the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Then, for every $N \in \mathbb{N}$, the following inequality holds $$\min \{f(x_k) - f^{*}: ~ \, k = 0, 1,\ldots, N \} \leq \frac{c}{\sqrt{\underline{\beta} (N+1)}} \|x_0 - x_{*}\|,$$ where $c\geq \max\{\|s_k\|:~ k=0, 1,\ldots \}$.

Applying Lemma \[Le:FejerConvPolyak\] with $x = x_*$ , where $f^* = f(x_*)$, we obtain $$\underline{\beta} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \|x_k-x_*\|^2 - \|x_{k+1}-x_*\|^2, \qquad\qquad k=0, 1, \ldots.$$ Performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we conclude that $$\sum_{k=0}^{N} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \frac{1}{\underline{\beta}} \|x_0-x_*\|^2.$$ Since $\{x_k\}$ is bounded, by using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Thus, we have $$\sum_{k=0}^{N} \left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2.$$ Therefore, $$(N+1) \min \{\left[f(x_k)-f^*\right]^2:~ \, k = 0, 1,\ldots, N \} \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2,$$ which is equivalent to the desired inequality.

Analysis of the subgradient-InexP method with dynamic stepsize {#Sec:Analyisdynamic}
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Next we consider the Subgradient-InexP method employing the dynamic stepsize Rule \[Dynamic.Step\], which guarantees that $\{f_{k}^{lev}\} $ converges to the optimum value $f^*$. In the following we present formally the algorithm which compute $f_{k}^{lev}$. This scheme was introduced in [@brannlund1993]; see also [@GoffinKiwiel1999].

Step 0.

:   Select $x_0\in C, \delta_0 > 0$, 