<?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_nep"><info><pubdate>04-Apr-2006</pubdate></info><refnamediv><refname>pl_me_nep</refname><refpurpose>Moreau envelope for convex functions, NEP algorithm</refpurpose></refnamediv>
  
   
  
   
  
   
  
   
  
   

  
   <refsynopsisdiv><title>Calling Sequence</title><synopsis>[M,P] = pl_me_nep(X,f,S)</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>proximal mapping, P(j)=Argmin(||S(j) - X(i)||^2 + f(i) | over all indexes i)
       </para>
    
            </listitem>
    
         </varlistentry>
  
      </variablelist>
  
   </refsection>
 
  
   <refsection><title>Description</title>
     
         <para>
	Compute numerically the discrete Moreau envelope of a set of planar points (X(i),f(i)) at slopes S(j), i.e. <!--<programlisting><![CDATA[
                                          2
       M(j) = min f(i) + || s(j) - x(i) ||.
               i
                                             2
       P(j) = Argmin f(i) + || s(j) - x(i) ||.
                i
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_me_nep/pl_me_nep_1.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
	<para>It uses the non-expansiveness of the proximal (NEP) mapping P to run in linear time theta(n+m) with n=length(X)=length(f) and m=length(S).</para>
	</para>
	
         <para>The algorithm only returns correct result when the proximal mapping P is <literal>nonexpansive</literal>. Otherwise, the algorithms may
     return an incorrect result. Classes of functions f that have a nonexpansive proximal mapping include <literal>convex</literal> functions and
     <literal>prox-regular</literal> functions.</para>
     
   </refsection>

  
   <refsection><title>Examples</title><programlisting role="example"><![CDATA[
    X=[-5:0.5:5]';
    Y=X.^2;
    S=(Y(2:size(Y,1))-Y(1:size(Y,1)-1))./(X(2:size(X,1))-X(1:size(X,1)-1));
    [M,p,P]=pl_me_nep(X,Y,S)
]]></programlisting></refsection>

  
   <refsection><title>See Also</title><simplelist type="inline">
    
      <member> 
         <link linkend="pl_me_nep_2d">pl_me_nep_2d</link> 
      </member>
    
      <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_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>
