## 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

<listitem>

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