Graph-matrix calculus (GPH), Moreau envelope (Moreau-Yoshida approximate)
gphme = gph_me(gph, lambda)
matrix. A function in GPH matrix form.
constant. A positive smoothing parameter.
matrix. The Moreau envelope of the function gph, as a GPH matrix.
Computes the Moreau Envelope of a convex PLQ function f, given in GPH matrix form. The Moreau envelope is defined as
For a convex PLQ function f with GPH matrix [x;s;y], the GPH matrix of the Moreau envelope with smoothing parameter L > 0 is
which can be derived from the Moreau envelope formula for convex functions (here * denotes the Legendre-Fenchel conjugate),
gph = [-1, 0, 0, 1, 2; .. -1, -1, 0, 0, 2; .. 1, 0, 0, 0, 1]; lambda = 3; gphme = gph_me(gph, lambda); scf(); gph_plot(gph, gphme); | ![]() | ![]() |
Bryan Gardiner
, University of British Columbia, BC, Canada