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CCA (Computational Convex Analysis) >> gph > gph_pa

# gph_pa

Graph-matrix calculus (GPH), Proximal Average (PA)

### Calling Sequence

    [G, X1, X2, Y1, Fq1, Fq2] = gph_pa(G1, G2, lambda)

### Parameters

G1

  matrix. The first input function f1, in GPH matrix form.

G2

  matrix. The second input function f2, in GPH matrix form.

lambda

  constant in [0,1]. lambda = lambda2 = the weighting of G2 in the PA. The weight of G1 is lambda1 = (1-lambda).

G

  matrix. The proximal average, in GPH matrix form.

X1, X2

  vectors. The points at which the infimum is attained; X = lambda1.*X1 + lambda2.*X2.

Y1

  vector. Y1=f1(X1), and S = Y1-X is the subdifferential of the PA.

Fq1, Fq2

  vectors. Images of X1 and X2 so the PA is F = lambda1.*Fq1 + lambda2.*Fq2 - X.^2/2.

### Description

Computes the proximal average (PA) of two convex functions in GPH matrix form. Computing the PA requires finding the points in each of the functions' domains that make the proximal average exact for a specific X value. This is done by parameterizing the graph with respect to X1, and determining the corresponding X2 values.

### Examples

    G1 = [-1 0 0 1; -1 -1 0 2; 1 0 0 1];
    G2 = [0 0; 0 1; 1 1];
    scf();
    for lambda = 0:0.1:1
        gph_plot(gph_pa(G1, G2, lambda));
        sleep(250);
    end;

### See Also

- gph_function

### Authors

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

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