sphere: Sphere Function for Benchmarking Optimization Algorithms

In EEEA: Explicit Exploration Strategy for Evolutionary Algorithms

Sphere Function for Benchmarking Optimization Algorithms

Description

The Sphere function is one of the simplest and most commonly used benchmark functions in optimization. It is a unimodal, convex function with a single global minimum, making it useful for testing the performance of optimization algorithms, especially in smooth, continuous search spaces.

Usage

sphere(x)

Arguments

x	A numeric vector representing the input variables for which the Sphere function is evaluated.

Value

Returns a numeric value representing the evaluation of the Sphere function at the given input vector x.

References

De Jong, K. A. (1975). Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph.D. dissertation, University of Michigan.

Examples

# Evaluation 1: Global minimum point in a four-dimensional space
x <- rep(0, 4)
sphere(x)

# Evaluation 2: A point in a six-dimensional space
x <- c(0, 0.24, 11, -1, -0.7, pi)
sphere(x)

# Contour Plot: Visualizing the Sphere Function
x1 <- seq(-10, 10, length.out = 100)
x2 <- seq(-10, 10, length.out = 100)
z <- outer(x1, x2, FUN = Vectorize(function(x, y) sphere(c(x, y))))
contour(x1, x2, z, nlevels = 20, main = "Contour of the Sphere Function")

# EDA.mnorm() example
res = EDA.mnorm(fun = sphere, lower = c(-10,-10), upper = c(10,10), n = 30, 
                k = 2, tolerance = 0.01, maxiter = 200)
res$sol

# Contour plot: Visualizing solution with EDA.mnorm()
x1 <- seq(-10, 10, length.out = 100)
x2 <- seq(-10, 10, length.out = 100)
z <- outer(x1, x2, FUN = Vectorize(function(x, y) sphere(c(x, y))))
contour(x1, x2, z, nlevels = 20, cex.axis = 0.8, 
        main = "Contour plot of the Sphere Function with EDA.mnorm solution")
points(res$sol[1], res$sol[2], col = "red", pch = 19)

EEEA documentation built on June 10, 2025, 9:13 a.m.