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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

- gph_function

### Authors

`Bryan Gardiner`, University of British Columbia, BC, Canada

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