ssFPA/DE: an efficient hybrid differential evolution–flower pollination algorithm based approach

Abstract

Evolutionary algorithm is a field of great interest to many researchers around the world. New algorithms are developed based on biological processes that exist in nature. In addition, different variants of the existing algorithms are also created with researchers working to find the most optimal method. This paper initially introduces Differential Evolution (DE) and Flower Pollination Algorithm (FPA). Subsequently, a description of the hybrid algorithm named ssFPA/DE that uses the search strategy of FPA and DE are explained along with their results.

Appendix

Benchmark functions used: Global optimization approaches should be verified with the benchmark functions or problems. A wide range of test functions are designed to signify the different parts of global optimization algorithm. They can be extended to arbitrary dimensionality to allow scaled testing. Unimodal and multimodal functions have been used in testing the algorithm. The various benchmark functions are discussed in detail:

• 1. Sphere function (f1)

The Sphere function has d local minima except for the global one. It is continuous, convex and unimodal. The maximum and minimum range is between (−5.12, 5.12).The equation is given as:

$$f(x) = \mathop \sum \limits_{i = 1}^{d} x_{i}^{2}$$

• 2. Beale function (f2)

The Beale function is multimodal, with sharp peaks at the corners of the input domain. The maximum and minimum range is between (−4.5, 4.5).

$$f(x) = (1.5 - x_{1} + x_{1} x_{2} )^{2} + (2.25 - x_{1} + x_{1} x_{2}^{2} )^{2} + (2.625 - x_{1} + x_{1} x_{2}^{3} )^{2}$$

• 3. Booth function (f3)

The function is usually evaluated on the square x_i ∈ [−10, 10], for all i = 1, 2. There are several local minima for this function. So it is a multimodal function.

$$f(x) = ( x_{1} + 2x_{2} - 7)^{2} + ( 2x_{1} + x_{2} - 5)^{2}$$

• 4. Schwefel function (f4)

The Schwefel function is complex, with many local minima. The plot shows the two-dimensional form of the function. The function is usually evaluated on the hypercube x_i ∈ [−500, 500], for all $i = 1, \ldots ,d$.

$$f(x) = 418.9829 d - \mathop \sum \limits_{i = 1}^{d} x_{i} \sin \left( {\sqrt {x_{i} } } \right)$$

• 5. Michalewicz function (f5)

The Michalewicz function has local minima, and it is multimodal. The parameter m defines the steepness of t valleys and ridges; a larger m leads to a more difficult search. The recommended value of m is m = 10.The maximum and minimum range is between (0, π).

$$f(x) = - \mathop \sum \limits_{i = 1}^{d} \sin (x_{i} )\sin^{2m} \left( {\frac{{i x_{i}^{2} }}{\pi }} \right)$$

• 6. Schaffner function N.2 (f6)

The second Schaffer function. It is shown on a smaller input domain in the second plot to show detail. The function is usually evaluated on the square x_i ∈ [−100, 100], for all i = 1, 2.

$$f(x,y) = 0.5 + \frac{{\sin^{2} (x^{2} - y^{2} ) - 0.5}}{{(1 + 0.001(x^{2} + y^{2} ))^{2} }}$$

• 7. Schaffner function N.4 (f7)

The fourth Schaffer function. It is shown on a smaller input domain in the second plot to show detail. The function is usually evaluated on the square x_i ∈ [−100, 100], for all i = 1, 2.

$$f(x,y) = 0.5 + \frac{{\cos^{2} (\sin^{2} (\left| {x^{2} - y^{2} } \right|)) - 0.5}}{{(1 + 0.001(x^{2} + y^{2} ))^{2} }}$$

• 8. HimmelBlau function (f8)

It is a multimodal function used to solve optimization problems. The function is evaluated on x_i ∈ [−5, 5], for all $i = 1, \ldots ,d$.

$$f(x,y) = (x^{2} + y - 11)^{2} + (y^{2} + x - 7)^{2}$$

• 9. Bird function (f9)

This is a bi-modal function with f(x*) 106.764537 in the search domain [2, 2]. The maximum and minimum range is given as (−2π, 2π).

$$f(x,y) = \sin (x)e^{{(1 - \cos (y))^{2} }} + \cos (y)e^{{(1 - \sin (x))^{2} }} + (x - y)^{2}$$

• 10. Extended cube function (f10)

This is a multimodal minimization problem for global optimization. Here, n represents the number of dimensions and the maximum and minimum range is between (−100, 100).

$$f(x) = \sum\limits_{i = 1}^{n} {100(x_{i + 1} - x_{i}^{3} )^{2} + (1 - x_{i} )^{2} }$$

• 11. Ackley function (f11)

This function is used mainly to test optimisation algorithms. The equation for this function is as given below:

$$f(x) = - a{\kern 1pt} \exp {\kern 1pt} \left( { - b\sqrt {\frac{1}{d}\sum\limits_{i = 1}^{d} {x_{i}^{2} } } } \right) - {\kern 1pt} \exp \left( {\frac{1}{d}\sum\limits_{i = 1}^{d} {\cos (cx_{i} )} } \right) + a + \exp (1){\kern 1pt}$$

The function is a risk for many optimisation problems to get trapped in one of its many local minima. The maximum and minimum range is between (−32, 32).

• 12. Goldstein-Price function (f12)

The Goldstein-Price function has several local minima. The function is usually evaluated on the square x_i ∈ [-2, 2], for all i = 1, 2. The maximum and minimum range is between (−2, 2).

$$f(x) = (1 + (x + y + 1)^{2} (19 - 14x + 3x^{2} - 14y + 6xy + 3y^{2} ))(30 + (2x - 3y)^{2} (18 - 32x + 12x^{2} + 48y - 36xy + 27y^{2} ))$$

• 13. Griewank function (f13)

The Griewank function has many widespread local minima, which are regularly distributed. The maximum and the minimum range is between (−600, 600).

$$f(x) = \mathop \sum \limits_{i = 1}^{d} \frac{{x_{i}^{2} }}{4000} - \mathop \prod \limits_{i = 1}^{d} \cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1$$

• 14. Rastrigen function (f14)

The Rastrigin function has several local minima. It is highly multimodal, but locations of the minima are regularly distributed. The maximum and minimum range is between (−15, 15).

$$f(x) = 10d + \mathop \sum \limits_{i = 1}^{d} [x_{i}^{2} - 10 \cos ( 2 \pi x_{i} )]$$

• 15. Rosenbrock function (f15)

The Rosenbrock function, also referred to as the Valley or Banana function, is a popular test problem for gradient-based optimization algorithms. The maximum and minimum range is between (−15, 15).

$$f(x) = \mathop \sum \limits_{i = 1}^{d - 1} \left[ { 100(x_{i + 1} - x_{i}^{2} )^{2} + ( x_{i} - 1) ^{2} } \right]$$

The function is unimodal, and the global minimum lies in a narrow, parabolic valley. However, even though this valley is easy to find, convergence to the minimum is difficult.