\section{Inexact AL Algorithm \label{sec:AL algorithm}}
To solve the equivalent formulation presented in Program~\eqref{eq:minmax}, we propose the inexact augmented Lagrangian algorithm, detailed in Algorithm~\ref{Algo:2}. Iteration $k$ of this algorithm calls a subroutine that finds an approximate stationary point of the augmented Lagrangian $\L_{\b_k}(\cdot,y_k)$ with increasing accuracy of $\epsilon_{k+1}$ (Step 2). The increasing sequence $\{\b_k\}_k$ and the dual update (Steps 4 and 5) are responsible for gradually enforcing the constraints in Program~\eqref{prob:01}. As we will see in the convergence analysis, the particular choice of dual step size $\s_k$ in Algorithm~\ref{Algo:2} ensures that the dual variable $y_k$ remains bounded.
\begin{algorithm}
\label{Algo:2}
Input: Parameters $\rho,\rho',\rho''>0$, initialization $x_{1}\in\RR^d$ such that $\|A(x_1)\|\le\rho$ and $\|x_1\|\le\rho'$, dual initialization $y_0\in\RR^m$ and initial dual step size $\s_1$, a non-decreasing, positive, unbounded sequence $\{\b_k\}_{k\ge1}$, stopping threshold $\tau$.