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CCA (Computational Convex Analysis) >> pl > pl_me_plt

pl_me_plt

Parametric Moreau Envelope, PME algorithm

Calling Sequence

[M, z] = pl_me_plt(X, f, df)

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 Y(i)=f(X(i)) for some function f.

df

column vector. The value of the derivative of the function f evaluated on X, which is also the slope.

M

column vector. Computed through the formula M = f(X) + lambda * S.^2 /2; ,where lambda = 0.5 and s = df(X)

z

column vector. Computed through the formula z = X + lambda * df(X); ,where lambda = 0.5

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

Here the conjugate is derived by the Parametric Legendre Transform.

Examples

function y=f(x), y=0.5 * x.^2, endfunction;
function s=df(x), s= x, endfunction;
b=%T;n=8;
X=(-n/2:1:n/2)';S=df(X);Y=f(X);
[M,z] = pl_me_plt(X, f, df);

See Also

Authors

Yves Lucet, University of British Columbia, BC, Canada


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