f(x_k)$, compute a stepsize $t_k>0$, and take the next iterate $x_{k+1}\in C$ as any feasible inexact projection of $x_k-t_ks_k$ onto $C$ relative to $x_{k}$ with error tolerance given by $\varphi_{ \gamma_k, \theta_k, \lambda_k}(x_k, x_k-t_ks_k,x_{k+1})  $, i.e., $x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right)$. We remark that if $\varphi_{ \gamma_k, \theta_k, \lambda_k}\equiv~0$, then ${\cal P}_C\left(0, x_k, x_k-t_ks_k\right)$ is the exact projection of $x_k-t_ks_k$ onto $C$, and Algorithm \[Alg:INP\] amounts to the projected subgradient method studied in [@AlberIusemSolodov1998]. Among the several possible choices that appeared in literature on subject, see for example [@Bertsekas1999; @nedic_bertsekas2001; @Shor1985], we studied three well known strategies, beginning with exogenous stepsize.

\[Exogenous.Step\] Let $ \mu\geq 0$. Take exogenous sequences $\{\alpha_k\}$ and $\{\epsilon_k\}$ of nonnegative real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{eq:ExogSeq}
        \sum_{k=0}^{\infty}\alpha_k=+\infty, \qquad \qquad  \sum_{k=0}^{\infty}\alpha_k^2<+\infty, \qquad \qquad   \epsilon_k\leq \mu \alpha_ k, \qquad \qquad  ~k=0, 1, \ldots.$$ Given $s_k \in \partial_{\epsilon_k} f(x_k)$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq:StepSize1}
        t_k:=\frac{\alpha_k}{\eta_k},   \qquad \qquad    \eta_k:= \max\left\{1, \| s_k\|\right\},  \qquad  \qquad  ~k=0, 1, \ldots.$$

The stepsize in Rule \[Exogenous.Step\] is one the most popular. It have been used in several paper for analyzing subgradient method; see for example, [@AlberIusemSolodov1998; 