<< lft_version CCA (Computational Convex Analysis) pl_me_pe_2d >>

CCA (Computational Convex Analysis) >> CCA (Computational Convex Analysis) > pl_me_pe

pl_me_pe

Moreau envelope, Parabolic Envelope algorithm

Calling Sequence

M = pl_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
It reduces computation to a 1,..,n grid with
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=pl_me_pe(-5,5,Y)

See Also

Authors

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

<< lft_version CCA (Computational Convex Analysis) pl_me_pe_2d >>