I'm interested in solving the following minimization problem:
$$ \min_X ||XD-b||_2^2 + ||X||_* $$
Where $D \in R^{n\times1}$, $ X \in R^{n\times n}$ and $b \in R^{n \times 1}$. That last term is the nuclear norm of $X$. I'm reading through this paper and they formulate the above problem in a more generalized way:
where $A(X)$ is a linear operator. They also define the adjoint as $A^*$. Now, when they differentiate the first term, they wrote it as: 
, which I don't understand is that how the adjoint operator involved in the derivative. If we assume that $A(X) = XD$, then the derivative of $.5||A(X) - b ||_2^2$ is $D^T(A(X) - b)$
My question: Are the adjoint linear operator $A^*(X) = D^TX$ and the linear operator $A(X) = XD$ correct definitions? I am getting error in my simulation so I am expecting I didn't correctly define $A$ and $A^*$