Thresholds of Prox-Boundedness of PLQ functions

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.