCInbgo&e=>). I'm unfamiliar with this approach, is there a good reference that describes different ways to scale A and c, and how to unscale the solution obtained from solving the scaled problem?
> 
> Also, what is the criterion that determines the solver should "give up?" Can I change any solver parameters to prolong the effort (the routine always "gives up" before my specified maximum iterations, and I'd the solver to see if it can chug on the problem a little bit longer).
> 
> For reference, I've attached a problem and CSDP cannot solve but I believe is solvable.
> 
> Thanks,
> Micah Fry
>  ​
>  fry_prob.dat-s <https://urldefense.proofpoint.com/v2/url?u=https-3A__drive.google.com_a_vt.edu_file_d_0B-2DbHmaZTGpTrX1QtOVFfZ2hsd1U_view-3Fusp-3Ddrive-5Fweb&d=DwMFaQ&c=Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=kv9YjHgx2Q3T85jYlx6ozVwGCbdfroU5MhEqr2aByKQ&m=NbDTYgDutbbTwGYD3E_9dY7-6hX7Wu_M825HFuuZexk&s=SmirDTcJ-Vr0lOxu9MZ87uoXYSkKlwPjqhSyM1_VYdE&e=>​
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