sequence $\{x_k\}$ generated by Algorithm \[Alg:INP\]. The obtained inequality in next lemma is its counterpart for unconstrained optimization provided in [@CorreaLemarecha1993 Lemma 1.1)]. As we shall see, this inequality will be the main tool in our asymptotic convergence analysis, as well as in the iteration-complexity analysis.

\[Le:FejerConv\] Let $\nu>0$ be as defined . For all $x \in C$, the following inequality holds $$\label{eq:MainIneq}
        \|x_{k+1}-x\|^2 \leq \|x_k-x\|^2 + \nu t_k^2\|s_k\|^2 - 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right], \qquad k=0, 1, \ldots.$$

Let $x \in C$. To simply the notations we set $z_k:= x_k-t_ks_k$. Due to $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, z_k\right)$ and $x_k \in C$, we apply Lemma \[pr:cond\] with $w=x_{k+1}$, $v=z_k$, $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$ to conclude $$\label{eq:mip}
        \|x_{k+1}-x\|^2 \leq \|z_k-x\|^2+ \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k ^2\|s_k\| ^2.$$ On the other hand, due to $z_k= x_k-t_ks_k$, after some algebraic manipulations, we obtain $$\|z_k-x\|^2 = \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 + 2t_k \langle s_k, x - x_k \rangle.$$ Since $s_k \in \partial_{\epsilon_k} f(x_k)$, the definition implies that $\langle s_{k}, z-x_k\rangle\leq  f(z)- f(x_k)+\epsilon_k$. Thus, $$\|z_k-x\|^2 \leq \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 +  2t_k\left[f(x) - f(x_k) + \epsilon_k\right].$$ Therefore, combining last inequality with we conclude that $$\begin{aligned}
    \|x_{k+1}-x\|^2 &\leq \|x_k-x\|^2 