A heuristic optimization method inspired by wolf preying behavior

Abstract

Optimization problems can become intractable when the search space undergoes tremendous growth. Heuristic optimization methods have therefore been created that can search the very large spaces of candidate solutions. These methods, also called metaheuristics, are the general skeletons of algorithms that can be modified and extended to suit a wide range of optimization problems. Various researchers have invented a collection of metaheuristics inspired by the movements of animals and insects (e.g., firefly, cuckoos, bats and accelerated PSO) with the advantages of efficient computation and easy implementation. This paper studies a relatively new bio-inspired heuristic optimization algorithm called the Wolf Search Algorithm (WSA) that imitates the way wolves search for food and survive by avoiding their enemies. The WSA is tested quantitatively with different values of parameters and compared to other metaheuristic algorithms under a range of popular non-convex functions used as performance test problems for optimization algorithms, with superior results observed in most tests.

Appendix: fitness functions

We used the following test functions to test our algorithms.

• 1. Griewangk’s function: In this test function, there are a lot of local minima,

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

  The global minimum is f _min = 0 at (0,…,0), where the −600 < x _i  < 600.

• 2. Sphere function:

  $$f(x) = \sum\limits_{i = 1}^{d} {x_{i}^{2} } ,\quad {\text{where}}\;{\text{the}}\; - 5.12 < x_{i} < 5.12$$

  The global minimum is f _min = 0 at (0,…,0).

• 3. Rastrigin’s function: This function is difficult due to its large search space and large number of local minima.

  $$f(x) = An + \sum\limits_{i = 1}^{n} {\left[ {x_{i}^{2} - A\cos (2\pi x_{i} )} \right]}$$

  where the A = 10, x _i  ∊ [−5.12, 5.12] and the global minimum is f _min = 0 at (0,…,0).

• 4. Schaffer F6:

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

  where x _i  ∊ [10, 10] and the global minimum is f _min = 0 at (0,…,0).

• 5. Moved axis parallel hyperellipsoid function:

  $$f(x) = \sum\limits_{i = 1}^{n} {5i \cdot x_{i}^{2} }$$

  where x _i  ∊ [−5.12, 5.12]and the global minimum is f _min = 0 at (0,…,0).

• 6. The third Bohachevsky function:

  $$f(x) = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos (3\pi x_{1} ) + 0.3\cos (4\pi x_{2} ) + 0.3$$

  This function only has two variables, where x ∈ [−10, 10], the global minimum is f _min = −0.24 located in [−0.24,0], [0,0.24].

• 7. Michalewicz’s function:

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

  where m = 10. x _i  ∊ [0, π]. When d = 2, the global minimum is f _min = −1.803 at position (2.0230, 2.0230).

• 8. Rosenbrock function:

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

  where the global minimum is f _min = 0 at positions (1,…1).