$\mu_h>0$ and possibly nondifferentiable.

[[Next]{}]{}, we [[present]{}]{} two instances of the applications of formulation .

Example problems
----------------

[**(i) Constrained nonlinear optimization:**]{} [[Consider a constrained]{}]{} convex optimization problem [[given]{}]{} as $$\begin{aligned}
\displaystyle \mbox{minimize}& \qquad h(x)\\
\mbox{subject to} &\qquad q_i(x) \leq 0, \ \hbox{for } i=1,\cdots,m\\
&\qquad x\in X\end{aligned}$$ where $X \subseteq \mathbb{R}^n$ [[is an easy-to-project constraint set]{}]{}, $h,q_i :\mathbb{R}^n \rightarrow \mathbb{R} $ for all $i=1,\cdots,m$ [[are convex (and possibly nonlinear) funcitons]{}]{}. This problem can be reformulated as by setting (cf. [@Solodov072]) $$\begin{aligned}
f(x) \triangleq \sum_{i=1}^{m} f_i(x)= \sum_{i=1}^{m} \max \{0, q_i(x)\}.\end{aligned}$$ [**(ii) Ill-posed distributed optimization:**]{} An [[optimization]{}]{} problem is called ill-posed when it has multiple optimal solutions or it is very sensitive to data perturbations [@Tikhonov77]. For instance, in applications arising in machine learning, consider the *empirical risk minimization problem* where the goal is to minimize the total loss $\sum_{i=1}^{m} \mathfrak{L} (a_ix,b_i)$, where $a_i$ is the input, $b_i$ is the output of $i$th observed datum and $\mathfrak{L}$ is the loss function. For example, in *logisitic loss regression*, $\mathfrak{L}$ is merely convex. In these cases, another criterion such as sparsity may be taken into account for the optimal solution. So, to induce sparsity, a secondary objective function $h$ is considered in the 