present our main result about the asymptotic convergence of $\{x_k\}$. It has as correspondent result in [@Polyak1969 Theorem 1]; see also [@nedic_bertsekas2001].

\[teo.MainConvPolyak\] The sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$.

Let $x\in \Omega^*$. Then, Lemma \[Le:FejerConvPolyak\] implies $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 $, for all $k=0, 1, \ldots$. Thus, $\{x_k\}$ is Fejér convergent to $\Omega^*$. Since $\Omega^* \neq \varnothing$, Theorem \[teo.qf\] implies that $\{x_k\}$ is bounded. By using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Then, from , after some algebra, we have $$\left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_k-x\|^2 - \|x_{k+1}-x\|^2\right), \qquad \qquad k=0, 1, \ldots.$$ Thus, performing the sum of the this inequality for $j=0, 1, \ldots, \ell$, we obtain $$\sum_{j=0}^{\ell}\left[f(x_j)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_{0}-x\|^2 - \|x_{ \ell+1}-x\|^2\right)\leq \frac{c^2}{\underline{\beta}} \|x_{0}-x\|^2.$$ Considering that this inequality holds for all $\ell\in \mathbb{N}$, we conclude that $\lim_{k\to +\infty}f(x_k)=f^*$. Let $x_*$ be a cluster point of $\{x_k\}$ and $\{x_{k_j}\}$ a subsequence of $\{x_k\}$ such that $\lim_{j\to +\infty}x_{k_j}=x_*$. Since $f$ is continuous, we have $f(x_*)= \lim_{j\to +\infty}f(x_{k_j})=f^*$. Therefore, $x_*\in\Omega^*$. Since $\{x_k\}$ is quasi-Fejér convergent to a set $\Omega^*$, it follows from Theorem \[teo.qf\] that $\{x_k\}$ converges $x_*$.

The next result presents an iteration-complexity bound, which is a version of [@Nesterov2014 Theorem 1].

Assume that 