plq_me_max_scale - Piecewise linear quadratic (plq), Maximum scale for Moreau envelope
This function computes the maximum smoothing parameter that may be passed as "lambda" or "mu" into plq_me such that a set of nonconvex functions have finite Moreau envelopes. Otherwise, if the smoothing parameter is too high, g* below will be infinite everywhere because a convex hull cannot be computed.
Moreau envelope formula: M[lambda](s) = (1/2lambda)||s||^2 - (1/lambda)g*[lambda](s), where g[lambda](x) = (1/2)*||x||^2 + lambda*f(x), and * represents the Fenchel Conjugate.
f1 = [0,-1,0,0;%inf,-2,0,0]; plq_me_max_scale(f1), # -1/2/(-2) == 0.25 f2 = [1,0,1,0;2,0,-1,2;%inf,-3,0,0]; plq_me_max_scale(f2), # -1/2/(-3) == 0.167 plq_me_max_scale(f1, f2); # 0.167
Bryan Gardiner, University of British Columbia, BC, Canada