Problem becomes DUAL_INFEASIBLE when add new constraints about linear SDP

[jingqiao hu]

Dear professor,

I have solved the linear SDP problem below by mosek successfully.

However, when I try to add constraints:

The problem can not  be solved caused DUAL_INFEASIBLE. The output.log is attached.

I don't know why. Why would the new constraints lead to DUAL_INFEASIBLE?

And how to do if I must add these new constraints?

Thanks in advance.

[Johan Löfberg]

you've added a new constraint and the problem simply became infeasible. Why is that strange? x>=0 is feasible, but if you now add x<=-1, the problem becomes infeasible

[jingqiao hu]

The new constraint is a manufacture restriction. Therefore, the minimal eigenvalues of C should be away from zero.

The linear SDP problem has nonlinear SDP expression either. How about I transform it to nonlinear SDP problem with the new constraints? The new nonlinear SDP problem might be solved by IPOPT or whatever......

[Johan Löfberg]

I have no idea what you are talking about. Supply reproducible example. C - delta*1>=0 does not say it is positive definite, it says it is larger than an all-ones matrix (which implies psd though). C - delta*eye(n)>=0 would be an eigenvalue constraint

The sentence "The linear SDP problem has nonlinear SDP expression" doesn't make sense

[jingqiao hu]

1. It is C - delta*eye(n)>=0. 

2. The original problem is below. It's a nonlinear SDP problem.

However,using the Schur complement theorem, it can be written as a linear SDP problem:

The sentence "The linear SDP problem has nonlinear SDP expression" is ambiguity. I mean I solve the problem using the original nonlinear SDP.

Sorry about my mistakes.

[Johan Löfberg]

How do you mean you solve the nonlinear SDP. The log you show is from mosek, so linear SDP

The problem can still become infeasible if delta is too large. If the problem only is marginally feasible, i.e. all feasible solutions are singular, then any delta>0 will render he problem infeasible. As I said, impossible to say anything without reproducible code

[jingqiao hu]

Thanks for reply.  I want to use IPOPT or some nonlinear SDP solver to solve original problem. I don't know if this way could solve the new constraints......

I attached my code. FMO.m is the main code, where line51 is the new constraint. 

The delta in line51 is supposed to be 25 as the reference paper said. However I tried to increase the delta to 1000 and the problem finally is solvable.

[Johan Löfberg]

cannot run code as it requires data

[jingqiao hu]

The required data is already defined between line11-32.

[Johan Löfberg]

The model is trivially infeasible with the setup and delta=25

 

If you simplify, you're trying to solve something much more constrained and complicated compared to this

 

E = sdpvar(3)

cons = [];

cons = [cons, E >= 0]; 

cons = [cons, trace(E) <= .1]; 

cons = [cons,E-ones(3)/25 >= 0];

optimize(cons)

        

infeasible. The smallest possible delta would be 30 (which probably is easy to prove)

 

sdpvar f

cons = [];

cons = [cons, E >= 0]; 

cons = [cons, trace(E) <= .1]; 

cons = [cons,E-ones(3)*f >= 0];

optimize(cons,-f)

value(1/f)   

 

Doing the same for your full model shows that the smallest possible delta is  90.0009

 

 

btw, the line  assign(E{i}, repmat(0.1 * u / 2, 3, 3)); makes no difference in your code

[jingqiao hu]

Thank you so much, professor.