+ 2 \langle v-w,u-w \rangle - 2 \|w-v\|^2 \\
                            &= \|v-u\|^2 - \|w-v\|^2 + 2 \langle v-w,u-w \rangle.
    \end{aligned}$$ Thus, due to $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $u \in C$, using , , $0\leq \theta < 1/2$ and $0 \leq \lambda <1/2$, we have $$\|w-u\|^2 \leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2 - \frac{1-2\theta}{1-2\lambda} \|w-v\|^2\leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2.$$ Therefore, combining the last inequality with , we obtain the desired inequality.

The conceptual subgradient method with feasible inexact projections for solving the Problem  is formally defined as follows:\

\[H\]

 Step 0.

:   Let $\{\epsilon_k\}$, $\{\theta_k\}$, and $\{\lambda_k\}$ be sequences of nonnegative real numbers. Let $x_0\in C$ and set $k=0$.

 Step 1.

:   If $0\in \partial f(x_k)$, then [**stop**]{}. Otherwise, choose a non-null element $s_k \in \partial_{\epsilon_k} f(x_k)$, compute a stepsize $t_k>0$, (to be specified later), and take the next iterate as any point such that $$x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right).$$

 Step 2.

:   Set $k\gets k+1$, and go to **Step 1**.

Let us describe the main features of the subgradient-InexP method. Firstly, we check if the current iterate $x_k$ is a solution of Problem . If $x_k$ is not a a solution, then we choose a non-null element $s_k \in \partial_{\epsilon_k} 