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Marcel Kronfeld 2008-03-27 14:52:43 +00:00
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resources/ESInitPopulationDOptimal.html
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<title>f_1 : Sphere function</title> <title>f_1 : Sphere function</title>
</head> </head>
<body> <body>
<EFBFBD>
<h1 align="center">ESInitPopulationSpaceFilling</h1> <h1 align="center">ESInitPopulationSpaceFilling</h1>
<center> <center>
</center><br> </center><br>

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resources/F1Problem.html
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<head> <head>
<title>f_1 : Sphere function</title> <title>f_1 : Sphere function</title>
</head> </head>
<body>  <body>
<h1 align="center">The Sphere function</h1> <h1 align="center">The F1 hyper-parabola function</h1>
<center> <center>
<img src="images/f1tex.jpg" width="85" height="95" border="0" align="center"> <img src="images/f1tex.jpg" width="85" height="95" border="0" align="center">
</center><br> </center><br>
The sphere function is a <i>n</i>-dimensional, axis-symmetric, continuously differentiable, convex function: The hyper-parabola function is a <i>n</i>-dimensional, axis-symmetric, continuously differentiable, convex function:
<p> <p>
Because of its simplicity every optimization-algorithm should be able to find its global minimum at <i>x</i>=[0, 0, ... , 0] Because of its simplicity every optimization-algorithm should be able to find its global minimum at <i>x</i>=[0, 0, ... , 0]
<p> <p>
@ -15,7 +15,7 @@ Because of its simplicity every optimization-algorithm should be able to find it
<img src="images/f1.jpg" width="480" height="360" border="2" align="middle"> <img src="images/f1.jpg" width="480" height="360" border="2" align="middle">
<hr> <hr>
More information about the sphere function can be found at: More information about the F1 function can be found at:
<p> <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. 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.

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resources/F2Problem.html
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<head> <head>
<title>Generalized Rosenbrock's function</title> <title>Generalized Rosenbrock's function</title>
</head> </head>
<body>  <body>
<h1 align="center">Generalized Rosenbrock's function</h1> <h1 align="center">Generalized Rosenbrock's function</h1>
<center> <center>
<img src="images/rosenbrocktex.jpg" width="500" height="78"> <img src="images/rosenbrocktex.jpg" width="500" height="78">
</center> </center>
<p> <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. This function is 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> <p>
<img src="images/f85.jpg" border="2"> <img src="images/f85.jpg" border="2">
<br> <br>
Rosenbrock's function within the co-domain -5 <= <i>x</i> <= 5. Rosenbrock's function within the co-domain -5 <= <i>x</i> <= 5.
<p> <p>
The global optimum is located in a prabolic formed valley (among the curve x^2 = x_1^2), which has a flatten ground. The global optimum is located in a parabolic formed valley (among the curve x^2 = x_1^2), which has a flat ground.
<br> <br>
<img src="images/f81.jpg" border="2"> <img src="images/f81.jpg" border="2">
<br> <br>

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resources/F3Problem.html
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<head> <head>
<title>The step function</title> <title>The step function</title>
</head> </head>
<body>  <body>
<h1 align="center">The step function</h1> <h1 align="center">The Step Function</h1>
<center> <center>
<img src="images/steptex.jpg" width="350" height="120" aling="center"> <img src="images/steptex.jpg" width="350" height="120" aling="center">
</center> </center>
<p> <p>
The idea of this function is the implementation of a flat plateau (slope 0)in an underlying continuous function.Its harder for optimization algortihms to find optimums because minor changes of the object variables don't affect the fitness. Therefore no conclusions about the search direction can be made. The idea of this function is the implementation of a flat plateau (slope 0)in an underlying continuous function. It's harder for optimization algortihms to find optima because minor changes of the object variables don't affect the fitness. Therefore no conclusions about the search direction can be made.
<p> <p>
<img src="images/step5.jpg" width="480" height="360" border="2" align="center"> <img src="images/step5.jpg" width="480" height="360" border="2" align="center">
<p> <p>
The step function is symmetric considering the underlying function (here: f(x,y) = f(y,x)), but between the bulk constant plateau-areas not continuously differentiable. The step function is symmetric considering the underlying function (here: f(x,y) = f(y,x)), but between the bulk constant plateau-areas not continuously differentiable.
<p> <p>
Its minimum-area is located in the intervalls: <i>f(x)</i>=<i>f</i>([-5.12,-5), ... , [-5.12,-5))=0. Its minimum-area is located in the intervals: <i>f(x)</i>=<i>f</i>([-5.12,-5), ... , [-5.12,-5))=0.
<p> <p>
<img src="images/stepopt.jpg" width="480" height="360" border="2" align="center"> <img src="images/stepopt.jpg" width="480" height="360" border="2" align="center">
<hr> <hr>

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resources/F5Problem.html
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<head> <head>
<title>Schwefel's double sum</title> <title>Schwefel's double sum</title>
</head> </head>
<body>  <body>
<h1 align="center">Schwefels double sum</h1> <h1 align="center">Schwefels double sum</h1>
<center> <center>
<img src="images/f2tex.jpg" width="220" height="102" border="0" align="center"> <img src="images/f2tex.jpg" width="220" height="102" border="0" align="center">
</center> </center>
<p> <p>
Schwefel's double sum is a quadratic minimization problem which difficulty increases by the dimension <i>n</i> in <i>O(n²)</i>. It is used for analysis of correlating mutations. Schwefel's double sum is a quadratic minimization problem. Its difficulty increases by the dimension <i>n</i> in <i>O(n^2)</i>. It is used for analysis of correlating mutations.
<p> <p>
It possesses specific symmetrical properties:<br> It possesses specific symmetrical properties:<br>

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resources/F6Problem.html
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<head> <head>
<title>Generalized Rastrigin's function</title> <title>Generalized Rastrigin's function</title>
</head> </head>
<body>  <body>
<h1 align="center">Generalized Rastrigin's function</h1> <h1 align="center">Generalized Rastrigin's function</h1>
<center> <center>
<img src="images/rastrigintex.jpg" width="500" height="101"> <img src="images/rastrigintex.jpg" width="500" height="101">
</center> </center>
<p> <p>
Rastrigin's function is symmetric. It is based on the simple <i>sphere function</i> (called f_1 in the JavaEva© context), but it is multimodal because a modulation term on the basis of the cosine function is added. This evokes hills and valleys which are misleading local optimums. Rastrigin's function is symmetric. It is based on the simple <i>parabola function</i> (called f1 in the JavaEvA context), but it is multimodal because a modulation term on the basis of the cosine function is added. This evokes hills and valleys which are misleading local optima.
<p> <p>
Values are used for the following illustrations: <i>A</i>=10, <i>&#969;</i>=2*&#960;, <i>n</i>=2. Values used for the following illustrations: <i>A</i>=10, <i>&#969;</i>=2*&#960;, <i>n</i>=2.
<br> <br>
<img src="images/rastrigin20.jpg" border="2"> <img src="images/rastrigin20.jpg" border="2">

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resources/F8Problem.html
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<head> <head>
<title>Ackley's function</title> <title>Ackley's function</title>
</head> </head>
<body>  <body>
<h1 align="center">Ackley's function</h1> <h1 align="center">Ackley's function</h1>
<center> <center>
<img src="images/ackleytex.jpg" width="500" height="58" aling="center"> <img src="images/ackleytex.jpg" width="500" height="58" aling="center">
</center> </center>
<p> <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> Ackley's function is multimodal and symmetrical. It is based on an exponential function and modulated by a cosine function.
The outside region is almost planar as to the growing influence of the exponential function.
In the center there is a steep hole as to the influence of the cosine function.<br>
Its minimum is at: <i>f(x)</i>=<i>f</i>([0, 0, ... , 0])=0. Its minimum is at: <i>f(x)</i>=<i>f</i>([0, 0, ... , 0])=0.
<p> <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. The difficulty for an optimization 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> <p>

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resources/GOParameters.html
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The GO parameter class is used to change main GO optimization settings. You may: The GO parameter class is used to change main GO optimization settings. You may:
<ul> <ul>
<li>Choose the optimizer. Check the optimizer object for further parameters and information.</li> <li>Choose the optimizer. Check the optimizer object for further parameters and information.</li>
<li>Choose an output file name. If "none" is used, no output file will be written.</li> <li>Set post-processing parameters or leave it turned off.</li>
<li>Select the problem to be optimized. Check the problem instance for further parameters and information. </li> <li>Select the problem to be optimized. Check the problem instance for further parameters and information. </li>
<li>Set a random seed. For the same seed, an optimization run should yield the same results. Set the seed to zero to use a dynamic seed for each run (using system time).</li> <li>Set a random seed. For the same positive seed, an optimization run should yield the same results. Set the seed to zero to use a dynamic seed for each run (using system time).</li>
<li>Define the termination criterion. Usually a maximum number of fitness evaluations is set, but <li>Define the termination criterion. Usually a maximum number of fitness evaluations is set, but
it is also possible to choose a maximum number of generations, an absolute fitness value to be reached, a it is also possible to choose a maximum number of generations, an absolute fitness value to be reached, a
convergence criterion measured in fitness change over time, or a combination of those.</li> convergence criterion or a combination of those.</li>
</ul> </ul>
<b>Note:</b> <br> <b>Note:</b> <br>
The evolutionary operators used by a strategy are tightly connected to the representation used. The evolutionary operators used by a strategy are tightly connected to the representation used.

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resources/GaussProcess.html
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<title>Gauss Process Regression Model</title> <title>Gauss Process Regression Model</title>
</head> </head>
<body> <body>
<EFBFBD>
<h1 align="center">Gauss Process Regression Model</h1> <h1 align="center">Gauss Process Regression Model</h1>
<center> <center>
</center><br> </center><br>

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resources/StatisticsParameterImpl.html
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<title>Statistics Parameter Panel</title> <title>Statistics Parameter Panel</title>
</head> </head>
<body> <body>
 
<h1 align="center">Statistics Parameter Panel</h1> <h1 align="center">Statistics Parameter Panel</h1>
<center> <center>
</center><br> </center><br>
Here you can edit the : Statistical options are:
<ul> <ul>
<li>Number of statistical independent runs.</li> <li>Convergence rate threshold. Provided the target value is zero,
<li>Name of result file.</li> convergence is assumed if a value smaller than this threshold is reached.
<li>Plot fitness of best, worse or both individuals.</li> For multi-run experiments, the number of hits is counted using this criterion.</li>
<li>Number of multi-runs. To achieve statistically meaningful results on how well
a certain optimizer works on a given problem, set this number to do several runs in a row.
The plot will be averaged, while all data can be collected in an output file or text window.</li>
<li>The fitness to plot. Define which fitness values to plot to the graph window after every
generation. Select "best", "worst" or both.</li>
<li>Result file name. If you want to collect optimization data, set this String to a desired file name.
Optimization results will be written to the indicated file in the working directory.</li>
<li>Show text output. If activated, the optimization data will also be shown in a graphical text
window for immediate viewing.</li>
</ul> </ul>
</body> </body>
</html> </html>