<?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="opt_pa">
  <info>
    <pubdate>17-May-2010</pubdate>
  </info>

  <refnamediv>
    <refname>opt_pa</refname>
    <refpurpose>Convex operators via optimization (OPT), Proximal average</refpurpose>
  </refnamediv>

   <refsynopsisdiv>
     <title>Calling Sequence</title>
     <synopsis>[y, argmin] = opt_pa(f1, df1, domf1, f2, df2, domf2, x, lambda, output)</synopsis>
   </refsynopsisdiv>

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

       <varlistentry>
         <term>f1, f2</term>
         <listitem>
           <para>Univariate functions, continuous but (potentially) nonsmooth.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>df1, df2</term>
         <listitem>
           <para>Univariate functions: derivatives of f1 and f2 (respectively).</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>domf1, domf2</term>
         <listitem>
           <para>Domains of functions f1, f2 (set of points where
           function is finite).  Domains are stored as intervals
           I=[lb,ub] with ub&lt;lb storing an empty set.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>x</term>
         <listitem>
           <para>Vector of points at which to evaluate the proximal average.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>lambda</term>
         <listitem>
           <para>real number.  lambda1 = 1 - lambda.  lambda2 = lambda.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>output</term>
         <listitem>
           <para>0: no output, 1: progress bar.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>y</term>
         <listitem>
           <para>The values of the proximal average evaluated at x.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>argmin</term>
         <listitem>
           <para>A point at which the proximal average attains its
           minimum, for each x.</para>
         </listitem>
       </varlistentry>

     </variablelist>
   </refsection>

   <refsection>
     <title>Description</title>

     <para>
       Computes the (lambda1,lambda2) proximal average of f1 and f2
       evaluated on vector x, giving the values y such that

       <!--<programlisting><![CDATA[
    y =          inf       [lambda1 f1(y1) + lambda2 f2(y2) + 1/2 lambda1 lambda2 ||y1-y2||^2] .
       (lambda1 y1+lambda2 y2=x)
]]></programlisting>-->
		<para>
			<inlinemediaobject>
				<imageobject>
					<imagedata fileref="opt_ImagesOfEquationsByLatex/opt_pa/opt_pa_1.png"/>
				</imageobject>
			</inlinemediaobject>
		</para>
     </para>
   </refsection>

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

</refentry>
