the optimum value is not known. In this case, we may modify the stepsize by replacing the optimum value with a suitable estimate in each iteration. This leads to the dynamic stepsize rule as follows.

\[Dynamic.Step\] 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 $$\label{betaDinamic}
        0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{2}{2\mu+\nu}, \qquad \qquad 0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f_{k}^{lev}], \quad \qquad  ~k=0, 1, \ldots,$$ where $f_{k}^{lev}$ will be specified later (see Section \[Sec:Analyisdynamic\]). Given $s_k \in \partial_{\epsilon_k} f(x_k)$ such that $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq.dynamicstep}
        t_k= \frac{\tilde{t}_k}{\left\|s_k\right\|}, \qquad\qquad  \tilde{t}_k = \beta_k\frac{f(x_k)-f_{k}^{lev}}{\left\|s_k\right\|}, \qquad\qquad  ~k=0, 1, \ldots.$$

The dynamic stepsize in Rule \[Dynamic.Step\] is based on the ideas of [@brannlund1993]; see also [@GoffinKiwiel1999]. This rule has been used in several papers, see for example [@nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018].

[*From now on we assume that the sequence $\{x_k\}$ is generated by Algorithm \[Alg:INP\], with one of the three above strategies for choosing the stepsize, is infinite.*]{}

Analysis of the subgradient-InexP method {#Sec:aca}
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In the following, we state and prove our first result to analyze the 