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OPT++

/** \page gensetGuide-format Choosing a Generating Set for OptGSS 

A generating set for <b>R<sup>n</sup></b> is a set of vectors that 
can generate any point in <b>R<sup>n</sup></b> by linear combinations with 
positive coefficients.
<br>
Currently, OPT++ contains three Generating Set Search options for OptGSS.
The first two can be used for n-dimensional problems, whereas
the third is applicable only for two-dimensional problems.

<hr>
<p>

<b>GenSetStd:</b>

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<img src="../images/gensetStd.gif">
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The standard generating set has 2<b>n</b> elements and
consists of the columns of the identity matrix and its negative. <br>
<center>{I, -I}</center>
The standard generating set gives rise to the well-known compass-search 
algorithm, which searches both sides of every Cartesian direction. 
</table>
<p>
 
<b>GenSetMin:</b>
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<img src="../images/gensetMin.gif">
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A generating set for <b>R<sup>n</sup></b> must
contain at least <b>n</b>+1 elements. The standard minimal 
generating set consists of the columns of the identity matrix plus 
the vector with all entries equal to -1.
<br>
<center>{I, -1}</center>
The standard minimal generating set is useful for problems with 
expensive function evaluations.
</table>
<p>

<b>GenSetBox2d:</b>
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<td>
<img src="../images/gensetBox.gif">
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The Box generating set augments the standard generating set
with the corners of a <b>n</b> dimensional hypercube (or "box").
In OPT++, the Box generating set has been implemented for <b>n</b>=2,
where the corner vectors are <br>
<center>{(1,1), (-1,1), (-1,-1), (1,-1)}.</center>
This generating set can be used instead of the standard set for
two dimensional problems with inexpensive function evaluations.
The additional directions can significantly improve the algorithm's
convergence rate.
</table>

<hr>
<p>
<b>References:</b> 
<br>

<ol>
<li> T. Kolda, R. Lewis, V. Torczon.
<i>Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods.</i>
 SIAM REVIEW Vol. 45, No.3, pp.385-482. 
<p>
<li>A. G. Buckley and H. Ma, 
<i>A Derivative-Free Algorithm for Parallel and Sequential Optimization,</i> 
Technical Report, Computer Science Department, University of Victoria, BC, Canada, 1994. 
</li>
</ol>

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Last revised <em> July 13, 2006 </em> 

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