GWO（灰色狼群优化）算法MATLAB源码详细中文注解

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《灰色狼群优化（GWO）算法在MATLAB中的应用详解》在现代优化领域，灰色狼群优化（Grey Wolf Optimizer，简称GWO）算法因其高效性和广泛的应用性而备受关注。GWO是一种模拟狼群行为的生物启发式算法，能够解决各种复杂优化问题。本文将基于MATLAB环境，详细解析GWO算法在支持向量机（SVM）参数优化中的应用，以及相关代码实现。我们需要理解GWO的基本原理。GWO由阿尔法（α）、贝塔（β）和德尔塔（δ）三只领导狼引领一群普通狼，通过不断调整位置来寻找最优解。这个过程中，阿尔法狼代表最优解，贝塔狼和德尔塔狼分别代表第二和第三优解，而其余的狼则跟随这三只领导狼进行搜索。算法的核心在于迭代过程中，狼群位置的更新策略，模拟了狼群狩猎时的群体动态。在MATLAB中，`GWO_SVM_exmp.m`和`GWO_SVR_exmp.m`是两个主要的源代码文件，分别用于SVM分类器和回归模型的参数优化。这两个文件会调用`initialization.m`进行初始化设置，包括设置问题维度、狼群规模、最大迭代次数等参数。`mymape.m`、`mymse.m`和`mymae.m`是评估函数，用于计算预测误差，如均方根误差（MSE）、平均绝对误差（MAE）和平均绝对百分比误差（MAPE），以便于评估模型性能。以`GWO_SVM_exmp.m`为例，代码首先定义SVM的参数c和g，然后初始化狼群的位置。接着进入主循环，每次迭代中，会更新每只狼的位置，并计算它们的适应度值，以确定新的阿尔法、贝塔和德尔塔狼。在每一轮迭代结束后，更新狼群的位置，并根据适应度值调整狼群的排名。这个过程持续到达到预设的最大迭代次数。返回最优参数和对应的模型性能。 `wine.mat`和`wndspd.mat`是数据集文件，用于训练和支持向量机的测试。`GWO_finalVersion.pdf`可能是算法的详细理论介绍或操作指南，提供更深入的理解。总结来说，GWO算法在MATLAB中的实现涉及生物启发式优化、SVM参数优化、误差评估等多个关键概念。通过理解并实践这些源代码，我们可以深入掌握GWO的工作机制，并将其应用于其他优化问题，提升模型的性能。同时，这样的实践也帮助我们更好地理解优化算法与机器学习模型之间的互动，为实际工程应用打下坚实基础。

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2016.09.08GWO.zip （9个子文件）

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S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey Wolf Optimizer, Advances in Engineering Software , vol. 69, pp.

46-61, 2014, DOI: http://dx.doi.org/10.1016/j.advengsoft.2013.12.007

Grey Wolf Optimizer

Seyedali Mirjalili,

Seyed Mohammad Mirjalili,

Andrew Lewis

School of Information and Communication Technology, Griffith University, Nathan, Brisbane, QLD 4111,

Australia

Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Shahid Beheshti

University, G. C. 1983963113, Tehran, Iran

seyedali.mirjalili@griffithuni.edu.au, mohammad.smm@gmail.com, a.lewis@griffith.edu.au

Abstract: This work proposes a new meta-heuristic called Grey Wolf Optimizer (GWO) inspired by grey wolves

(Canis lupus). The GWO algorithm mimics the leadership hierarchy and hunting mechanism of grey wolves in

nature. Four types of grey wolves such as alpha, beta, delta, and omega are employed for simulating the

leadership hierarchy. In addition, the three main steps of hunting, searching for prey, encircling prey, and

attacking prey, are implemented. The algorithm is then benchmarked on 29 well-known test functions, and the

results are verified by a comparative study with Particle Swarm Optimization (PSO), Gravitational Search

Algorithm (GSA), Differential Evolution (DE), Evolutionary Programming (EP), and Evolution Strategy (ES).

The results show that the GWO algorithm is able to provide very competitive results compared to these well-

known meta-heuristics. The paper also considers solving three classical engineering design problems

(tension/compression spring, welded beam, and pressure vessel designs) and presents a real application of the

proposed method in the field of optical engineering. The results of the classical engineering design problems and

real application prove that the proposed algorithm is applicable to challenging problems with unknown search

spaces.

Keywords: Optimization, optimization techniques, heuristic algorithm, Metaheuristics, Constrained

Optimization, GWO

1. Introduction

Meta-heuristic optimization techniques have become very popular over the last two decades. Surprisingly,

some of them such as Genetic Algorithm (GA) [1], Ant Colony Optimization (ACO) [2], and Particle Swarm

Optimization (PSO) [3] are fairly well-known among not only computer scientists but also scientists from

different fields. In addition to the huge number of theoretical works, such optimization techniques have been

applied in various fields of study. There is a question here as to why meta-heuristics have become remarkably

common. The answer to this question can be summarized into four main reasons: simplicity, flexibility,

derivation-free mechanism, and local optima avoidance.

First, meta-heuristics are fairly simple. They have been mostly inspired by very simple concepts. The

inspirations are typically related to physical phenomena, animals’ behaviors, or evolutionary concepts. The

simplicity allows computer scientists to simulate different natural concepts, propose new meta-heuristics,

hybridize two or more meta-heuristics, or improve the current meta-heuristics. Moreover, the simplicity assists

other scientists to learn meta-heuristics quickly and apply them to their problems.

Second, flexibility refers to the applicability of meta-heuristics to different problems without any special

changes in the structure of the algorithm. Meta-heuristics are readily applicable to different problems since they

mostly assume problems as black boxes. In other words, only the input(s) and output(s) of a system are

important for a meta-heuristic. So, all a designer needs is to know how to represent his/her problem for meta-

heuristics.

Third, the majority of meta-heuristics have derivation-free mechanisms. In contrast to gradient-based

optimization approaches, meta-heuristics optimize problems stochastically. The optimization process starts with

random solution(s), and there is no need to calculate the derivative of search spaces to find the optimum. This

makes meta-heuristics highly suitable for real problems with expensive or unknown derivative information.

Finally, meta-heuristics have superior abilities to avoid local optima compared to conventional optimization

techniques. This is due to the stochastic nature of meta-heuristics which allow them to avoid stagnation in local

solutions and search the entire search space extensively. The search space of real problems is usually unknown

and very complex with a massive number of local optima, so meta-heuristics are good options for optimizing

these challenging real problems.

S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey Wolf Optimizer, Advances in Engineering Software , vol. 69, pp.

46-61, 2014, DOI: http://dx.doi.org/10.1016/j.advengsoft.2013.12.007

The No Free Lunch (NFL) theorem [4] is worth mentioning here. This theorem has logically proved that

there is no meta-heuristic best suited for solving all optimization problems. In other words, a particular meta-

heuristic may show very promising results on a set of problems, but the same algorithm may show poor

performance on a different set of problems. Obviously, NFL makes this field of study highly active which

results in enhancing current approaches and proposing new meta-heuristics every year. This also motivates our

attempts to develop a new meta-heuristic with inspiration from grey wolves.

Generally speaking, meta-heuristics can be divided into two main classes: single-solution-based and

population-based. In the former class (Simulated Annealing [5] for instance) the search process starts with one

candidate solution. This single candidate solution is then improved over the course of iterations. Population-

based meta-heuristics, however, perform the optimization using a set of solutions (population). In this case the

search process starts with a random initial population (multiple solutions), and this population is enhanced over

the course of iterations. Population-based meta-heuristics have some advantages compared to single solution-

based algorithms:

 Multiple candidate solutions share information about the search space which results in sudden jumps

toward the promising part of search space

 Multiple candidate solutions assist each other to avoid locally optimal solutions

 Population-based meta-heuristics generally have greater exploration compared to single solution-based

algorithms

One of the interesting branches of the population-based meta-heuristics is Swarm Intelligence (SI). The

concepts of SI was first proposed in 1993 [6]. According to Bonabeau et al. [1], SI is “The emergent collective

intelligence of groups of simple agents”. The inspirations of SI techniques originate mostly from natural

colonies, flock, herds, and schools. Some of the most popular SI techniques are ACO [2], PSO [3], and Artificial

Bee Colony (ABC) [7]. A comprehensive literature review of the SI algorithms is provided in the next section.

Some of the advantages of SI algorithms are:

 SI algorithms preserve information about the search space over the course of iteration, whereas

Evolutionary Algorithms (EA) discard the information of the previous generations

 SI algorithms often utilize memory to save the best solution obtained so far

 SI algorithms usually have fewer parameters to adjust

 SI algorithms have less operators compared to evolutionary approaches (crossover, mutation, elitism,

and so on)

 SI algorithms are easy to implement

Regardless of the differences between the meta-heuristics, a common feature is the division of the search

process into two phases: exploration and exploitation [8-12]. The exploration phase refers to the process of

investigating the promising area(s) of the search space as broadly as possible. An algorithm needs to have

stochastic operators to randomly and globally search the search space in order to support this phase. However,

exploitation refers to the local search capability around the promising regions obtained in the exploration phase.

Finding a proper balance between these two phases is considered a challenging task due to the stochastic nature

of meta-heuristics. This work proposes a new SI technique with inspiration from the social hierarchy and

hunting behavior of grey wolf packs. The rest of the paper is organized as follows:

Section 2 presents a literature review of SI techniques. Section 3 outlines the proposed GWO algorithm. The

results and discussion of benchmark functions, semi-real problems, and a real application are presented in

Section 4, Section 5, and Section 6, respectively. Finally, Section 7 concludes the work and suggests some

directions for future studies.

2. Literature review

Meta-heuristics may be classified into three main classes: evolutionary, physics-based, and SI algorithms.

EAs are usually inspired by the concepts of evolution in nature. The most popular algorithm in this branch is

GA. This algorithm was proposed by Holland in 1992 [13] and simulates Darwnian evolution concepts. The

engineering applications of GA were extensively investigated by Goldberg [14]. Generally speaking, the

optimization is done by evolving an initial random solution in EAs. Each new population is created by the

combination and mutation of the individuals in the previous generation. Since the best individuals have higher

probability of participating in generating the new population, the new population is likely to be better than the

previous generation(s). This can guarantee that the initial random population is optimized over the course of

generations. Some of the EAs are Differential Evolution (DE) [15], Evolutionary Programing (EP) [16, 17], and

S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey Wolf Optimizer, Advances in Engineering Software , vol. 69, pp.

46-61, 2014, DOI: http://dx.doi.org/10.1016/j.advengsoft.2013.12.007

Evolution Strategy (ES) [18, 19], Probability-Based Incremental Learning (PBIL), Genetic Programming (GP)

[20], and Biogeography-Based Optimizer (BBO) [21].

As an example, the BBO algorithm was first proposed by Simon in 2008 [21]. The basic idea of this

algorithm has been inspired by biogeography which refers to the study of biological organisms in terms of

geographical distribution (over time and space). The case studies might include different islands, lands, or even

continents over decades, centuries, or millennia. In this field of study different ecosystems (habitats or

territories) are investigated for finding the relations between different species (habitants) in terms of

immigration, emigration, and mutation. The evolution of ecosystems (considering different kinds of species

such as predator and prey) over migration and mutation to reach a stable situation was the main inspiration of

the BBO algorithm.

The second main branch of meta-heuristics is physics-based techniques. Such optimization algorithms

typically mimic physical rules. Some of the most popular algorithms are Gravitational Local Search (GLSA)

[22], Big-Bang Big-Crunch (BBBC) [23], Gravitational Search Algorithm (GSA) [24], Charged System Search

(CSS) [25], Central Force Optimization (CFO) [26], Artificial Chemical Reaction Optimization Algorithm

(ACROA) [27], Black Hole (BH) [28] algorithm, Ray Optimization (RO) [29] algorithm, Small-World

Optimization Algorithm (SWOA) [30], Galaxy-based Search Algorithm (GbSA) [31], and Curved Space

Optimization (CSO) [32]. The mechanism of these algorithms is different from EAs, in that a random set of

search agents communicate and move throughout search space according to physical rules. This movement is

implemented, for example, using gravitational force, ray casting, electromagnetic force, inertia force, weights,

and so on.

For example, the BBBC algorithm was inspired by the big bang and big crunch theories. The search agents of

BBBC are scattered from a point in random directions in a search space according to the principles of the big

bang theory. They search randomly and then gather in a final point (the best point obtained so far) according to

the principles of the big crunch theory. GSA is another physics-based algorithm. The basic physical theory from

which GSA is inspired is Newton’s law of universal gravitation. The GSA algorithm performs search by

employing a collection of agents that have masses proportional to the value of a fitness function. During iteration,

the masses are attracted to each other by the gravitational forces between them. The heavier the mass, the bigger

the attractive force. Therefore, the heaviest mass, which is possibly close to the global optimum, attracts the other

masses in proportion to their distances.

The third subclass of meta-heuristics is the SI methods. These algorithms mostly mimic the social behavior

of swarms, herds, flocks, or schools of creatures in nature. The mechanism is almost similar to physics-based

algorithm, but the search agents navigate using the simulated collective and social intelligence of creatures. The

most popular SI technique is PSO. The PSO algorithm was proposed by Kennedy and Eberhart [3] and inspired

from the social behavior of birds flocking. The PSO algorithm employs multiple particles that chase the position

of the best particle and their own best positions obtained so far. In other words, a particle is moved considering

its own best solution as well as the best solution the swarm has obtained.

Another popular SI algorithm is ACO, proposed by Dorigo et al. in 2006 [2]. This algorithm was inspired

by the social behavior of ants in an ant colony. In fact, the social intelligence of ants in finding the shortest path

between the nest and a source of food is the main inspiration of ACO. A pheromone matrix is evolved over the

course of iteration by the candidate solutions. The ABC is another popular algorithm, mimicking the collective

behavior of bees in finding food sources. There are three types of bees in ABS: scout, onlooker, and employed

bees. The scout bees are responsible for exploring the search space, whereas onlooker and employed bees

exploit the promising solutions found by scout bees. Finally, the Bat-inspired Algorithm (BA), inspired by the

echolocation behavior of bats, has been proposed recently [33]. There are many types of bats in the nature. They

are different in terms of size and weight, but they all have quite similar behaviors when navigating and hunting.

Bats utilize natural sonar in order to do this. The two main characteristics of bats when finding prey have been

adopted in designing the BA algorithm. Bats tend to decrease the loudness and increase the rate of emitted

ultrasonic sound when they chase prey. This behavior has been mathematically modeled for the BA algorithm.

The rest of the SI techniques proposed so far are as follows:

 Marriage in Honey Bees Optimization Algorithm (MBO) in 2001 [34]

 Artificial Fish-Swarm Algorithm (AFSA) in 2003 [35]

 Termite Algorithm in 2005 [36]

 Wasp Swarm Algorithm in 2007 [37]

 Monkey Search in 2007 [38]

 Bee Collecting Pollen Algorithm (BCPA) in 2008 [39]

 Cuckoo Search (CS) in 2009 [40]

 Dolphin Partner Optimization (DPO) in 2009 [41]

S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey Wolf Optimizer, Advances in Engineering Software , vol. 69, pp.

46-61, 2014, DOI: http://dx.doi.org/10.1016/j.advengsoft.2013.12.007

 Firefly Algorithm (FA) in 2010 [42]

 Bird Mating Optimizer (BMO) in 2012 [43]

 Krill Herd (KH) in 2012 [44]

 Fruit fly Optimization Algorithm (FOA) in 2012 [45]

This list shows that there are many SI techniques proposed so far, many of them inspired by hunting and

search behaviors. To the best of our knowledge, however, there is no SI technique in the literature mimicking

the leadership hierarchy of grey wolves, well known for their pack hunting. This motivated our attempt to

mathematically model the social behavior of grey wolves, propose a new SI algorithm inspired by grey wolves,

and investigate its abilities in solving benchmark and real problems.

3. Grey Wolf Optimizer (GWO)

In this section the inspiration of the proposed method is first discussed. Then, the mathematical model is

provided.

3.1. Inspiration

Grey wolf (Canis lupus) belongs to Canidae family. Grey wolves are considered as apex predators, meaning

that they are at the top of the food chain. Grey wolves mostly prefer to live in a pack. The group size is 5-12 on

average. Of particular interest is that they have a very strict social dominant hierarchy as shown in Fig. 1.

Fig. 1. Hierarchy of grey wolf (dominance decreases from top down)

The leaders are a male and a female, called alphas. The alpha is mostly responsible for making decisions

about hunting, sleeping place, time to wake, and so on. The alpha’s decisions are dictated to the pack. However,

some kind of democratic behavior has also been observed, in which an alpha follows the other wolves in the

pack. In gatherings, the entire pack acknowledges the alpha by holding their tails down. The alpha wolf is also

called the dominant wolf since his/her orders should be followed by the pack [46]. The alpha wolves are only

allowed to mate in the pack. Interestingly, the alpha is not necessarily the strongest member of the pack but the

best in terms of managing the pack. This shows that the organization and discipline of a pack is much more

important than its strength.

The second level in the hierarchy of grey wolves is beta. The betas are subordinate wolves that help the

alpha in decision-making or other pack activities. The beta wolf can be either male or female, and he/she is

probably the best candidate to be the alpha in case one of the alpha wolves passes away or becomes very old.

The beta wolf should respect the alpha, but commands the other lower-level wolves as well. It plays the role of

an advisor to the alpha and discipliner for the pack. The beta reinforces the alpha's commands throughout the

pack and gives feedback to the alpha.

The lowest ranking grey wolf is omega. The omega plays the role of scapegoat. Omega wolves always have

to submit to all the other dominant wolves. They are the last wolves that are allowed to eat. It may seem the

omega is not an important individual in the pack, but it has been observed that the whole pack face internal

fighting and problems in case of losing the omega. This is due to the venting of violence and frustration of all

wolves by the omega(s). This assists satisfying the entire pack and maintaining the dominance structure. In

some cases the omega is also the babysitters in the pack.

If a wolf is not an alpha, beta, or omega, he/she is called subordinate (or delta in some references). Delta

wolves have to submit to alphas and betas, but they dominate the omega. Scouts, sentinels, elders, hunters, and

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S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey Wolf Optimizer, Advances in Engineering Software , vol. 69, pp.

46-61, 2014, DOI: http://dx.doi.org/10.1016/j.advengsoft.2013.12.007

caretakers belong to this category. Scouts are responsible for watching the boundaries of the territory and

warning the pack in case of any danger. Sentinels protect and guarantee the safety of the pack. Elders are the

experienced wolves who used to be alpha or beta. Hunters help the alphas and betas when hunting prey and

providing food for the pack. Finally, the caretakers are responsible for caring for the weak, ill, and wounded

wolves in the pack.

In addition to the social hierarchy of wolves, group hunting is another interesting social behavior of grey

wolves. According to Muro et al. [47] the main phases of grey wolf hunting are as follows:

 Tracking, chasing, and approaching the prey

 Pursuing, encircling, and harassing the prey until it stops moving

 Attack towards the prey

These steps are shown in Fig. 2.

Fig. 2. Hunting behaviour of grey wolves: (A) chasing, approaching, and tracking prey (B-D) pursuiting, harassing, and encircling

(E) stationary situation and attack [47]

In this work this hunting technique and the social hierarchy of grey wolves are mathematically modeled in

order to design GWO and perform optimization.

3.2. Mathematical model and algorithm

In this subsection the mathematical models of the social hierarchy, tracking, encircling, and attacking prey

are provided. Then the GWO algorithm is outlined.

3.2.1. Social hierarchy:

In order to mathematically model the social hierarchy of wolves when designing GWO, we consider the

fittest solution as the alpha (). Consequently, the second and third best solutions are named beta () and delta

() respectively. The rest of the candidate solutions are assumed to be omega (). In the GWO algorithm the

hunting (optimization) is guided by , , and . The  wolves follow these three wolves.

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2017-11-03

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