me_pe - Moreau envelope, PE algorithm

Calling Sequence

M = me_pe(x0,xn,f)

Parameters

• x0 : real number. Lower bound of the interval.
• xn : real number. Upper bound of the interval.
• f : column vector. The value of the function on the grid X=(x0:h:xn)'; where n=length(f), and h=(xn-x0)/(n-1); so f(i)=fu(X(i)) for some function fu.
• 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)=X(j), i.e.

                                          2
       M(j) = min f(i) + || s(j) - x(i) ||.
               i

           2               2                    2
    M(j)= h * min [ ||j-i|| + f(x_0 + i * h) / h  ]
               i

The algorithms runs in linear time Theta(n) with n=length(X)=length(f) by computing the lower parabolic envelope (PE).

Examples

    X=[-5:0.5:5]';
    Y=X.^2;
    M=me_pe(-5,5,Y)

See Also

me_pe2d,  me_direct,  me_llt,  me_nep,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography