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Marcel Kronfeld
2007-12-11 16:38:11 +00:00
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<html>
<head>
<title>Ackley's function</title>
</head>
<body><EFBFBD>
<h1 align="center">Ackley's function</h1>
<center>
<img src="images/ackleytex.jpg" width="500" height="58" aling="center">
</center>
<p>
Ackley's function is intense multimodal and symmetrical. It refers to an exponential function which is modulated through a cosine function. The outside region is almost planar by the growing influence of the exponential function. In the center it possesses a hole by the influence of the cosine function.<br>
Its minimum is at: <i>f(x)</i>=<i>f</i>([0, 0, ... , 0])=0.
<p>
The difficulty for an optmization algorithm is mid-graded because a simple optimization-algorithm like <i>hill-climbing</i> would get stuck in a local minimum. The optimization algorithm has to search a broader environ to overcome the local minimum and get closer to the global optima.
<p>
<img src="images/ackley.jpg" width="480" height="360" border="2" align="center">
<br>
Ackley's function within the co-domain -20 >= <i>x</i> >= 20, <i>a</i>=20, <i>b</i>=0.2, <i>c</i>=2*&#960;, <i>n</i>=2.
<p>
<img src="images/ackleyopt.jpg" width="480" height="360" border="2" align="center">
<br>
Ackley's function close to the optimum.
<hr>
More information about Ackley's function can be found at:
<p>
David. H. Ackley. <i>A connection machine for genetic hillclimbing.</i> Kluwer Academic Publishers, Boston, 1987.
<p>
Thomas Baeck. <i>Evolutionary Algorithms in Theory and Practice.</i> Oxford University Press, 1996.
</body>
</html>