
Journal of Convex Analysis 23 (2016), No. 3, 691718 Copyright Heldermann Verlag 2016 Thresholds of ProxBoundedness of PLQ Functions Warren L. Hare Dept. of Mathematics, University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada V1V 1V7 warren.hare@ubc.ca Chayne Planiden Dept. of Mathematics, University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada V1V 1V7 chayneplaniden@hotmail.com Introduced in the 1960s, the Moreau envelope has grown to become a key tool in nonsmooth 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 proxbounded, 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 proxboundedness (proxthreshold) of the function f. In this paper, we seek to understand the proxthresholds of piecewise linearquadratic (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 proxthreshold for a PLQ function, and further analyzes the behavior of the Moreau envelope of the function using the proxthreshold. We provide several examples to illustrate the techniques and challenges. Keywords: Moreau Envelope, piecewise linearquadratic, PLQ, proxthreshold. MSC: 49J52, 49J53, 49N10 [ Fulltextpdf (300 KB)] for subscribers only. 