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CCA (Computational Convex Analysis) >> pl > pl_ll

# pl_ll

The Lasry-Lions double envelope approximation

### Calling Sequence

    L = pl_ll(X,Y,S,lambda,mu,algo)

### Parameters

X

  column vector. A grid of points on which the function is sampled.

Y

  column vector. The value of the function on the grid X: usually f(i)=fu(X(i)) for some function fu.

S

  column vector. The grid on which we want to compute the conjugate: f* is evaluated on S.

lambda

  constant. This value determines how close the Lasry-Lions envelope will be to the original function.

mu

  constant. This value determines the smoothness factor inside the nonconvex function (i.e. not on the convex hull). Note: mu > lambda.

algo

  text. Name of the fast algorithm used to compute the Moreau envelope

### Description

Compute numerically the Lasry-Lions double envelope of a set of planar points (X(i),Y(i)) at slopes S, i.e., (M is the Moreau envelope)

It has a complexity which is dependent on the type of algorithm used to compute the Moreau Envelope.

### Examples

    X=[-2:0.01:2]';
    Y=abs(abs(X)-1);
    S=X;
    mu=0.7;
    lambda=0.4;

    scf(0);clf;
    L = pl_ll(X,Y,S,mu,lambda,pl_me_direct);
    plot2d([abs(abs(X)-1),L]);
    xtitle("Lasry Lions double envelope");

### See Also

- pl_me_direct
- pl_me_llt
- pl_me_nep
- pl_me_pe

### Authors

`Yves Lucet`, University of British Columbia, BC, Canada

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