Slow progress termination of Mosek in Julia JuMP

[Huanwen Shen]

Hi there,

I tried to use Mosek to solve the following SDP in Julia JuMP

, and Mosek cannot solve this problem to optimality on some real data set. The solver terminates with the status: MathOptInterface.SLOW_PROGRESS. 

My questions:

1.From the website, I find that Mosek returns slow progress status if the problem is badly scaled, near feasible or near infeasible. Under each cases, what should I do respectively to make sure that I am able to solve the problem to optimality?

2. Constraint X-xx^T in PSDCone() is not allowed due to the error: this constraint is not positive definite, but the solver works when  I replace it by [1 x^T;  x  X] in PSDCone(). Why does this happen?

Thanks!

[Erling D. Andersen]

1. If there was simple generic method for verifying this we would have stated it in documentation. You have to use your knowledge about the problem to investigate this.

2. xx^T is quadratic i.e. in the one dimensional case it is x^2. That is most likely what JuMP complains about since you have to have  <affine expression> in PSDcone. In any case this is a JuMP issue.

If you post the Mosek log output we might be able to shed additional light on the problem.

[Huanwen Shen]

Thank you for the clarification. Here is Mosek log output.

 

[Erling D. Andersen]

My best guess is Mosek computes a fairly accurate solution. And most likely it is good enough for all practical purposes.

You are aware you always get an approximate solution so even if Mosek had not said slow progress you would have to evaluate whether the solution is accurate enough.

Btw you can see Mosek warns about a large number in the objective. That is because such things makes your problem harder numerically.

[Huanwen Shen]

Thank you!