How to convert an element of a variable to a convex constraint using binary variables?

Question

I defined a complex variable in cvx, but I want to restrict the first element of the variable to be larger than the max of the variable, but it doesn't work. Someone told me to transform it using a binary variable, but I don't know how to do that? The code is in matlab:

cvx_begin quiet
variable wt(Mt-2) complex %variable
minimize(norm(cur_Qt*wt,1));
subject to
At_tar’*wt == 1;
abs(wt(1)) >= max(abs(wt))*0.0001; %%canstrain the first element of the variable
abs(wt(end)) >= max(abs(wt))*0.0001; %%canstrain the first element of the variable
for sl=1:length(sl_ind)
abs(Ats(:,sl)’*wt) <= 10^(desired_psll/20);
end
cvx_end

The error is: Disciplined convex programming error:

   Invalid constraint: {convex} >= {convex}

error  >=  (line 21)
b = newcnstr( evalin( 'caller', 'cvx_problem', '[]' ), x, y, '>=' );

error hybrid_weight_location_minpsll (line 54)
            abs(wt(1)) >= max(abs(wt))*0.0001;

Answer 1

Impossible in cvx since absolute value of a complex expression effectively is something represented using second-order cones/quadratics, and you thus have a nonconvex quadratic constraints since you are bounding that term from below, and that is not supported in cvx.

Not possible to represent using reformulations with binary variables either. That would be the case if the variable was real and it was absolute values instead of 2-norms.