$\lambda$ which is difficult to determine in most of cases. To address this issue, Mangasarian et al. [@Mangasarian79; @Mangasarian85] introduced *exact regularization*. A solution of problem is called *exact* when it is in the set $X^*$. The main drawback of this approach is that the threshold below which for any $\lambda$ the regularization is exact, is very difficult to determine a priori [[(see [@Tseng07])]{}]{}.

[**(ii) Iterative regularization:**]{} In this approach[[,]{}]{} the idea is to develop a single-loop scheme where the regularization parameter is updated iteratively during the algorithm. In [@Solodov07], an *explicit* descent algorithm is proposed, where problem is solved as [[a single]{}]{}-level unconstrained problem iteratively. [[In the smooth case,]{}]{} the convergence is shown when $\sum_{k=1}^{\infty} \lambda_k =\infty$ and $\lim_{k\rightarrow \infty} \lambda_k=0$. For nonsmooth cases, a bundle method was proposed, which has a descent step for the weighted combination of objective functions in the lower and upper levels [@Solodov072]. Another algorithm called *hybrid steepest descent method* (HSDM) and its extensions were developed in [@Yamada11; @Neto11]. The main drawback of all these works is that no complexity analysis was provided for the [[underlying algrotihm]{}]{}. In [@Yousefian17], the rate of $\mathcal{\cal O} (1/k^{1/6-\epsilon})$ is derived for a special case of problem [[,]{}]{} 