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CCA (Computational Convex Analysis) >> CCA (Computational Convex Analysis) > pl_ph_direct

# pl_ph_direct

Discrete Pasch-Hausdorff envelope

### Calling Sequence

    ph = pl_ph_direct(X, Y, S, lambda)

### Parameters

X

  column vector of size m.

Y

  column vector of size m.

S

  column vector of size n.

lambda

  constant. Scaling parameter for the norm function.

ph

  The Pasch-Hausdorff envelope evaluated at S.

### Description

Computes the Pasch-Hausdorff envelope of a function described by the vectors X and Y, where y(i)=f(x(i)). Returns the envelope evaluated at the points in the vector S. The Pasch-Hausdorff envelope is the inf-convolution of the function with a scaled norm term,

    (f ic lambda||.||)(s) = inf [f(x) + lambda||x-s||].
                             x

The discrete Pasch-Hausdorff envelope is computed using the formula

    S = min [y_i + lambda |x_i - s_i|].
         i

This function runs in O(mn) time.

### Examples

    lambda = 0.5;
    X = [-2;-1;0;1;2];
    Y = abs(abs(X)-1);
    S = X;
    ph = pl_ph_direct(X,Y,S,lambda),

### Authors

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

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