objective function for an arbitrary small $\epsilon>0$. [[We]{}]{} present the performance of the algorithm on a binary text classification problem.'
author:
- 'Mostafa Amini$^{1}$ and Farzad Yousefian$^{2}$[^1]'
title: '**An Iterative Regularized Incremental Projected Subgradient Method for a Class of Bilevel Optimization Problems** '
---

Introduction
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In this paper, we consider a class of bilevel optimization problems as follows $$\begin{aligned}
\label{def:SL-INC} \tag{$P_f^h$}
\displaystyle \mbox{minimize}& \qquad h(x)\\
\mbox{subject to} & \qquad x \in X^* \triangleq \arg\min_{y\in X}f(y) \notag.\end{aligned}$$ where $f,h :\mathbb{R}^n \rightarrow \mathbb{R}$ [[denote the lower and upper level objective functions, respectively, and $X \subseteq \mathbb{R}^n $ is a constraint set]{}]{}. This is called the *selection problem* ([@Tseng07; @Sabach17]) as we are selecting among optimal solutions of a lower level problem, one that minimizes the objective function $h$. [[In particular]{}]{}, we [[consider the case where the lower level objective function is]{}]{} given as $f(x) \triangleq \sum_{i=1}^{m} f_i(x)$, where $f_i:\mathbb{R}^n \rightarrow \mathbb{R}$ is the $i$th component function for $i=1,\cdots,m$. We make the following basic assumptions.

\[assum:properties\] 

The set $X \subset \mathbb{R}^n$ is nonempty, compact and convex; also $X\subseteq int(dom(f) \cap dom(h))$.

The functions $f_i(x)$ for $i=1,\cdots, m$ are proper, closed, convex, and possibly nondifferentiable.

The function $h$ is strongly convex with parameter 