  Computing the proximal operator of the sparsity-promoting piece-wise
exponential (PiE) penalty $1-e^{-|x|/\sigma}$ with a given shape parameter
$\sigma>0$, which is treated as a popular nonconvex surrogate of $\ell_0$-norm,
is fundamental in feature selection via support vector machines, image
reconstruction, zero-one programming problems, compressed sensing, etc. Due to
the nonconvexity of PiE, for a long time, its proximal operator is frequently
evaluated via an iteratively reweighted $\ell_1$ algorithm, which substitutes
PiE with its first-order approximation, however, the obtained solutions only
are the critical point. Based on the exact characterization of the proximal
operator of PiE, we explore how the iteratively reweighted $\ell_1$ solution
deviates from the true proximal operator in certain regions, which can be
explicitly identified in terms of $\sigma$, the initial value and the
regularization parameter in the definition of the proximal operator. Moreover,
the initial value can be adaptively and simply chosen to ensure that the
iteratively reweighted $\ell_1$ solution belongs to the proximal operator of
PiE.
