How to treat abs() in a quasi-socp problem

[叶超天]

Hello,Johan Löfberg.

 I have encountered this problem, which has some constraints like

abs(A(l,:)*F(:,l))>=norm(C*vec(F),2);

abs(B(l,:)*F(:,l))>=norm(D*vec(F),2); 

It is these constraints which make the problem like SOCP, but not a SOCP actually.

My question is how to treat abs() to make the whole problem a SOCP, even approximation is OK.

Thank you very much.

[Johan Löfberg]

You cannot, as it is a nonconvex disconnected set in general. If YALMIP encounters this model, it will use a MILP-representation of the absolute value (assuming the argument is real)

[叶超天]

What is a MILP-representation?

As far as you know, is there any paper which has debated how to treat this situation?(F is complex)

[Johan Löfberg]

Mixed-integer linear representation, i.e, binary variables to handle the two cases A(l,:)*F(:,l)>=0 or A(l,:)*F(:,l)<=0

Not possible in your case though as you have complex data, so abs actually means norm(). There is no nice way to deal with that. You have a nonconvex quadratic constraint (A(l,:)*F(:,l))'*(A(l,:)*F(:,l)) >= (C*vec(F))'*(C*vec(F))

[叶超天]

Thank you very much. I understand.