Hartmann 3D Function (F19)

Source: R/F19.R

A multimodal test function with 4 local minima in 3 dimensions. Fixed dimension of 3.

F19(x)

Arguments

x

Numeric vector of length 3 (3-dimensional input).

Value

Numeric scalar representing the function value.

Details

Formula: $$f(x) = -\sum_{i=1}^{4} c_i \exp\left(-\sum_{j=1}^{3} a_{ij}(x_j - p_{ij})^2\right)$$

where \(a_{ij}\), \(c_i\), and \(p_{ij}\) are predefined constants.

Global minimum: \(f(0.114614, 0.555649, 0.852547) \approx -3.86278\)

Characteristics:

  • Type: Multimodal

  • Separable: No

  • Differentiable: Yes

  • Fixed dimension: 3

  • Number of local minima: 4

  • Default bounds: \([0, 1]^3\)

The Hartmann functions are a family of multimodal test functions commonly used in optimization benchmarks.

See also

test-functions for an overview of all test functions, get_function_details to retrieve function parameters, F20 for the 6D version.

Examples

F19(c(0.114614, 0.555649, 0.852547))  # Returns approximately -3.86278
#> [1] -3.862782
F19(c(0.5, 0.5, 0.5))  # Returns a value > -3.86
#> [1] -0.6280221