modified: docs/src/example.md

	modified:   src/benchmark/CEC2005.jl
	modified:   test/benchfunctions.jl

Author: abcdvvvv

docs/src/example.md

@@ -9,22 +9,11 @@ with ``a = 20``, ``b = 0.2``, ``c = 2π``, and global minimum at ``x=0``.
 
 ```@example
 using MetaheuristicsAlgorithms
-using Statistics: mean
-
-function Ackley(x::Vector{Float64})::Float64
-    a = 20.0
-    b = 0.2
-    c = 2π
-
-    term1 = -a * exp(-b * sqrt(mean(x.^2)))
-    term2 = -exp(mean(cos.(c * x)))
-
-    return term1 + term2 + a + exp(1)
-end
+import MetaheuristicsAlgorithms as MA
 
 lb = [-32.768 for _ = 1:5]
 ub = [ 32.768 for _ = 1:5]
-result = AEO(Ackley, lb, ub, 100, 1000)
+result = AEO(MA.Ackley, lb, ub, 100, 1000)
 convergence_curve(result)
 ```
 

src/benchmark/CEC2005.jl

@@ -159,6 +159,42 @@ function F9(x)
     return sum(x.^2 .- 10 .* cos.(2π .* x)) + 10 * dim
 end
 
+"""
+    Ackley(x)
+
+Ackley function.
+
+A popular multimodal benchmark with a nearly flat outer region and many local minima.
+Commonly used to test global optimization algorithms.
+
+Equation (with a = 20, b = 0.2, c = 2π):
+
+```math
+f(\\mathbf{x}) = -a\\,\\exp\\left(-b\\,\\sqrt{\\tfrac{1}{n}\\sum_{i=1}^n x_i^2}\\right)
+                 - \\exp\\left(\\tfrac{1}{n}\\sum_{i=1}^n \\cos(c\\,x_i)\\right)
+                 + a + e
+```
+
+Properties:
+- Domain: any dimension `n`; typical bounds `x_i ∈ [-32.768, 32.768]` (sometimes `[-5, 5]`).
+- Global minimum: `f(0,…,0) = 0` at `x = 0`.
+- Output: `Float64` scalar value of the function at `x`.
+"""
+function Ackley(x::AbstractVector{<:Real})::Float64
+    n = length(x)
+    a = 20.0
+    b = 0.2
+    c = 2π
+
+    sum1 = sum(xi^2 for xi in x)
+    sum2 = sum(cos(c * xi) for xi in x)
+
+    term1 = -a * exp(-b * sqrt(sum1 / n))
+    term2 = -exp(sum2 / n)
+
+    return term1 + term2 + a + exp(1)
+end
+
 """
     F10(x)
 
@@ -175,11 +211,7 @@ f(\\mathbf{x}) = -20 \\exp\\left(-0.2 \\sqrt{\\frac{1}{n} \\sum_{i=1}^n x_i^2}\\
                 + 20 + e
 ```
 """
-function F10(x)
-    dim = length(x)
-    return -20 * exp(-0.2 * sqrt(sum(x.^2) / dim)) -
-           exp(sum(cos.(2π .* x)) / dim) + 20 + exp(1)
-end
+F10 = Ackley
 
 """
     F11(x)

test/benchfunctions.jl

@@ -1,18 +1,3 @@
-function Ackley(x::AbstractVector{<:Real})::Float64
-    n = length(x)
-    a = 20.0
-    b = 0.2
-    c = 2 * π
-
-    sum1 = sum(xi^2 for xi in x)
-    sum2 = sum(cos(c * xi) for xi in x)
-
-    term1 = -a * exp(-b * sqrt(sum1 / n))
-    term2 = -exp(sum2 / n)
-
-    return term1 + term2 + a + exp(1)
-end
-
 function Griewank(x::AbstractVector{<:Real})::Float64
     sum_term = sum(xi^2 / 4000 for xi in x)
     prod_term = prod(cos(xi / sqrt(i + 1)) for (i, xi) in enumerate(x))