Package 'marinepredator' reference manual

Title:	Marine Predators Algorithm
Description:	Implementation of the Marine Predators Algorithm (MPA) in R. MPA is a nature-inspired optimization algorithm that follows the rules governing optimal foraging strategy and encounter rate policy between predator and prey in marine ecosystems. Based on the paper by Faramarzi et al. (2020) <doi:10.1016/j.eswa.2020.113377>.
Authors:	Marc Grossouvre [trl, cre] (R implementation), Afshin Faramarzi [aut] (Original MATLAB code), Seyedali Mirjalili [aut] (Original MATLAB code)
Maintainer:	Marc Grossouvre <[email protected]>
License:	MIT + file LICENSE
Version:	0.0.1
Built:	2026-06-05 09:17:53 UTC
Source:	https://github.com/urbs-dev/marinepredator

Sphere Function (F01)

Description

The Sphere function is a simple unimodal test function commonly used to evaluate the performance of optimization algorithms. It has a single global minimum at the origin.

Usage

F01(x)
F01(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \sum_{i=1}^{n} x_i^2$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Unimodal
Separable: Yes
Differentiable: Yes
Convex: Yes
Default bounds: $[-100, 100]^n$
Default dimensions: 50

Value

Numeric scalar representing the function value.

Examples

F01(c(0, 0, 0))    # Returns 0 (global minimum)
F01(c(1, 2, 3))    # Returns 1 + 4 + 9 = 14
F01(rep(0, 50))    # Returns 0 in 50 dimensions

F01(c(0, 0, 0))    # Returns 0 (global minimum)
F01(c(1, 2, 3))    # Returns 1 + 4 + 9 = 14
F01(rep(0, 50))    # Returns 0 in 50 dimensions

Sum of Absolute Values and Products (F02)

Description

A unimodal test function that combines the sum of absolute values and the product of absolute values.

Usage

F02(x)
F02(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \sum_{i=1}^{n} |x_i| + \prod_{i=1}^{n} |x_i|$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Unimodal
Separable: No (due to product term)
Differentiable: No (at points where any x_i = 0)
Convex: Yes
Default bounds: $[-10, 10]^n$
Default dimensions: 50

Value

Numeric scalar representing the function value.

Examples

F02(c(0, 0))      # Returns 0 (global minimum)
F02(c(1, 2))      # Returns |1| + |2| + |1|*|2| = 1 + 2 + 2 = 5
F02(c(-1, -2))    # Returns 1 + 2 + 2 = 5

F02(c(0, 0))      # Returns 0 (global minimum)
F02(c(1, 2))      # Returns |1| + |2| + |1|*|2| = 1 + 2 + 2 = 5
F02(c(-1, -2))    # Returns 1 + 2 + 2 = 5

Sum of Squared Cumulative Sums (F03)

Description

A unimodal test function that calculates the sum of squared cumulative sums, also known as the Schwefel 2.22 variant.

Usage

F03(x)
F03(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \sum_{i=1}^{n} \left(\sum_{j=1}^{i} x_j\right)^2$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Unimodal
Separable: No
Differentiable: Yes
Convex: Yes
Default bounds: $[-100, 100]^n$
Default dimensions: 50

This function is non-separable because each term depends on all previous variables, making it useful for testing algorithms' ability to handle variable dependencies.

Value

Numeric scalar representing the function value.

Examples

F03(c(0, 0, 0))    # Returns 0 (global minimum)
F03(c(1, 2, 3))    # Returns 1^2 + (1+2)^2 + (1+2+3)^2 = 1 + 9 + 36 = 46

F03(c(0, 0, 0))    # Returns 0 (global minimum)
F03(c(1, 2, 3))    # Returns 1^2 + (1+2)^2 + (1+2+3)^2 = 1 + 9 + 36 = 46

Maximum Absolute Value (F04)

Description

A unimodal test function that returns the maximum absolute value in the input vector, also known as the Schwefel 2.21 function.

Usage

F04(x)
F04(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \max_i |x_i|$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Unimodal
Separable: No
Differentiable: No
Convex: Yes
Default bounds: $[-100, 100]^n$
Default dimensions: 50

This function is particularly challenging because it only depends on the single variable with the largest absolute value, making gradient-based information less useful.

Value

Numeric scalar representing the function value.

Examples

F04(c(0, 0, 0))    # Returns 0 (global minimum)
F04(c(-1, 2, -3))  # Returns 3
F04(c(5, -5, 5))   # Returns 5

F04(c(0, 0, 0))    # Returns 0 (global minimum)
F04(c(-1, 2, -3))  # Returns 3
F04(c(5, -5, 5))   # Returns 5

The Rosenbrock function (also known as Rosenbrock's valley or banana function) is a classic test function with a narrow, curved valley. While often described as unimodal, it is technically multimodal for $n \geq 4$ .

Usage

F05(x)
F05(x)

Arguments

x

Numeric vector of input values (minimum length 2).

Details

Formula:

$f(x) = \sum_{i=1}^{n-1} \left[100(x_{i+1} - x_i^2)^2 + (x_i - 1)^2\right]$

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

Characteristics:

Type: Unimodal (for n < 4), Multimodal (for n >= 4)
Separable: No
Differentiable: Yes
Convex: No
Default bounds: $[-30, 30]^n$
Default dimensions: 50

The global minimum lies inside a long, narrow, parabolic-shaped flat valley. Finding the valley is trivial, but converging to the global minimum is difficult.

Value

Numeric scalar representing the function value.

Examples

F05(c(1, 1))       # Returns 0 (global minimum)
F05(c(0, 0))       # Returns 1
F05(c(1, 1, 1, 1)) # Returns 0 (global minimum in 4D)

F05(c(1, 1))       # Returns 0 (global minimum)
F05(c(0, 0))       # Returns 1
F05(c(1, 1, 1, 1)) # Returns 0 (global minimum in 4D)

Step Function / Shifted Sphere (F06)

Description

A shifted version of the Sphere function with the minimum at $(-0.5, -0.5, ..., -0.5)$ , also known as the Step function.

Usage

F06(x)
F06(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \sum_{i=1}^{n} (x_i + 0.5)^2$

Global minimum: $f(-0.5, -0.5, ..., -0.5) = 0$

Characteristics:

Type: Unimodal
Separable: Yes
Differentiable: Yes
Convex: Yes
Default bounds: $[-100, 100]^n$
Default dimensions: 50

This function tests the algorithm's ability to find optima that are not located at the origin.

Value

Numeric scalar representing the function value.

Examples

F06(c(-0.5, -0.5))       # Returns 0 (global minimum)
F06(c(0, 0))             # Returns 0.5
F06(rep(-0.5, 50))       # Returns 0 in 50 dimensions

F06(c(-0.5, -0.5))       # Returns 0 (global minimum)
F06(c(0, 0))             # Returns 0.5
F06(rep(-0.5, 50))       # Returns 0 in 50 dimensions

Quartic Function with Noise (F07)

Description

A unimodal test function with quartic terms and added uniform random noise. The noise makes this a stochastic function, useful for testing robustness.

Usage

F07(x)
F07(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \sum_{i=1}^{n} i \cdot x_i^4 + \text{random}[0,1)$

Global minimum: $f(0, 0, ..., 0) \approx 0$ (stochastic, depends on noise)

Characteristics:

Type: Unimodal (with noise)
Separable: Yes (deterministic part)
Differentiable: Yes
Stochastic: Yes (random noise added)
Default bounds: $[-1.28, 1.28]^n$
Default dimensions: 50

The random noise component makes function evaluations non-deterministic, testing an algorithm's ability to handle noisy objective functions.

Value

Numeric scalar representing the function value.

Examples

F07(c(0, 0, 0))  # Returns a value close to 0 (with some noise)
# Multiple calls may return different values due to noise:
replicate(5, F07(c(0, 0, 0)))

F07(c(0, 0, 0))  # Returns a value close to 0 (with some noise)
# Multiple calls may return different values due to noise:
replicate(5, F07(c(0, 0, 0)))

Schwefel Function (F08)

Description

A multimodal test function with many local minima. The global minimum is geometrically distant from the next best local minima, making this function deceptive and challenging for optimization algorithms.

Usage

F08(x)
F08(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = -\sum_{i=1}^{n} x_i \sin(\sqrt{|x_i|})$

Global minimum: $f(420.9687, ..., 420.9687) \approx -418.9829 \times n$

Characteristics:

Type: Multimodal
Separable: Yes
Differentiable: Yes (except at x_i = 0)
Default bounds: $[-500, 500]^n$
Default dimensions: 50

The Schwefel function is deceptive in that the global minimum is geometrically distant from the next best local minima. This tests an algorithm's ability to escape local optima and explore widely.

Value

Numeric scalar representing the function value.

Examples

F08(c(420.9687, 420.9687))  # Returns approximately -837.97 (near global minimum)
F08(c(0, 0))                # Returns 0

F08(c(420.9687, 420.9687))  # Returns approximately -837.97 (near global minimum)
F08(c(0, 0))                # Returns 0

Rastrigin Function (F09)

Description

A highly multimodal test function with many local minima arranged in a regular lattice pattern. The global minimum is at the origin.

Usage

F09(x)
F09(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = 10n + \sum_{i=1}^{n} \left[x_i^2 - 10\cos(2\pi x_i)\right]$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Multimodal
Separable: Yes
Differentiable: Yes
Number of local minima: $\approx 10^n$
Default bounds: $[-5.12, 5.12]^n$
Default dimensions: 50

The Rastrigin function is a typical example of non-linear multimodal function. The large number of local minima makes it difficult for optimization algorithms to find the global minimum.

Value

Numeric scalar representing the function value.

Examples

F09(c(0, 0))    # Returns 0 (global minimum)
F09(c(1, 1))    # Returns approximately 2
F09(rep(0, 10)) # Returns 0 in 10 dimensions

F09(c(0, 0))    # Returns 0 (global minimum)
F09(c(1, 1))    # Returns approximately 2
F09(rep(0, 10)) # Returns 0 in 10 dimensions

Ackley Function (F10)

Description

A widely used multimodal test function characterized by a nearly flat outer region with a large central hole at the origin. It poses a risk for optimization algorithms to be trapped in local minima.

Usage

F10(x)
F10(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = -20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}\right) - \exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_i)\right) + 20 + e$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Default bounds: $[-32, 32]^n$
Default dimensions: 50

The Ackley function has an exponential term covering its surface with numerous local minima. The function poses a risk of premature convergence for hill-climbing algorithms.

Value

Numeric scalar representing the function value.

Examples

F10(c(0, 0))  # Returns approximately 0 (global minimum)
F10(c(1, 1))  # Returns approximately 3.6

F10(c(0, 0))  # Returns approximately 0 (global minimum)
F10(c(1, 1))  # Returns approximately 3.6

Griewank Function (F11)

Description

A multimodal test function with many regularly distributed local minima. The complexity increases with the number of dimensions.

Usage

F11(x)
F11(x)

Arguments

x

Numeric vector of input values.

Details

Formula:

$f(x) = \frac{1}{4000}\sum_{i=1}^{n}x_i^2 - \prod_{i=1}^{n}\cos\left(\frac{x_i}{\sqrt{i}}\right) + 1$

Global minimum: $f(0, 0, ..., 0) = 0$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Default bounds: $[-600, 600]^n$
Default dimensions: 50

The Griewank function has a product term that introduces interdependence among the variables. As dimension increases, the function becomes more difficult to optimize.

Value

Numeric scalar representing the function value.

Examples

F11(c(0, 0))  # Returns 0 (global minimum)
F11(c(1, 1))  # Returns approximately 0.007

F11(c(0, 0))  # Returns 0 (global minimum)
F11(c(1, 1))  # Returns approximately 0.007

Penalized Function 1 (F12)

Description

A multimodal test function with penalty terms that create a complex search landscape. Uses the Ufun helper function.

Usage

F12(x)
F12(x)

Arguments

x

Numeric vector of input values (minimum length 2).

Details

Formula:

$f(x) = \frac{\pi}{n}\left\{10\sin^2(\pi y_1) + \sum_{i=1}^{n-1}(y_i-1)^2(1 + 10 \sin^2(\pi y_{i+1})) + (y_n-1)^2\right\} + \sum_{i=1}^{n}u(x_i,10,100,4)$

where $y_i = 1 + (x_i + 1)/4$ and $u(x,a,k,m)$ 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

The penalty term $u(x,a,k,m)$ adds a cost when variables exceed the threshold $a$ , helping to constrain solutions to a feasible region.

Value

Numeric scalar representing the function value.

Examples

F12(c(-1, -1, -1))  # Returns approximately 0 (near global minimum)
F12(c(0, 0, 0))     # Returns a small positive value

F12(c(-1, -1, -1))  # Returns approximately 0 (near global minimum)
F12(c(0, 0, 0))     # Returns a small positive value

Penalized Function 2 (F13)

Description

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

Usage

F13(x)
F13(x)

Arguments

x

Numeric vector of input values (minimum length 2).

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.

Value

Numeric scalar representing the function value.

Examples

F13(c(1, 1, 1))  # Returns approximately 0 (near global minimum)
F13(c(0, 0, 0))  # Returns a small positive value

F13(c(1, 1, 1))  # Returns approximately 0 (near global minimum)
F13(c(0, 0, 0))  # Returns a small positive value

Shekel's Foxholes Function (F14)

Description

A multimodal test function with 25 local minima of different depths, arranged in a grid pattern. Fixed dimension of 2.

Usage

F14(x)
F14(x)

Arguments

x

Numeric vector of length 2 (2-dimensional input).

Details

Formula:

$f(x) = \left(\frac{1}{500} + \sum_{j=1}^{25}\frac{1}{j + \sum_{i=1}^{2}(x_i - a_{ij})^6}\right)^{-1}$

where $a_{ij}$ are predefined constants forming a 5x5 grid.

Global minimum: $f(-32, -32) \approx 0.998$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 2
Number of local minima: 25
Default bounds: $[-65.536, 65.536]^2$

The 25 minima are located at points from a 5x5 grid with coordinates $(-32, -16, 0, 16, 32)$ in each dimension.

Value

Numeric scalar representing the function value.

Examples

F14(c(-32, -32))  # Returns approximately 0.998 (global minimum)
F14(c(0, 0))      # Returns approximately 10

F14(c(-32, -32))  # Returns approximately 0.998 (global minimum)
F14(c(0, 0))      # Returns approximately 10

Kowalik Function (F15)

Description

A multimodal test function used for testing optimization algorithms, based on fitting experimental data. Fixed dimension of 4.

Usage

F15(x)
F15(x)

Arguments

x

Numeric vector of length 4 (4-dimensional input).

Details

Formula:

$f(x) = \sum_{i=1}^{11}\left(a_i - \frac{x_1(b_i^2 + b_i x_2)}{b_i^2 + b_i x_3 + x_4}\right)^2$

where $a_i$ and $b_i$ are predefined constants from experimental data.

Global minimum: $f(0.1928, 0.1908, 0.1231, 0.1358) \approx 0.0003075$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 4
Default bounds: $[-5, 5]^4$

This function is derived from a curve-fitting problem and has several local minima near the global minimum.

Value

Numeric scalar representing the function value.

Examples

F15(c(0.1928, 0.1908, 0.1231, 0.1358))  # Returns approximately 0.0003
F15(c(0, 0, 0, 0))  # Returns a larger value

F15(c(0.1928, 0.1908, 0.1231, 0.1358))  # Returns approximately 0.0003
F15(c(0, 0, 0, 0))  # Returns a larger value

Six-Hump Camel Back Function (F16)

Description

A multimodal test function with six local minima, two of which are global. Fixed dimension of 2.

Usage

F16(x)
F16(x)

Arguments

x

Numeric vector of length 2 (2-dimensional input).

Details

Formula:

$f(x) = 4x_1^2 - 2.1x_1^4 + \frac{x_1^6}{3} + x_1 x_2 - 4x_2^2 + 4x_2^4$

Global minimum: $f(\pm 0.0898, \mp 0.7126) \approx -1.0316$

There are two global minima at approximately $(0.0898, -0.7126)$ and $(-0.0898, 0.7126)$ .

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 2
Number of local minima: 6
Number of global minima: 2
Default bounds: $[-5, 5]^2$

The function has a shape resembling a camel's back with six humps (minima).

Value

Numeric scalar representing the function value.

Examples

F16(c(0.0898, -0.7126))   # Returns approximately -1.0316 (global minimum)
F16(c(-0.0898, 0.7126))   # Returns approximately -1.0316 (global minimum)
F16(c(0, 0))              # Returns 0

F16(c(0.0898, -0.7126))   # Returns approximately -1.0316 (global minimum)
F16(c(-0.0898, 0.7126))   # Returns approximately -1.0316 (global minimum)
F16(c(0, 0))              # Returns 0

Branin Function (F17)

Description

A multimodal test function with three global minima, commonly used for testing optimization algorithms. Fixed dimension of 2.

Usage

F17(x)
F17(x)

Arguments

x

Numeric vector of length 2 (2-dimensional input).

Details

Formula:

$f(x) = \left(x_2 - \frac{5.1}{4\pi^2}x_1^2 + \frac{5}{\pi}x_1 - 6\right)^2 + 10\left(1 - \frac{1}{8\pi}\right)\cos(x_1) + 10$

Global minimum: $f^* \approx 0.397887$ at three locations:

$(-\pi, 12.275)$
$(\pi, 2.275)$
$(9.42478, 2.475)$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 2
Number of global minima: 3
Default bounds: $x_1 \in [-5, 10]$ , $x_2 \in [0, 15]$

The Branin function is often used as a benchmark because it has three global minima that are well-separated.

Value

Numeric scalar representing the function value.

Examples

F17(c(-pi, 12.275))      # Returns approximately 0.398 (global minimum)
F17(c(pi, 2.275))        # Returns approximately 0.398 (global minimum)
F17(c(9.42478, 2.475))   # Returns approximately 0.398 (global minimum)

F17(c(-pi, 12.275))      # Returns approximately 0.398 (global minimum)
F17(c(pi, 2.275))        # Returns approximately 0.398 (global minimum)
F17(c(9.42478, 2.475))   # Returns approximately 0.398 (global minimum)

Goldstein-Price Function (F18)

Description

A multimodal test function with four local minima and one global minimum. Fixed dimension of 2.

Usage

F18(x)
F18(x)

Arguments

x

Numeric vector of length 2 (2-dimensional input).

Details

Formula:

$f(x) = (1 + (x_1 + x_2 + 1)^2(19 - 14x_1 + 3x_1^2 - 14x_2 + 6x_1 x_2 + 3x_2^2)) \times (30 + (2x_1 - 3x_2)^2(18 - 32x_1 + 12x_1^2 + 48x_2 - 36x_1 x_2 + 27x_2^2))$

Global minimum: $f(0, -1) = 3$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 2
Number of local minima: 4
Default bounds: $[-2, 2]^2$

The Goldstein-Price function has a complex landscape with the global minimum surrounded by local minima of increasing value.

Value

Numeric scalar representing the function value.

Examples

F18(c(0, -1))  # Returns 3 (global minimum)
F18(c(0, 0))   # Returns 600

F18(c(0, -1))  # Returns 3 (global minimum)
F18(c(0, 0))   # Returns 600

Hartmann 3D Function (F19)

Description

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

Usage

F19(x)
F19(x)

Arguments

x

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

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.

Value

Numeric scalar representing the function value.

Examples

F19(c(0.114614, 0.555649, 0.852547))  # Returns approximately -3.86278
F19(c(0.5, 0.5, 0.5))  # Returns a value > -3.86

F19(c(0.114614, 0.555649, 0.852547))  # Returns approximately -3.86278
F19(c(0.5, 0.5, 0.5))  # Returns a value > -3.86

Hartmann 6D Function (F20)

Description

A multimodal test function with 6 local minima in 6 dimensions. Fixed dimension of 6.

Usage

F20(x)
F20(x)

Arguments

x

Numeric vector of length 6 (6-dimensional input).

Details

Formula:

$f(x) = -\sum_{i=1}^{4} c_i \exp\left(-\sum_{j=1}^{6} a_{ij}(x_j - p_{ij})^2\right)$

where $a_{ij}$ , $c_i$ , and $p_{ij}$ are predefined constants.

Global minimum: $f(0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573) \approx -3.32237$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 6
Number of local minima: 6
Default bounds: $[0, 1]^6$

The Hartmann 6D function is more challenging than the 3D version due to the higher dimensionality.

Value

Numeric scalar representing the function value.

Examples

F20(c(0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573))
# Returns approximately -3.32237

F20(c(0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573))
# Returns approximately -3.32237

Shekel 5 Function (F21)

Description

A multimodal test function from the Shekel family with 5 local minima. Fixed dimension of 4.

Usage

F21(x)
F21(x)

Arguments

x

Numeric vector of length 4 (4-dimensional input).

Details

Formula:

$f(x) = -\sum_{i=1}^{5} \frac{1}{(x - a_i)^T(x - a_i) + c_i}$

where $a_i$ are 4-dimensional vectors and $c_i$ are scalars.

Global minimum: $f(4, 4, 4, 4) \approx -10.1532$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 4
Number of local minima: 5
Default bounds: $[0, 10]^4$

The Shekel functions are parameterized by the number of terms m (here m=5). As m increases, the function becomes more challenging.

Value

Numeric scalar representing the function value.

Examples

F21(c(4, 4, 4, 4))  # Returns approximately -10.15 (near global minimum)
F21(c(0, 0, 0, 0))  # Returns a value close to 0

F21(c(4, 4, 4, 4))  # Returns approximately -10.15 (near global minimum)
F21(c(0, 0, 0, 0))  # Returns a value close to 0

Shekel 7 Function (F22)

Description

A multimodal test function from the Shekel family with 7 local minima. Fixed dimension of 4.

Usage

F22(x)
F22(x)

Arguments

x

Numeric vector of length 4 (4-dimensional input).

Details

Formula:

$f(x) = -\sum_{i=1}^{7} \frac{1}{(x - a_i)^T(x - a_i) + c_i}$

where $a_i$ are 4-dimensional vectors and $c_i$ are scalars.

Global minimum: $f(4, 4, 4, 4) \approx -10.4029$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 4
Number of local minima: 7
Default bounds: $[0, 10]^4$

The Shekel 7 function has more local minima than Shekel 5, making it slightly more challenging.

Value

Numeric scalar representing the function value.

Examples

F22(c(4, 4, 4, 4))  # Returns approximately -10.40 (near global minimum)
F22(c(0, 0, 0, 0))  # Returns a value close to 0

F22(c(4, 4, 4, 4))  # Returns approximately -10.40 (near global minimum)
F22(c(0, 0, 0, 0))  # Returns a value close to 0

Shekel 10 Function (F23)

Description

A multimodal test function from the Shekel family with 10 local minima. Fixed dimension of 4.

Usage

F23(x)
F23(x)

Arguments

x

Numeric vector of length 4 (4-dimensional input).

Details

Formula:

$f(x) = -\sum_{i=1}^{10} \frac{1}{(x - a_i)^T(x - a_i) + c_i}$

where $a_i$ are 4-dimensional vectors and $c_i$ are scalars.

Global minimum: $f(4, 4, 4, 4) \approx -10.5364$

Characteristics:

Type: Multimodal
Separable: No
Differentiable: Yes
Fixed dimension: 4
Number of local minima: 10
Default bounds: $[0, 10]^4$

The Shekel 10 function is the most challenging of the Shekel family due to having the most local minima.

Value

Numeric scalar representing the function value.

Examples

F23(c(4, 4, 4, 4))  # Returns approximately -10.54 (near global minimum)
F23(c(0, 0, 0, 0))  # Returns a value close to 0

F23(c(4, 4, 4, 4))  # Returns approximately -10.54 (near global minimum)
F23(c(0, 0, 0, 0))  # Returns a value close to 0

Get Function Details

Description

Retrieves the details (bounds, dimension, and function object) for a specific benchmark test function.

Usage

get_function_details(function_name)
get_function_details(function_name)

Arguments

function_name

Name of the test function (e.g., "F01", "F02", ..., "F23")

Value

A list containing:

lb

Lower bounds for the function

ub

Upper bounds for the function

dim

Number of dimensions

fobj

The objective function

Examples

# Get details for the Sphere function (F01)
details <- get_function_details("F01")
str(details)

# Get details for the Sphere function (F01)
details <- get_function_details("F01")
str(details)

Initialize Population

Description

Initializes the first population of search agents (prey) for the Marine Predators Algorithm.

Usage

initialize_population(SearchAgents_no, dim, ub, lb)
initialize_population(SearchAgents_no, dim, ub, lb)

Arguments

SearchAgents_no

Number of search agents

dim

Number of dimensions

ub

Upper bounds for each dimension

lb

Lower bounds for each dimension

Value

A matrix of initialized positions

Examples

# Initialize a population of 25 agents in 30 dimensions with bounds [-100, 100]
population <- initialize_population(25, 30, 100, -100)

# Initialize a population of 25 agents in 30 dimensions with bounds [-100, 100]
population <- initialize_population(25, 30, 100, -100)

Levy Flight Random Step Generator

Description

Generates Levy flight random steps for optimization algorithms. Levy flight is characterized by a heavy-tailed power-law distribution and is used to model the movement patterns of marine predators, allowing for occasional long jumps that help escape local optima.

Usage

levy(n, m, beta)
levy(n, m, beta)

Arguments

n

Number of rows (typically the number of search agents).

m

Number of columns (typically the number of dimensions).

beta

Power law exponent controlling the tail heaviness of the distribution. Must satisfy 1 < beta < 2. Lower values produce heavier tails (more frequent long jumps). The value 1.5 is commonly used.

Details

The Levy flight step is computed using Mantegna's algorithm:

$L = u / |v|^{1/\beta}$

where $u \sim N(0, \sigma_u^2)$ and $v \sim N(0, 1)$ , with:

$\sigma_u = \left( \frac{\Gamma(1+\beta) \cdot \sin(\pi \beta / 2)} {\Gamma((1+\beta)/2) \cdot \beta \cdot 2^{(\beta-1)/2}} \right)^{1/\beta}$

Levy flights are used in MPA during Phase 2 (for half the population) and Phase 3 (for all agents) to balance exploration and exploitation.

Value

A numeric matrix of dimension n x m containing Levy flight step values. Each element represents a random step drawn from the Levy distribution.

References

Mantegna, R. N. (1994). Fast, accurate algorithm for numerical simulation of Levy stable stochastic processes. Physical Review E, 49(5), 4677.

Yang, X. S., & Deb, S. (2013). Multiobjective cuckoo search for design optimization. Computers & Operations Research, 40(6), 1616-1624.

Examples

# Generate 10 Levy flight steps in 2 dimensions with beta = 1.5
steps <- levy(10, 2, 1.5)
dim(steps)  # 10 x 2

# Visualize the distribution of Levy steps
steps_1d <- levy(1000, 1, 1.5)
hist(steps_1d, breaks = 50, main = "Levy Flight Distribution")

# Generate 10 Levy flight steps in 2 dimensions with beta = 1.5
steps <- levy(10, 2, 1.5)
dim(steps)  # 10 x 2

# Visualize the distribution of Levy steps
steps_1d <- levy(1000, 1, 1.5)
hist(steps_1d, breaks = 50, main = "Levy Flight Distribution")

Marine Predators Algorithm (MPA)

Description

Implementation of the Marine Predators Algorithm (MPA) in R. MPA is a nature-inspired optimization algorithm that follows the rules governing optimal foraging strategy and encounter rate policy between predator and prey in marine ecosystems.

Usage

mpa(SearchAgents_no, Max_iter, lb, ub, dim, fobj, logFile = NULL, ...)
mpa(SearchAgents_no, Max_iter, lb, ub, dim, fobj, logFile = NULL, ...)

Arguments

SearchAgents_no

Number of search agents (predators). At least 2. Typical values range from 20 to 50.

Max_iter

Maximum number of iterations. At least 3. Typical values range from 100 to 500 depending on problem complexity.

lb

Lower bounds for each dimension. Can be a single value (applied to all dimensions) or a vector of length dim.

ub

Upper bounds for each dimension. Can be a single value (applied to all dimensions) or a vector of length dim.

dim

Number of dimensions (decision variables) in the optimization problem.

fobj

Objective function to minimize. Must accept a numeric vector of length dim and return a single numeric value.

logFile

A path for logging (text file). Defaulted to NULL. If NULL, no logging is performed.

...

Additional arguments. Currently supports prefix for log message prefixing. See Details.

Details

This is a minimization algorithm. To maximize a function, negate its output (e.g., use function(x) -f(x) instead of f).

Value

An object of class mpa_result, which is a list containing:

Top_predator_fit

Best fitness value found (numeric scalar)

Top_predator_pos

Best position found (numeric vector of length dim)

Convergence_curve

Convergence curve over iterations (numeric vector of length Max_iter)

Algorithm Phases

The MPA algorithm operates in three distinct phases based on the iteration count:

Phase 1 (iterations 0 to Max_iter/3): High velocity ratio - The prey moves faster than the predator. Exploration is emphasized using Brownian motion. This phase promotes global search across the solution space.
Phase 2 (iterations Max_iter/3 to 2*Max_iter/3): Unit velocity ratio - Predator and prey move at similar speeds. The population is split: half uses Brownian motion (exploitation), half uses Levy flight (exploration). This balances exploration and exploitation.
Phase 3 (iterations 2*Max_iter/3 to Max_iter): Low velocity ratio - The predator moves faster than the prey. Levy flight is used for all agents, focusing on exploitation around the best solution found.

Internal Parameters

The algorithm uses two internal parameters that are not exposed:

FADs: Fish Aggregating Devices effect parameter, set to 0.2. Controls the probability of applying the FADs effect which helps escape local optima.
P: Prey movement probability, set to 0.5. Controls the step size scaling factor during position updates.

Memory Mechanism

MPA implements a memory mechanism (Marine Memory) that preserves the best positions found by each agent. If a new position has worse fitness than the previous one, the agent reverts to its previous position.

Additional Arguments

The ... parameter currently accepts:

prefix: A character string to prefix log messages. Defaults to empty string if not provided.

References

Faramarzi, A., Heidarinejad, M., Mirjalili, S., & Gandomi, A. H. (2020). Marine Predators Algorithm: A Nature-inspired Metaheuristic. Expert Systems with Applications, 152, 113377. doi:10.1016/j.eswa.2020.113377

Examples

# Basic usage with the Sphere function (F01)
result <- mpa(
  SearchAgents_no = 25, Max_iter = 100, lb = -100, ub = 100,
  dim = 30, fobj = F01
)
print(result)

# Using different bounds per dimension
result2 <- mpa(
  SearchAgents_no = 20, Max_iter = 50,
  lb = c(-5, 0), ub = c(10, 15),
  dim = 2, fobj = F17
)

# Maximization example (negate the objective function)
maximize_f <- function(x) -sum(x^2)
result3 <- mpa(
  SearchAgents_no = 20, Max_iter = 50,
  lb = -10, ub = 10, dim = 5,
  fobj = function(x) -maximize_f(x)
)
# The actual maximum value is -result3$Top_predator_fit

# Basic usage with the Sphere function (F01)
result <- mpa(
  SearchAgents_no = 25, Max_iter = 100, lb = -100, ub = 100,
  dim = 30, fobj = F01
)
print(result)

# Using different bounds per dimension
result2 <- mpa(
  SearchAgents_no = 20, Max_iter = 50,
  lb = c(-5, 0), ub = c(10, 15),
  dim = 2, fobj = F17
)

# Maximization example (negate the objective function)
maximize_f <- function(x) -sum(x^2)
result3 <- mpa(
  SearchAgents_no = 20, Max_iter = 50,
  lb = -10, ub = 10, dim = 5,
  fobj = function(x) -maximize_f(x)
)
# The actual maximum value is -result3$Top_predator_fit

Print method for MPA results

Description

Prints a summary of the Marine Predators Algorithm optimization results.

Usage

## S3 method for class 'mpa_result'
print(x, ...)
## S3 method for class 'mpa_result'
print(x, ...)

Arguments

x

An object of class mpa_result returned by [mpa()].

...

Additional arguments (currently unused).

Value

Invisibly returns the input object x.

Examples

result <- mpa(SearchAgents_no = 10, Max_iter = 20, lb = -10, ub = 10,
              dim = 5, fobj = F01)
print(result)

result <- mpa(SearchAgents_no = 10, Max_iter = 20, lb = -10, ub = 10,
              dim = 5, fobj = F01)
print(result)

Test Functions for Optimization

Description

Collection of benchmark test functions for evaluating optimization algorithms. These functions are commonly used in the literature to test the performance of metaheuristic optimization algorithms.

Details

The following test functions are implemented:

Unimodal Functions:

F01 - Sphere function: Simple quadratic function with a single minimum at the origin. F01
F02 - Sum of absolute values and products: Combines sum and product of absolute values. F02
F03 - Sum of squared terms: Sum of squared cumulative sums. F03
F04 - Maximum absolute value: Returns the maximum absolute value in the vector. F04
F05 - Rosenbrock function: Classic test function with a banana-shaped valley. F05
F06 - Shifted sphere function: Sphere function shifted to (-0.5, -0.5, ...). F06
F07 - Quartic function with noise: Quartic terms with added random noise. F07

Multimodal Functions:

F08 - Schwefel function: Multimodal function with many local minima. F08
F09 - Rastrigin function: Many local minima arranged in a regular pattern. F09
F10 - Ackley function: Global minimum at origin with many local minima. F10
F11 - Griewank function: Many local minima regularly distributed. F11
F12 - Penalized function 1: Complex landscape with penalty terms. F12
F13 - Penalized function 2: Different penalty terms creating complex landscape. F13
F14 - Shekel's Foxholes: 25 minima of different depths. F14
F15 - Kowalik function: Multimodal function for testing optimization. F15
F16 - Six-Hump Camel Back: Six local minima, two global. F16
F17 - Branin function: Commonly used for testing optimization algorithms. F17
F18 - Goldstein-Price function: Four local minima. F18
F19 - Hartman 3D function: 3 dimensions with multiple local minima. F19
F20 - Hartman 6D function: 6 dimensions with multiple local minima. F20
F21 - Shekel 5 function: Based on Shekel's family with 5 terms. F21
F22 - Shekel 7 function: Based on Shekel's family with 7 terms. F22
F23 - Shekel 10 function: Based on Shekel's family with 10 terms. F23

References

Faramarzi, A., Heidarinejad, M., Mirjalili, S., & Gandomi, A. H. (2020). Marine Predators Algorithm: A Nature-inspired Metaheuristic. Expert Systems with Applications, 113377. DOI: 10.1016/j.eswa.2020.113377

Examples

# Get details for a specific function
details <- get_function_details("F01")

# Use a test function directly
F01(c(0, 0, 0))  # Returns 0 (global minimum)

# List all available test functions
available_functions <- c("F01", "F02", "F03", "F04", "F05", "F06", "F07", "F08", "F09", "F10",
                       "F11", "F12", "F13", "F14", "F15", "F16", "F17", "F18", "F19", "F20",
                       "F21", "F22", "F23")

# Get details for a specific function
details <- get_function_details("F01")

# Use a test function directly
F01(c(0, 0, 0))  # Returns 0 (global minimum)

# List all available test functions
available_functions <- c("F01", "F02", "F03", "F04", "F05", "F06", "F07", "F08", "F09", "F10",
                       "F11", "F12", "F13", "F14", "F15", "F16", "F17", "F18", "F19", "F20",
                       "F21", "F22", "F23")

Package 'marinepredator'

Help Index

Sphere Function (F01)

Description

Usage

Arguments

Details

Value

See Also

Examples

Sum of Absolute Values and Products (F02)

Description

Usage

Arguments

Details

Value

See Also

Examples

Sum of Squared Cumulative Sums (F03)

Description

Usage

Arguments

Details

Value

See Also

Examples

Maximum Absolute Value (F04)

Description

Usage

Arguments

Details

Value

See Also

Examples

Rosenbrock Function (F05)

Description

Usage

Arguments

Details

Value

See Also

Examples

Step Function / Shifted Sphere (F06)

Description

Usage

Arguments

Details

Value

See Also

Examples

Quartic Function with Noise (F07)

Description

Usage

Arguments

Details

Value

See Also

Examples

Schwefel Function (F08)

Description

Usage

Arguments

Details

Value

See Also

Examples

Rastrigin Function (F09)

Description

Usage

Arguments

Details

Value

See Also

Examples

Ackley Function (F10)

Description

Usage

Arguments

Details

Value