of [[the]{}]{} lower level problem in and $x^*$ shows any element of this set . Likewise, $x^*_h$ [[denotes]{}]{} the optimal solution of problem [[, and]{}]{} $x^*_{\lambda}$ denotes the optimal solution of problem . [[Also, we let $\mathcal{P}_X(x)$ denote the Euclidean projection of vector $x$ onto the set $X$.]{}]{}

The rest of this paper is organized as follows. In Section \[sec: alg out\], we present the algorithm outline. Then, we discuss the convergence analysis in Section \[sec: conv ana\], and derive the convergence rate in Section \[sec:rate result\]. We present the numerical results in Section \[sec:num\], and conclude in Section \[sec:rem\].

Algorithm outline {#sec: alg out}
=================

In this section, we introduce the *iterative regularized incremental projected (sub)gradient* (IR-IG) for generating a sequence that converges to the unique optimal solution of . See Algorithm \[algorithm:IRIG\].

. $$\begin{aligned}
\label{mainstep}
        x_{k,i+1} := \mathcal{P}_X \left(x_{k,i} - \gamma_k\left(g_{f_{i+1}}(x_{k,i}) + \frac{\lambda_k}{m} g_h(x_{k,i})\right) \right).
        \end{aligned}$$ $$\begin{aligned}
        S_{k+1}:=S_k+\gamma_{k+1}^r, \quad          \bar x_{k+1}:=\frac{S_k \bar x_k+\gamma_{k+1}^r x_{k+1}}{S_{k+1}}.\label{def:averagingII} 
        \end{aligned}$$ $\bar x_{N}.$

IR-IG method includes two main steps. First, the agents update their iterates in an incremental fashion similar to the standard IG method. This step takes a circle around the all components of function $f$ to update the iterate. However, The main 