Scilab Function
Last update : 3/11/2006

me_nep - Moreau envelope for convex functions, NEP algorithm

Calling Sequence

[M,P] = me_nep(X,f,S)

Parameters

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
It uses the non-expansiveness of the proximal (NEP) mapping P to run in linear time theta(n+m) with n=length(X)=length(f) and m=length(S).

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

Y. Lucet, 2006, Fast Moreau Envelope Computation I: Numerical Algorithms, Numerical Algorithms, 43 (2006), 235-249