Penalized Function 2 (F13)

Source: R/F13.R

A multimodal test function with different penalty terms than F12, creating a complex search landscape. Uses the Ufun helper function.

F13(x)

Arguments

x

Numeric vector of input values (minimum length 2).

Value

Numeric scalar representing the function value.

Details

Formula: $$f(x) = 0.1\left\{\sin^2(3\pi x_1) + \sum_{i=1}^{n-1}(x_i-1)^2(1+\sin^2(3\pi x_{i+1})) + (x_n-1)^2(1+\sin^2(2\pi x_n))\right\} + \sum_{i=1}^{n}u(x_i,5,100,4)$$

where \(u(x,a,k,m)\) is a penalty function.

Global minimum: \(f(1, 1, ..., 1) = 0\)

Characteristics:

  • Type: Multimodal

  • Separable: No

  • Differentiable: Yes (in the interior)

  • Penalized: Yes (boundary penalty)

  • Default bounds: \([-50, 50]^n\)

  • Default dimensions: 50

This function differs from F12 in the sinusoidal terms and penalty threshold, resulting in different landscape characteristics.

See also

test-functions for an overview of all test functions, get_function_details to retrieve function parameters, Ufun for the penalty function.

Examples

F13(c(1, 1, 1))  # Returns approximately 0 (near global minimum)
#> [1] 1.349784e-32
F13(c(0, 0, 0))  # Returns a small positive value
#> [1] 0.3