<?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="plq_ll">
  <info>
    <pubdate>18-May-2010</pubdate>
  </info>

  <refnamediv>
    <refname>plq_ll</refname>
    <refpurpose>Piecewise linear quadratic (plq), Lasry-Lions double envelope</refpurpose>
  </refnamediv>

   <refsynopsisdiv>
     <title>Calling Sequence</title>
     <synopsis>plqg = plq_ll(plqf, lambda, mu)</synopsis>
   </refsynopsisdiv>

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

       <varlistentry>
         <term>plqf</term>
         <listitem>
           <para>matrix. A PLQ function.</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>plqg</term>
         <listitem>
           <para>matrix.  The resulting PLQ function.</para>
         </listitem>
       </varlistentry>

     </variablelist>
   </refsection>

   <refsection>
     <title>Description</title>

     <para>
       Computes the Lasry-Lions double envelope of a given PLQ function, using the formula (M is the Moreau envelope)
       <!--<programlisting><![CDATA[
    h[mu,lambda]=-M[mu](-M[lambda](x)).
]]></programlisting>-->
	   <para>
			<inlinemediaobject>
				<imageobject>
					<imagedata fileref="plq_ImagesOfEquationsByLatex/plq_ll/plq_ll_1.png"/>
				</imageobject>
			</inlinemediaobject>
		</para>
     </para>

     <para>
       This algorithm, like its subroutines, runs in linear time.
     </para>
   </refsection>

   <refsection>
     <title>Examples</title>
     <programlisting role="example"><![CDATA[
plqf = [-1,0,-1,-1; 0,0,1,1; 1,0,-1,1; %inf,0,1,-1];
mu = 0.7;
lambda = 0.4;
plqg = plq_ll(plqf, lambda, mu);
plq_plot(plqf, plqg, -3, 3);
xtitle("Lasry-Lions double envelope");
]]></programlisting>
   </refsection>

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

   <refsection>
     <title>Authors</title>
     <para>
       <literal>Bryan Gardiner</literal>, University of British Columbia, BC, Canada
     </para>
   </refsection>

</refentry>
