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Schwefel's (sine root) function is highly multimodal and has no global basin of attraction. The optimum at a fitness of f(x*)=0 lies at x*=420.9687. Schwefel's sine root is a tough challenge for any global optimizer due to the multiple distinct optima. Especially, there is a deceptive nearly optimal solution close to x=(-420.9687)n. @@ -14,7 +14,7 @@ Schwefel's (sine root) function is highly multimodal and has no global basin of
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Schwefels's sine root function in 2D within the co-domain -500 <= x <= 500.
diff --git a/resources/html/F1Problem.html b/resources/html/F1Problem.html index 1cfb1fe6..500eeaf3 100644 --- a/resources/html/F1Problem.html +++ b/resources/html/F1Problem.html @@ -5,14 +5,14 @@
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Because of its simplicity every optimization-algorithm should be able to find its global minimum at x=[0, 0, ... , 0]
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This function is unimodal and continuous, but the global optimum is hard to find, because of independence through the term (x_(i+1) - x_i*x_i)^2 between contiguous parameters.
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Rosenbrock's function within the domain -5 <= x <= 5.
The global optimum is located in a parabolic formed valley (among the curve x^2 = x_1^2), which has a flat ground.
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The function close to its global optimum, which is: f(x) = f(1, 1, ... , 1) = 0.
diff --git a/resources/html/F3Problem.html b/resources/html/F3Problem.html index 7b187e07..20655a6b 100644 --- a/resources/html/F3Problem.html +++ b/resources/html/F3Problem.html @@ -5,18 +5,18 @@
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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.
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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.
Its minimum-area is located in the intervals: f(x)=f([-5.12,-5), ... , [-5.12,-5))=0.
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diff --git a/resources/html/F5Problem.html b/resources/html/F5Problem.html index 9697f9f7..33760602 100644 --- a/resources/html/F5Problem.html +++ b/resources/html/F5Problem.html @@ -5,18 +5,18 @@
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Schwefel's double sum is a quadratic minimization problem. Its difficulty increases by the dimension n in O(n^2). It is used for analysis of correlating mutations.
It possesses specific symmetrical properties:
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Its minimum is located at: f(x)=f([0, 0, ... , 0])=0
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Rastrigin's function is symmetric. It is based on the simple parabola function (called f1 in the EvA 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.
@@ -13,14 +13,14 @@ Rastrigin's function is symmetric. It is based on the simple parabola functio
Values used for the following illustrations: A=10, ω=2*π, n=2.
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Rastrigin's function within the co-domain -20>=x>=20
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Rastrigin's function within the co-domain -5>=x>=5
@@ -29,7 +29,7 @@ Rastrigin's function within the co-domain -5>=x>=5
Like Ackley's function a simple evolutionary algorithm would get stuck in a local optimum, while a broader searching algorithm would get out of the local optimum closer to the global optimum, which in this case is: f(x) = f(0, 0, ... , 0) = 0.
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Rastrigin's function close to its optimum.
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Ackley's function is multimodal and symmetrical. It is based on an exponential function and modulated by a cosine function. @@ -17,12 +17,12 @@ The difficulty for an optimization algorithm is mid-graded because a simple opti
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Ackley's function within the co-domain -20 >= x >= 20, a=20, b=0.2, c=2*π, n=2.
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Ackley's function close to the optimum.