<?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_me_generic"><info><pubdate>04-Apr-2006</pubdate></info><refnamediv><refname>pl_me_generic</refname><refpurpose>Generic Moreau envelope</refpurpose></refnamediv>
  
   
  
   
  
   
  
   
  
   
  
   <refsynopsisdiv><title>Calling Sequence</title><synopsis>[M] = pl_me_generic(X,Y,S)
[M,p,P] = pl_me_generic(X,Y,S)
[M,g,Conjpartial,Conj] = pl_me_generic(Xr,Xc,f,Sr,Sc)</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>f</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>M</term>
    
            <listitem>
       
               <para>column vector. Contains the value of the Moreau envelope M of the function f evaluated on at the points S(j). In other words: M(j) = Min(||S(j) - X(i)||^2 + f(i) | over all indexes i)
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>p</term>
    
            <listitem>
       
               <para>selection of the proximal mapping, p(j) is in  Argmin(||S(j) - X(i)||^2 + f(i) | over all indexes i)
       </para>
    
            </listitem>
    
         </varlistentry>

    
         <varlistentry>
    
            <term>P</term>
    
            <listitem>
       
               <para>proximal mapping, P(j)=Argmin(||S(j) - X(i)||^2 + f(i) | over all indexes i)
       </para>
    
            </listitem>
    
         </varlistentry>
     
         <varlistentry>
     
            <term>Xr</term>
     
            <listitem>
        
               <para>column vector of length n.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>Xc</term>
     
            <listitem>
        
               <para>column vector of length m.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>f</term>
     
            <listitem>
        
               <para>matrix of size nxm. The function f is sampled on a grid Xr x Xc so f(i,j)=fu(Xr(i),Xc(j)) for some function fu.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>Sr</term>
     
            <listitem>
        
               <para>column vector of size m1.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>Sc</term>
     
            <listitem>
        
               <para>column vector of size m2. The Moreau envelope is computed on a grid SrxSc.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>M</term>
     
            <listitem>
        
               <para>matrix of size m1xm2 containing the Moreau envelope of the function f.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>g</term>
     
            <listitem>
        
               <para>matrix of size nxm containing the function g (see formula below).
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>Conjpartial</term>
     
            <listitem>
        
               <para>matrix containing the partial conjugate of the function g
     i.e. the one dimensional conjugate applied to each row.
        </para>
     
            </listitem>
     
         </varlistentry>
 
     
         <varlistentry>
     
            <term>Conj</term>
     
            <listitem>
        
               <para>matrix of size m1xm2 containing the conjugate of the function g (see pl_me_llt_2d for formula).
        </para>
     
            </listitem>
    
         </varlistentry>
  
      </variablelist>
  
   </refsection>
 
  
   <refsection><title>Description</title>
     
         <para>
	The moreau function is a wrapper for three different moreau envelope computations depending on the calling sequence. 
	The first calling sequence [M] = pl_me_generic(X,Y,S) uses pl_me_llt function for the 1D me computation. 
	The second calling sequence [M,p,P] = pl_me_generic(X,Y,S) uses the pl_me_direct function for the 1D me computation.
	The last calling sequence [M,g,Conjpartial,Conj] = pl_me_generic(Xr,Xc,f,Sr,Sc) uses pl_me_llt_2d function for the 2D me computation.
	For a more detailed description see links below.
    </para>
    
   </refsection>
  
   <refsection><title>Examples</title><programlisting role="example"><![CDATA[
//Example 1. First Calling Sequence
X = (1:10)';S = X;
F=X.^2;
[M] = pl_me_generic(X, F, S);

//Example 2. Second Calling Sequence
X = (1:10)';S = X;
F=X.^2;
[M,p,P] = pl_me_generic(X,F,S);

//Example 3. Third Calling Sequence
Xr = [1 2.2 3]'; Sr = Xr;
Xc = [1 2 3]'; Sc = Xc;
F = [3 100 100;100 100 100;100 100 2 ];
[MY,g,Conjpartial,Conj] = pl_me_generic(Xr, Xc, F, Sr, Sc);
  ]]></programlisting></refsection>

   <refsection><title>See Also</title><simplelist type="inline">	
	
      <member> 
         <link linkend="pl_me_llt">pl_me_llt</link> 
      </member>
	
      <member> 
         <link linkend="pl_me_direct">pl_me_direct</link> 
      </member>
	
      <member> 
         <link linkend="pl_me_llt_2d">pl_me_llt_2d</link> 
      </member>

   </simplelist></refsection>
	
  
   <refsection><title>Authors</title><para>
	  
      <literal>Mike Trienis</literal>, University of British Columbia, BC, Canada
  </para></refsection>

</refentry>
