the same target level $f_k^{lev} = f_{k(\ell)}^{rec} - \delta_\ell$, for $k \in K_\ell$. Also, note that the target level is update only if sufficient descent or oscillation is detected (Step 3 or Step 4, respectively). Whenever $\sigma_k$ exceeds the upper bound $R$, the parameter $\delta_\ell$ is decreased, which increases the target level $f_k^{lev}$.

From now on, we assume that Algorithm \[Alg:INPDyn\] generates an infinite sequence. In the next theorem we present the result about the asymptotic convergence of the sequence $\{x_k\}$. It is the versions of [@GoffinKiwiel1999 Theorem 1] and [@nedic_bertsekas2001 Proposition 2.7] by using inexact projections.

\[teo.MainConvDynamic\] There holds $\inf_{k \geq 0} f(x_k)= f^*$.

Since $x_k \in C$ and $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, x_k - t_ks_k\right)$, by the first equality in and applying Lemma \[pr:cond\] with $w=x_{k+1}$, $v=x_k - t_ks_k$, $x=x_k$ , $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$, we conclude that $$\|x_{k+1}-x_{k}\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\,t_k\|s_k\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, \tilde{t}_k, \qquad k=0, 1, \ldots.$$ We claim that the index $\ell$ goes to $+\infty$ and either $\inf_{k \geq 0} f(x_k) =-\infty$ or $\lim_{l\to\infty} \delta_\ell = 0$. Indeed, assume that $\ell$ takes only a finite number of values, i.e., $\ell < \infty$. 