problem .

Motivated by the existing lack in the literature and inspired by the advantages of incremental approaches, in this paper, we let the lower level objective function to be a summation of $m$ components. Then we use the idea of incremental subgradient optimization to address problem . We let functions in both levels to be nondifferentiable. We then prove the convergence of our proposed algorithm as well as the $\mathcal{\cal O} (1/k^{0.5-\epsilon})$ rate of convergence.

An interesting research question that is remained as a future direction to our research is if we can establish the convergence of iterative regularized IAG method in solving problem or similarly SAG, SAGA and MISO in stochastic regimes.

[**Notation**]{} The inner product of two vectors $x,y \in \mathbb{R}^n$, is shown as $x^Ty$. Also, $\|\cdot\|$ denotes Euclidean norm known as $\|\cdot\|_2$. For a convex function $f$ with the domain dom$(f)$, any vector $g_f$ with $f(x)+g_f^T(y-x) \leq f(y)$ for all $x,y \in \hbox{dom}(f)$, is called a subgradient of $f$ at $x$. We let $\partial f(x)$ and $\partial  h(x)$ denote the set of all subgradients of functions $f$ and $h$ at $x$. Let $f^*$ be the optimal value and $X^*$ represent the set of all optimal solutions 