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