cca_ll - The Lasry-Lions double envelope approximation

Calling Sequence

L = cca_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

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)).
                 

Examples

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

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

See Also

me_direct,  me_llt,  me_nep,  me_pe,  

Author

Yves Lucet, University of British Columbia, BC, Canada