using Convex
n = 20
p = 2
A = rand(n,p) + im*randn(n,p)
x = rand(p) + im*randn(p)
b = abs(A*x) + rand(n)

M = diagm(b)*(eye(n)-A*ctranspose(A))*diagm(b)
U = ComplexVariable(n,n)
objective = inner_product(U,M)
c1 = diag(U) == 1 
c2 = U in :SDP
p = minimize(objective,c1,c2)
solve!(p)
U.value

# Verify if the rank of U is 1
B, C = eig(U.value);
println(length([e for e in B if(abs(real(e))>1e-4)]))
#Decompose U = uu*
# u is the phase of Ax
u = C[:,1];
for i in 1:n
    u[i] = u[i]/abs(u[i])
end
u