definitions, and results used throughout the paper. We are interested in $$\label{eq:OptP}
    \min \{ f(x) :~  x\in C\},$$ where $C$ is a closed and convex subset of $\mathbb{R}^n$, $f:\mathbb{R}^n \to \mathbb{R}$ is a convex function. We denote by $$\label{eq:ValueOpt}
f^*:= \inf_{x\in C} f(x),$$ its infimal value (possibly $-\infty$) and by $\Omega^*$ its solution set (possibly $\Omega^*= \varnothing$). The next concept will be useful in the analysis of the sequence generated by the subgradient method to solve .

\[def:QuasiFejer\] A sequence $\{y_k\}\subset \mathbb{R}^n$ is said to be quasi-Fejér convergent to a nonempty set $W\subset \mathbb{R}^n$ if, for every $w\in W$, there exists a sequence $\{\delta_k\}\subset\mathbb{R}$ such that $\delta_k\geq 0$, $\sum_{k=1}^{\infty}\delta_k<+\infty$, and $$\|y_{k+1}-w\|^2\leq \|y_k-w\|^2+\delta_k, \qquad \forall~k=0, 1, \ldots.$$ When, $\delta_k= 0$, for all $k=0, 1, \ldots.$, $\{y_k\}$ is called Fejér convergent to a set $W$.

The main property of the quasi-Fejér convergent sequence is stated in the next result, and its proof can be found in [@burachik1995full].

\[teo.qf\] Let $\{y_k\}$ be a sequence in $\mathbb{R}^n$. If $\{y_k\}$ is quasi-Fejér convergent to a nomempty set $W\subset  \mathbb{R}^n$, then $\{y_k\}$ is bounded. If furthermore, a cluster point $y$ of $\{y_k\}$ belongs to $W$, then $\lim_{k\rightarrow\infty}y_k=y$.

To describe the method for solving the problem  we 