I have to figure out the Lagrange dual of a constrained matrix $\ell_1$-norm minimization problem:
$$\operatorname{minimize}_{\ Y} ||A(Y)||_1$$ subject to \begin{align} \operatorname{diag}(Y) &= \vec{c}\\ Y &\succcurlyeq 0, \end{align}
where $Y$ and $A(Y)$ are square matrices in $R^{n\times n}$, $A(Y)$ is linear in $Y$
I know the dual norm of $\ell_1$-norm is the infinity norm. But I am struggling with the operator to extract the diagonal entries of $Y$. This problem appears in several contexts like matrix completion, filtering etc. Can some one please help.