need to define, for each $\epsilon \geq 0$, the $\epsilon$-subdifferential $\partial_{\epsilon} f(x)$ of a convex function $f$ at $x\in {\mathbb R}^n$, $$\label{eq:e-subdif}
    \partial_{\epsilon} f(x):=\{ s\in {\mathbb R}^n:~f(y)\geq f(x)+\langle s, y-x\rangle -\epsilon, ~\forall y\in {\mathbb R}^n\}.$$ We end this section by presenting important properties of the set $\epsilon$-subdifferential of a convex function, which proofs follow by combining [@Bertsekas2003 Proposition 4.3.1(a)] and [@UrrutyLemarechal1993_II Proposition 4.1.1, Proposition 4.1.2].

\[pr:CompE-subdif\] Let $f:\mathbb{R}^n \to \mathbb{R}$ be a convex function and $\epsilon \geq 0$. The set $\partial_{\epsilon} f(x)$ is nonempty, convex, and compact. Moreover, if $B\subset  \mathbb{R}^n$ is a bounded set, then there exists a real number $L>0$ such that $\|s\|<L$, for all $s\in \cup_{x\in B} \partial_{\epsilon} f(x)$. In addition, if $\{\epsilon_k\}$ is a bounded sequence of nonnegative real numbers, the sequence $\{x_k\}$ converges to $x \in \mathbb{R}^n$, and $s_k \in \partial_{\epsilon_k} f(x_k)$ for all $k$, then the sequence $\{s_k\}$ is bounded.

Subgradient-InexP method {#Sec:SubInexProj}
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Next, we present the subgradient method with a feasible inexact projections, which will be called [*Subgradient-InexP method*]{}. We begin by presenting the concept of relative feasible inexact projection, which is a variation of those presented in [@OrizonFabianaGilson2018; @VillaSalzBaldassarre2013].

\[def:InexactProj\] Let $C\subset {\mathbb R}^n$ be a closed convex set and $\varphi_{\gamma, 