Scalar constraints not respected in a SOS problem

[Pan Zhao]

Dear Johan,

I am trying to solve a SOS problem with scalar constraints (the last one involving w_lower in the following codes). 

F = [F sos(yMy) sos(y_W_Wlower_y) sos(y_Wupper_W_y) w_lower>= 1e-4 ]

[sol,v,Q,res] = solvesos(F,obj,ops,paras);

After solving using Mosek solver, I got 

max(res) = 8.2626e-05

w_lower  = 7.3624e-05

Obviously, the scalar constraint w_lower>=1e-4 is not satisfied. I wonder whether the SOS constraint has higher priority than other constraints including scalar ones in solving a SOS problem. If so, any suggestions to ensure that the scalar constraint is satisfied. Thanks. 

Best

Pan

[Pan Zhao]

Another question, for the same problem, I often found that the results could change drastically sometimes when I change some parameters slightly. This made me feel that my problem is susceptible to numerical issues. Do you have some common suggestions for mitigating numerical issues in SOS problems? Thank you very much 

Best

Pan

[Johan Löfberg]

You are simply in the region of solver tolerances, probably made worse by a numerically bad problem

There are no trivial trick to resort this, sos problems are often nice in theory but a bad idea in practice, you will simply have to read the various Wiki examples and tutorials and see if you find anything that could help in your case