{convex} - {convex} error

Nonconvex
1

I want to solve the minimization problem min(norm(y-Af,2)^2+lambda*norm(d(abs(f))/dx,1))

where y, f are complex vectors, A is a complex matrix, lambda is a constant.

when f=[f1, f2, … , fn], d(abs(f))/dx can be expressed as d(abs(f))/dx=[abs(f1)-abs(f2),abs(f2)-abs(f3),…,abs(fn-1)-abs(fn)].

So, when I run this problem in CVX

cvx_begin
variable f(n) complex;
expressions f_dx1(n-1) f_dx2(n-1);
f_dx1=f(1:n-1);
f_dx2=f(2:n);
minimize( sum_square_abs(A * f-y)+lambda * (norm(abs(f_dx1)-abs(f_dx2),1)) );
cvx_end

I received an error that {convex} - {convex}.

abs(f_dx1) - abs(f_dx2) is not convex. but norm(abs(f_dx1)-abs(f_dx2),1) is convex.

How can I express this convex problem so that it can be executed in cvx?

2

Let f(x,y) = norm(abs(x)-abs(y))

Then
f(1,-1) = f(1,1) = 0
f((1+1)/2,(-1+1)/2) = 1

f((1+1)/2,(-1+1)/2) \nleq 1/2*f(1,-1)+1/2*f(1,1)
This proves f(x,y) is not convex.