i.e., for all $i=1,\cdots, m$ and $x,y\in X$ $$\begin{aligned}
        |f_i(x)-f_i(y)| \leq C_f \|x-y\|, \quad   |h(x)-h(y)| \leq C_h \|x-y\|.
        \end{aligned}$$ [[(c) Assumption \[assum:properties\](a,b) imply that the optimal solution set, $X^*$, is nonempty.]{}]{}

Here, we start with a lemma which helps bound the error of optimal solutions of the problem for two different values of $\lambda$. We will make use of this lemma in the convergence analysis. The proof for this lemma can be done in a same fashion to that of Proposition 1 in [@Yousefian17].

\[lemma:err\_inequalitygeneral\] Let Assumption \[assum:properties\] hold. Suppose $\{x_{\lambda_k}^* \}$ be the sequence of the optimal solutions of problem with parameter $\lambda:= \lambda_k$. Then,

-   $\|x_{\lambda_k}^*-x_{\lambda_{k-1}}^*\|\leq \frac{C_h}{\mu_h}\left  |1-\frac{\lambda_{k-1}}{\lambda_k} \right|$.

-   If $\lambda_k \rightarrow 0$, then the sequence $\{x_{\lambda_k}^*\}$ converges to the unique optimal solution of problem , i.e., $x_{h}^*$.

To get started, we also need a recursive upper bound on the term $ \|x_{k+1} - x_{\lambda_k}^*  \|$. This is provided by the following lemma and will be used in Proposition \[conv x\_k INC\] to prove the convergence of sequence $\{x_k\}$ generated by the algorithm to $x_h^*$.

\[[**A recursive error bound**]{}\] \[lemma: an error bound INC\] Let Assumption \[assum:properties\] hold and $0<\mu_k\lambda_k \mu_h \leq 2m$. Then, for the sequence 