+ t_k ^2\|s_k\|^2 + 2t_k\left[f(x) - f(x_k) + \epsilon_k\right] + \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k^2 \|s_k\|^2\\
                           &= \|x_k-x\|^2 + \frac{1+2\gamma_k}{1-2\lambda_k}t_k^2 \|s_k\|^2- 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right].
    \end{aligned}$$ Considering that $0 \leq \lambda_k < \bar{\lambda}<1/2$ and $0 \leq \gamma_k < \bar{\gamma}$, and using , we obtain .

Analysis of the subgradient-InexP method with exogenous stepsize {#Sec:AnalyisExog}
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In this section we will analyze the subgradient-InexP method with stepsizes satisfying Rule \[Exogenous.Step\]. For that, [*throughout this section we assume also that $\{x_k\}$ is a sequence generated by Algorithm \[Alg:INP\] with the stepsize given by Rule \[Exogenous.Step\] and, define $$\label{eq:rho}
    \rho:= \nu + 2\mu > 0.$$* ]{} First of all, note that under the above assumptions, Lemma \[Le:FejerConv\] becomes as follows.

\[Le:FejerConvExog\] Let $\rho>0$ be as in . For all $x\in C$, the following inequality holds $$\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2 - 2 \frac{\alpha_k}{\eta_k} \left[f(x_k)-f(x)\right], \qquad k=0, 1, \ldots.$$

The definition of $t_k$ in implies $t_k\leq \alpha_k$, which combined with the last inequality in yields $2t_k\epsilon_k\leq 2\mu \alpha_k^2$. Moreover, also implies that $t_k^2\|s_k\|^2\leq \alpha_k^2$. Therefore, using , the desired inequality follows directly from .

To proceed with the analysis of Algorithm \[Alg:INP\], we also need the following auxiliary set $$\label{eq:DefOmega}
    \Omega:=\left\{x\in C:~f(x)\leq \inf_kf(x_k)\right\}.$$ It is worth 