  In this paper, we initiate the study of a triple $(X,\Delta,D)$ which
consists of a pair $(X,\Delta)$ and a polarizing pseudoeffective divisor $D$.
The adjoint asymptotic multiplier ideal sheaf $\mathcal{J}(X,\Delta;\lVert D
\rVert)$ associated to the triple gives a simultaneous generalization of the
multiplier ideal sheaf $\mathcal{J}(D)$ and asymptotic multiplier ideal sheaf
$\mathcal{J}(\lVert D \rVert)$. We describe the closed set defined by the ideal
sheaf $\mathcal{J}(X,\Delta;\lVert D \rVert)$ in terms of the minimal model
program. We also characterize the case where $\mathcal{J}(X,\Delta;\lVert D
\rVert)=\mathcal{O}_X$. Lastly, we also prove a Nadel type vanishing theorem of
cohomology using $\mathcal{J}(X,\Delta;\lVert D \rVert)$.
