@CorreaLemarecha1993; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018].

[*From now on we assume that there exist $0\leq \bar{\theta} < 1/2$ and $0\leq \bar{\lambda} < 1/2$, such that $\{\theta_k\}\subset [0, {\bar \theta})$, $\{\gamma_k\}\subset [0, {\bar \gamma})$ and $\{\lambda_k\}\subset [0, {\bar \lambda})$. For future references define*]{}

$$\label{eq:nu}
    \nu := \frac{1+2{\bar \gamma}}{1-2{\bar \lambda}}> 0.$$

To define the next stepsize, we need to known the optimum value $f^*$ given in . In [@PolyakBook p.142], is present some examples of problems for which the optimum value are known. The statement of the Polyak’s stepsize is as follows.

\[Poliak.Step\] Assume that $\Omega^*\neq\varnothing$ and the optimal value $f^*>-\infty$ is known. Let $\mu\geq 0$, $ \underline{\beta} >0$, ${\bar \beta} >0$ and take exogenous sequences $\{\beta_k\}$ and $\{\epsilon_k\}$ of real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and

$$\label{beta}
        0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{1}{2\mu +\nu}, \qquad\qquad  0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f^*], \quad \qquad  ~k=0, 1, \ldots.$$

Given $s_k \in \partial_{\epsilon_k} f(x_k)$, $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{StepsizePolyak}
        t_k=\beta_k\frac{f(x_k)-f^*}{\left\|s_k\right\|^2},   \qquad \qquad  ~k=0, 1, \ldots.$$

The stepsize in Rule \[Poliak.Step\] was introduced in [@Polyak1969] and has been used in several papers, including the ones [@nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018]. In general, in practical problems 