me_plt - Parametric Moreau Envelope, PME algorithm
Compute numerically the discrete Moreau envelope of a set of planar points (X(i),f(i)) at slopes S(j), i.e.
2 M(j) = min f(i) + || s(j) - x(i) ||. iIt reduces computation to computing the Legendre conjugate through the formula
2 2 M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) + f(i)) ], i where the conjugate is derived by the Parametric Legendre Transform.
function y = f(x), y=0.5 * x.^2, endfunction; function s = df(x), s= x, endfunction; b=%T;n=8; X=(-n/2:1:n/2)';S=df(X);Y=f(X); [M,z] = me_plt(X, f, df);
me_direct, me_llt, me_nep, me_pe,
Yves Lucet, University of British Columbia, BC, Canada