Name

pl_me_direct — [For comparison only] Moreau envelope, direct computation

Calling Sequence

[M,p,P] = pl_me_direct(X,f,S)

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 f(i)=fu(X(i)) for some function fu.

S

column vector. The grid on which we want to compute the conjugate: f* is evaluated on S.

M

column vector. Contains the value of the Moreau envelope M of the function f evaluated on at the points S(j). In other words: M(j) = Min(||S(j) - X(i)||^2 + f(i) | over all indexes i)

p

selection of the proximal mapping, p(j) is in Argmin(||S(j) - X(i)||^2 + f(i) | over all indexes i)

P

proximal mapping, P(j)=Argmin(||S(j) - X(i)||^2 + f(i) | over all indexes i)

Description

<listitem>

Warning: This function is provided only for comparison purposes and unit testing, use more efficient linear-time algorithms for faster computation.

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
                                              2
       p(j) in Argmin f(i) + || s(j) - x(i) ||.
                 i
                                             2
       P(j) = Argmin f(i) + || s(j) - x(i) ||.
                i

It uses straight computation for a quadratic-time algorithm theta(n*m) with n=length(X)=length(f) and m=length(S).

</listitem>

Examples

    X=[-5:0.5:5]';
    Y=X.^2;
    S=(Y(2:size(Y,1))-Y(1:size(Y,1)-1))./(X(2:size(X,1))-X(1:size(X,1)-1));
    [M,p,P]=pl_me_direct(X,Y,S)

See Also

pl_me_direct_2d , pl_me_llt , pl_me_nep , pl_me_pe , pl_me_brute_2d

Authors

Yves Lucet, University of British Columbia, BC, Canada