me_nep - Moreau envelope for convex functions, NEP algorithm

Calling Sequence

[M,P] = me_nep(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 : proximal mapping, P(j)=Argmin(||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
       P(j) = Argmin f(i) + || s(j) - x(i) ||.
                i

The algorithm only returns correct result when the proximal mapping P is nonexpansive. Otherwise, the algorithms may return an incorrect result. Classes of functions f that have a nonexpansive proximal mapping include convex functions and prox-regular functions.

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]=me_nep(X,Y,S)

See Also

me_nep2d,  me_direct,  me_llt,  me_pe,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography