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CCA (Computational Convex Analysis) >> gph > gph_me

gph_me

Graph-matrix calculus (GPH), Moreau envelope (Moreau-Yoshida approximate)

Calling Sequence

gphme = gph_me(gph, lambda)

Parameters

gph

matrix. A function in GPH matrix form.

lambda

constant. A positive smoothing parameter.

gphme

matrix. The Moreau envelope of the function gph, as a GPH matrix.

Description

Computes the Moreau Envelope of a convex PLQ function f, given in GPH matrix form. The Moreau envelope is defined as

For a convex PLQ function f with GPH matrix [x;s;y], the GPH matrix of the Moreau envelope with smoothing parameter L > 0 is

which can be derived from the Moreau envelope formula for convex functions (here * denotes the Legendre-Fenchel conjugate),

Examples

gph = [-1,  0, 0, 1, 2; ..
       -1, -1, 0, 0, 2; ..
        1,  0, 0, 0, 1];
lambda = 3;
gphme = gph_me(gph, lambda);
scf(); gph_plot(gph, gphme);

See Also

Authors

Bryan Gardiner, University of British Columbia, BC, Canada


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