## Name

pl_ll — The Lasry-Lions double envelope approximation

## Calling Sequence

    L = pl_ll(X,Y,S,mu,lambda,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.

mu

  constant. This value determines the smoothness factor inside the nonconvex function (i.e. not on the convex hull)

lambda

  constant. This value determines how close the Lasry-Lions envelope will be to the original function (Note: Mu > Lambda)

algo

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

## Description

<listitem>

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

                     h[mu,lambda]=-F[mu](-F[lambda](x)).

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

</listitem>

## 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