Name

pl_me_plt — Parametric Moreau Envelope, PME algorithm

Calling Sequence

[M, z] = pl_me_plt(X, f, df)

Parameters

X

column vector. A grid of points on which the function is sampled.

f

column vector. The value of the function on the grid X: usually Y(i)=f(X(i)) for some function f.

df

column vector. The value of the derivative of the function f evaluated on X, which is also the slope.

Description

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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) ||.
               i

It 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.

</listitem>

Examples

    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] = pl_me_plt(X, f, df);
  

See Also

pl_me_direct , pl_me_llt , pl_me_nep , pl_me_pe

Authors

Yves Lucet, University of British Columbia, BC, Canada