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Marcel Kronfeld
2007-12-11 16:38:11 +00:00
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<html>
<head>
<title>Generalized Rosenbrock's function</title>
</head>
<body><EFBFBD>
<h1 align="center">Generalized Rosenbrock's function</h1>
<center>
<img src="images/rosenbrocktex.jpg" width="500" height="78">
</center>
<p>
This function i unimodal and continuous, but the global optimum is hard to find, because of independence through the term (<i>x</i>_(<i>i</i>+1) - <i>x_i</i>*<i>x_i</i>)^2 between contiguous parameters.
<p>
<img src="images/f85.jpg" border="2">
<br>
Rosenbrock's function within the co-domain -5 <= <i>x</i> <= 5.
<p>
The global optimum is located in a prabolic formed valley (among the curve x^2 = x_1^2), which has a flatten ground.
<br>
<img src="images/f81.jpg" border="2">
<br>
The function close to its global optimum, which is: f(<i>x</i>) = f(1, 1, ... , 1) = 0.
<p>
Rosenbrock' function is not symmetric, not convex and not linear.
<hr>
More information about Rosenbrock's function can be found at:
<p>
Kenneth De Jong. <i>An analysis of the behaviour of a class of genetic adaptive systems.</i> Dissertation, University of Michigan, 1975. Diss. Abstr. Int. 36(10), 5140B, University Microflims No. 76-9381.
<p>
Hans Paul Schwefel. <i>Evolution and optimum seeking.</i> Sixth-Generation Computer Technology Series. John Wiley & Sons, INC., 1995.
<p>
Darrell Whitley, Soraya Rana, John Dzubera, Keith E. Mathias. <i>Evaluating Evolutionary Algorithms. Artificial Intelligence</i>, 85(1-2):245-276. 1996.
<p>
Eberhard Schoeneburg, Frank Heinzmann, Sven Feddersen. <i>Genetische Algorithmen und Evolutionstrategien - Eine Einfuehrung in Theorie und Praxis der simulierten Evolution.</i> Addison-Wesley, 1994.
</body>
</html>