  Introduced in the 1960s, the Moreau envelope has grown to become a key tool
in non\-smooth analysis and optimization. Essentially an infimal convolution
with a parametrized norm squared, the Moreau envelope is used in many
applications and optimization algorithms. An important aspect in applying the
Moreau envelope to nonconvex functions is determining if the function is
prox-bounded, that is, if there exists a point $x$ and a parameter $r$ such
that the Moreau envelope is finite. The infimum of all such $r$ is called the
threshold of prox-boundedness (prox-threshold) of the function $f.$ In this
paper, we seek to understand the prox-thresholds of piecewise linear-quadratic
(PLQ) functions. (A PLQ function is a function whose domain is a union of
finitely many polyhedral sets, and that is linear or quadratic on each piece.)
The main result provides a computational technique for determining the
prox-threshold for a PLQ function, and further analyzes the behavior of the
Moreau envelope of the function using the prox-threshold. We provide several
examples to illustrate the techniques and challenges.
