<?xml version="1.0" encoding="ISO-8859-1"?><refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" version="5.0-subset Scilab" xml:lang="en" xml:id="pl_ll"><info><pubdate>02-May-2006</pubdate></info><refnamediv><refname>pl_ll</refname><refpurpose>The Lasry-Lions double envelope approximation</refpurpose></refnamediv>
  
   
  
   
  
   
  
   
  
   

  
   <refsynopsisdiv><title>Calling Sequence</title><synopsis>L = pl_ll(X,Y,S,lambda,mu,algo)</synopsis></refsynopsisdiv>

  
   <refsection><title>Parameters</title>
  
      <variablelist>

    
         <varlistentry>
    
            <term>X</term>
    
            <listitem>
       
               <para>column vector. A grid of points on which the function is sampled.
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>Y</term>
    
            <listitem>
       
               <para>column vector. The value of the function on the grid X: usually f(i)=fu(X(i)) for some function fu.
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>S</term>
    
            <listitem>
       
               <para>column vector. The grid on which we want to compute the conjugate: f* is evaluated on S.  
       </para>
    
            </listitem>
    
         </varlistentry>
  
  
    
         <varlistentry>
    
            <term>lambda</term>
    
            <listitem>
       
               <para>constant. This value determines how close the Lasry-Lions envelope will be to the original function.
       </para>
    
            </listitem>
    
         </varlistentry> 

    
         <varlistentry>
    
            <term>mu</term>
    
            <listitem>
       
               <para>constant. This value determines the smoothness factor inside the nonconvex function (i.e. not on the convex hull).  Note: mu &gt; lambda.
       </para>
    
            </listitem>
    
         </varlistentry>
    
    
         <varlistentry>
    
            <term>algo</term>
    
            <listitem>
       
               <para>text. Name of the fast algorithm used to compute the Moreau envelope
       </para>
    
            </listitem>
    
         </varlistentry>  
    
  
      </variablelist>
  
   </refsection>
 
 
 
 
  
   <refsection><title>Description</title>
     
         <para>
            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) <!--<programlisting><![CDATA[
    
                 h[mu,lambda]=-M[mu](-M[lambda](x)).
                 
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_ll/pl_ll_1.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
			<para>
				It has a complexity which is dependent on the type of algorithm used to compute the Moreau Envelope.
			</para>
     </para>
  
   </refsection>

  
   <refsection><title>Examples</title><programlisting role="example"><![CDATA[
    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");
  ]]></programlisting></refsection>

  
   <refsection><title>See Also</title><simplelist type="inline">
  	
      <member> 
         <link linkend="pl_me_direct">pl_me_direct</link> 
      </member>
        
      <member> 
         <link linkend="pl_me_llt">pl_me_llt</link> 
      </member>
        
      <member> 
         <link linkend="pl_me_nep">pl_me_nep</link> 
      </member>
        
      <member> 
         <link linkend="pl_me_pe">pl_me_pe</link> 
      </member>
  
   </simplelist></refsection>

  
   <refsection><title>Authors</title><para>
	  
      <literal>Yves Lucet</literal>, University of British Columbia, BC, Canada
  </para></refsection>

</refentry>
