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CCA (Computational Convex Analysis) >> pl > pl_me_llt

pl_me_llt

Moreau envelope, LLT algorithm

Calling Sequence

M = pl_me_llt(X,f,S,isconvex,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.

isconvex

Optional. Boolean. If not Specified pre-assumption is %f (false)

fusionopt

Optional. integer. Select the implementation of the fusion algorithm. fusionopt=1 (or omitted) selects _pl_fusionsci, a fast implementation using scilab syntax but with nonlinear complexity. Any over value selects the _pl_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.

It reduces computation to computing the Legendre conjugate through the formula

Thereby resulting in a theta(n + m) linear-time algorithm with n=length(X)=length(f) and m=length(S).

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=pl_me_llt(X,Y,S)

See Also

Authors

Yves Lucet, University of British Columbia, BC, Canada

Used Functions

The core computation is done by pl_lft_llt.


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