given problem. For instance, the well-known *elastic net* regularization can be used as function $h$. Hence, to address ill-posedness, the following bilevel optimization model is considered [@Tseng07; @Sabach17]: $$\begin{aligned}
\displaystyle \mbox{minimize}& \qquad \|x\|_1 + \mu \|x\|_2^2\\
\mbox{subject to} &\qquad x \in \arg \min_{y \in X} \sum_{i=1}^{m} \mathfrak{L} (a_i^T y,b_i),\end{aligned}$$\
where $\mu>0$ regulates the trade-off between $\ell_1$ and $\ell_2$ norms.

\[fig:fiveplots\]

Existing methods
----------------

Problem [[,]{}]{} that is also referred to as *hierarchical optimization*[[,]{}]{} is a particular case of *mathematical program with generalized equation (or equilibrium) constraint* [@Kocvara04; @Pang96]. There has been a few approaches to tackle this problem. Note that in all approaches the following minimization problem and its minimizer have been extensively utilized.

\[def:reg minimizer\] [[Given a parameter $\lambda > 0$]{}]{}, the regularized problem corresponding to , [[is defined]{}]{} as $$\begin{aligned}
\label{reg form INC}
    \displaystyle \mbox{minimize}& \qquad f_\lambda (x) \triangleq f(x)+ \lambda h(x)\\
    \mbox{subject to} &\qquad x \in X\notag.
    \end{aligned}$$ Also, let $x_\lambda^*$ denote the unique minimizer of this problem. We may categorize the existing algorithms as follows.

[**(i) Exact regularization:**]{} The regularization technique has been highly used in some applications such as signal processing with $h(x)= \|x\|_2^2$ or $h(x)= \|x\|_1$ [@Tikhonov77; @Beck09]. This technique needs a proper parameter 