<BODY bgcolor="#aaeebb"> <H3><CENTER>Generalized and Softened Definition of Optimization</CENTER></H3> <P><CENTER><IMG SRC="Lect2_0.gif" WIDTH="343" HEIGHT="145" ALIGN="BOTTOM" NATURALSIZEFLAG="3"></CENTER> <UL> <LI><B>When G and S are singletons as in traditional optimization we are trying to hit one speeding bullet with another.</B><BR> <BR> <LI><B>Under softened definition, we are attempting to hit a truck with a shotgun.</B> </UL> <BR> <BR> <tt> <A HREF="Slide02.html" target="_parent">To Next Slide</A> / <A HREF="../OOTOC.html" target="_parent">Table of Content</A> </tt> <BR> <BR> <hr> <hr> <P> More precisely, the purpose of our OO study is to quantify, make explicit, and further exploit the implications of &quot;softening&quot;. and the replacement of value by order. To do this let us introduce some definitions and notations. Let <b>G</b> denote the <B>good enough</B> subset of a search space <b><i>Q</i></b>. To fix ideas, let us say <b>G</b> is top-<i>n</i>% of the search space and |<b>G</b>| = <i>g</i>, the size of <b>G</b>. Let <b>S</b> be the <B>selected</B> set which again could be the observed top-<i>m</i>% of the samples taken from the search space and |<b>S</b>| = <i>s</i>, the size of <b>S</b>. Note <b>S</b> and <b>G</b> are generalizations of the concept of optimum and the estimated optimum in traditional optimization. There <b>G</b> and <b>S</b> are singletons. In our case, the softened and analogous requirement is that <B>|<IMG SRC="GcapS.gif" WIDTH="40" HEIGHT="19" ALIGN="MIDDLE" NATURALSIZEFLAG="3">| not equal to 0</B>. Again the metaphors in the transparency emphasizes the advantages of softening which we shall quantify shortly.<BR> <BR> Now for a given problem, a given parameter search space <b><i>Q</i></b>, a given sample size <i>N</i>, a given model for estimating the performance of the system (which implies certain error of the estimate sigma<sup>2</sup>), a given &quot;good enough&quot; set <b>G</b>, and a given procedure for selecting <b>S</b>, we define the alignment probability as ...<BR> </BODY> </HTML>