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CCA (Computational Convex Analysis) >> CCA (Computational Convex Analysis) > pl_me_llt

pl_me_llt

Moreau envelope, LLT algorithm

Calling Sequence

M = pl_me_llt(X,f,S,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.

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.

2
M(j) = min f(i) + || s(j) - x(i) ||.
i
It reduces computation to computing the Legendre conjugate through the formula
2                                                      2
M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) +  f(i)) ]
                         i
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.

<< lft_version CCA (Computational Convex Analysis) pl_me_llt_2d >>