me_llt - Moreau envelope, LLT algorithm

Calling Sequence

M = me_llt(X,f,S,fusionopt)

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.
• fusionopt : Optional. integer. Select the implementation of the fusion algorithm. fusionopt=1 (or omitted) selects fusionsci, a fast implementation using scilab syntax but with nonlinear complexity. Any over value selects the fusion implementation, a (slower) loop-based implementation that runs in linear-time.
• 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)

Description

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                                                      2
M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) +  f(i)) ]
                                    i

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=me_llt(X,Y,S)

See Also

me_llt2d,  me_direct,  me_nep,  me_pe,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

Used Function