Abstract
This paper presents a new hybrid global optimization algorithm, which is based on the wind driven optimization (WDO) and differential evolution (DE), named WDO-DE algorithm. The WDO-DE algorithm is based on a double population evolution strategy, the individuals in a population evolved by wind driven optimization algorithm, and a population of individuals evolved from difference operation. The populations of individuals both in WDO and DE employ an information sharing mechanism to implement coevolution. This paper chose fifteen benchmark functions to have a test. The experimental results show that the proposed algorithm can be feasible in both low-dimensional and high-dimensional cases. Compared to GA-PSO, WDO, DE, PSO, and BA algorithm, the convergence speed and precision of WDO-DE are higher. This hybridization showed a better optimization performance and robustness and significantly improves the original WDO algorithm.
1. Introduction
Nature always makes people produce a lot of inspiration. In the past, many natural heuristic algorithms have been proposed for solving real-world and large scale problems which are very difficult to solve by traditional methods, for example, genetic algorithm (GA) [1], particle swarm optimization (PSO) [2, 3], ant colony optimization (ACO) [4], differential evolution (DE) [5], firefly algorithm (FA) [6], bat algorithm (BA) [7], cuckoo search (CS) [8], flower pollination algorithm (FPA) [9], wind driven optimization (WDO) [10].
Because some heuristic algorithms are not very satisfactory in many respects, in the recent years, according to the characteristics of some algorithms, the heuristic algorithm combined with each other is an increasingly popular strategy to improve algorithms. These hybridizations have been shown to be effective global optimization algorithms and have been applied to solve application problems. For example, a hybrid of genetic algorithm (GA) and particle swarm optimization (PSO) is applied to recurrent neural/fuzzy network design [11]. A hybrid metaheuristic differential evolution (DE) and cuckoo search (CS) algorithm is implemented to solve the UCAV path planning problem [12]. A hybrid particle swarm optimization (PSO) and ant colony optimization (ACO) algorithm is implemented to solve hierarchical classification [13].
The wind driven optimization (WDO) is a novel nature-inspired technique. It is a population based iterative heuristic process. The WDO was proposed by Bayraktar et al. in 2010 [10]. The inspiration of the WDO derives from the atmosphere, where wind blows in an attempt to balance the atmospheric pressure. Due to the fact that the WDO has few parameters in need of control and it is very easy to be carried out, it has been paid more attention by the academic community. But, in the early stage of solving optimization problems, the convergence speed of the WDO is quicker than others; when all the individuals are close to the optimal one in the late stage of solving optimization problems, it will lead to the loss of population diversity, and it is easy to fall into local optimum in finding better solutions.
In 1997, Storn and Price have introduced the DE algorithm [5]. In the next time, the DE algorithm has been applied to solve optimization problems in diverse fields. After the population initialization, the population through mutation, crossover, and selection operators generates new population, it is able to maintain diversity of population, which can achieve the search of global optimal solution after several iterations. But it also has flaws. Its disadvantages contain the slow pace of convergence, and it is easy to fall into local optimum.
WDO algorithm can easily suffer from the premature convergence when solving global optimization problems. It is an important method for relieving the premature convergence to control the population diversity. In order to overcome the deficiencies of a single algorithm in solving a global optimization problem, in this paper, we propose a new hybrid global optimization algorithm based on the wind driven optimization and differential evolution. This evolution strategy allows WDO and DE algorithms to give full play to their respective advantages. DE algorithm enables keeping the diversity of the population. This can be a good remedy defect of WDO algorithm, so it can avoid falling into a local optimum due to the loss of population diversity. And WDO algorithm converges faster; it can be good to make up for the shortcomings of DE algorithm in convergence speed, utilizing the individual of DE algorithm to guide the evolution of the individual of WDO algorithm, which reduces the risk of falling into a local optimal solution. It is not only to ensure the accuracy of the algorithm, but also to guarantee the speed of solving problems. Finally, the fifteen benchmark functions are tested; the experimental results show that the proposed algorithm can be feasible in both low-dimensional and high-dimensional cases. Compared to GA-PSO, WDO, DE, PSO, and BA algorithm, the convergence speed and precision of WDO-DE are higher. This hybridization showed a better optimization performance and robustness and significantly improves the original WDO algorithm.
2. A Brief Introduction on WDO and DE Algorithm
2.1. Wind Driven Optimization
The inspiration of the proposed WDO derives from the atmosphere. In the atmosphere, wind blows in an attempt to balance the imbalance of pressure. It flows from high pressure areas to low pressure areas at a velocity. Depending on the above analysis, some theoretical assumptions are formulated in derivation of the WDO algorithm. The starting point of WDO algorithm is Newton’s second law of motion, which is used to provide accurate results for the analysis of atmospheric motion in the Lagrangian description [14, 15] where is the acceleration, is the air density for an infinitesimal air parcel, and are all the forces acting on the air parcel. In order to let air pressure establish the equation relationship with the air parcel’s density and temperature, the ideal gas law is given bywhere is the pressure, is the universal gas constant, and is the temperature.
The cause of the air movement is due to the combination of many forces, mainly including gravitational force (), pressure gradient force (), Coriolis force (), and friction force (). The physical equations of the above mentioned forces are as follows:where is finite volume of the air, represents the gravitational acceleration, represents the pressure gradient, is rotation of the Earth, represents the velocity vector of the wind, and is the friction coefficient.
The forces mentioned above can be added to (1). The equation can be described as inwhere the acceleration in (1) is rewritten as ; for simplicity, set ; for an infinitesimal air parcel, set , which simplifies (4) to
On the basis of (2), the density can be written in terms of the pressure; thus (5) can be rewritten aswhere is the pressure of current location. It is assumed in the WDO algorithm that velocity and position of the air parcel are changing at each iteration. Thus, can be written as , where represents the velocity in next iteration and is the velocity at the current iteration. and are vectors, they can be broken down in direction and magnitude as , , is the optimum pressure point that has been found so far, is the optimum location that has been found so far, and is the current location; updating (6) with the new equations, (6) can be rewritten as
Finally, there are three additional substitutions needed. Firstly, the influence of the Coriolis force is replaced by the velocity influence from another dimension . Secondly, all the coefficients are combined together; that is, . Thirdly, in some cases where the pressure is extremely large, the updated velocities are too large to become meaningless and the efficiency of the WDO algorithm will be reduced. So the actual pressure value is replaced by rank among all air parcels based on their pressure values, the resulting equation of updating the velocity can be described as in (8), and the equation of updating the location can be described as in (9):where is the ranking among all air parcels and represents the new location for the next iteration.
WDO is similar to other nature-inspired optimization algorithms, but compared to other optimization algorithms, the code of WDO is more simple and easy to implement; it has less few control variables that need adjustment.
2.2. Differential Evolution
Differential evolution is introduced by Storn and Price in 1997 [5]. DE is an effective and simple global optimization algorithm.
First of all, a population is generated randomly, it may be represented as , NP is the number of population, is the number of dimensions, and denotes the generation of the population. There are three operators—mutation, crossover, and selection. Then the original population will through three operators generate a new population. The main progress of DE in detail is as follows.
(1) Mutation. After initialization, mutation operation is used to generate the mutant vectors . Mutation operation usually has the following five most frequently implemented strategies.
DE/rand/1:
DE/best/1:
DE/current-to-best/1:
DE/best/2:
DE/rand/2:The subscripts are mutually exclusive integers within the range and is the scale factor of difference vector.
(2) Crossover. After the mutation operation is completed, DE will utilize crossover operation to generate a trial vector :where ; it is a crossover constant. is a random integer within the range .
(3) Selection. After completing the first two operators, according the fitness value of trial vectors, DE utilizes selection operation to select the best one for the next generation:After gradual iteration, DE can achieve the search of global optimal solution.
3. WDO-DE Algorithm
This section focuses on the rationale of the WDO-DE algorithm. WDO and DE are based on the population of the global search techniques. And the WDO-DE is based on a double population evolution strategy [16]. The individuals both in WDO and DE employ an information sharing mechanism to implement coevolution. The strategy makes WDO-DE enjoy the advantages of two algorithms. It can maintain diversity of the populations, and the WDO-DE algorithm has the capability to jump out of the local optimal solution. Based on the above description, the main procedure of WDO-DE is as shown in Algorithm 1.
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4. Experimental Results
4.1. Experimental Setup
All algorithms are implemented in MATLAB R2012a, and experiments are performed on a Pentium 3.00 GHz Processor with 4.0 GB of memory, Windows 7 operating system.
4.2. Benchmark Test Functions
To test the performance of WDO-DE algorithm, we use 15 benchmark functions [17, 18] which have been widely used in the test. Among these benchmarks, part I contains the nine high-dimensional functions and part II contains six low-dimensional functions. Table 1 has shown the benchmark functions.
4.3. Parameters Setting
In this section, the parameters setting are presented. Bayraktar et al. did a lot of research for the parameters set of WDO [19]. The parameters set of WDO and DE is based on the practical experience to take the appropriate value. Tables 2 and 3 represent the necessary parameters in our experiment [20].
4.4. Algorithm Performance Comparison
In this section, in order to test the performance of WDO-DE algorithm, WDO-DE algorithm has been compared with the algorithms GA-PSO, WDO, DE, BA, and PSO in low dimension and high dimension. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs for are shown in Table 4. Bold and italicized results mean that WDO-DE is better. Population size of other algorithms is 100. Max number of iterations of all tests is 1000.
For the low-dimensional case, according to Table 4, test results of WDO-DE are better than the other algorithms except , , and . For and , the DE algorithm gives the better results. Although the result of function is worse than GA-PSO, DE, and PSO algorithm, it has already reached the theoretical optimal value. What is more, WDO-DE can find the theoretical optimum values for twelve benchmark functions (, ) and has a very strong robustness. The novel hybrid global optimization algorithm is better than the original algorithm.
In the last, we calculated the average rank based on these fifteen functions’ ranking [18]. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.
For the benchmark function , the solution of WDO-DE algorithm is the closest to the theoretical optimal solution; for the benchmark function , the original WDO algorithm cannot be close to the theoretical optimal solution; however the WDO-DE algorithm is close to the theoretical optimal solution (Table 5). And we can find that the convergence speed of the WDO-DE algorithm is quicker than other algorithms. In benchmark functions and , the WDO-DE algorithm has converged within 100 generations. And in benchmark functions and , the WDO-DE algorithm also has converged within 200 generations.
Meanwhile, Figures 1–15 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 16–30 are the ANOVA tests of the global minimum. As can be seen from Figures 1–15, WDO-DE algorithm can converge within the maximum number of iterations except and , and it has a faster global convergence speed in many functions and higher convergence precision. From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 16–30, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is an effective and feasible solution for optimization problems in low-dimensional case.
In order to test the optimization ability of the algorithms in high-dimensional space, this paper selects several different dimensions for tests [15]. Among them, and were set to 100 dimensions, and set to 300 dimensions, and set to 500 dimensions, and , , and set to 1000 dimensions. In all the tests, the max number of iterations is 1000, and the set of other parameters is the same. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs for are shown in Table 6.
For the high-dimensional case, as seen from Table 6, test results of WDO-DE are better than the other algorithms except . For , the results of function are secondary to DE algorithm; although the DE algorithm gives better results, it has not reached the theoretical optimal value. WDO-DE can find the theoretical optimum values for seven benchmark functions (, ) and has a very strong robustness. This indicates that WDO-DE is very robust and efficient.
The same as before, we calculated the average rank based on these nine functions’ ranking. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.
The same as before, Figures 31–39 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 40–48 are the ANOVA tests of the global minimum. As can be seen from Figures 31–39, WDO-DE algorithm has the fastest global convergence speed and the highest convergence precision in all of these functions only except . From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 40–48, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is also an effective and feasible solution for optimization problems in high-dimensional case.
5. Conclusion and Future Research
In this paper, we present a new hybrid global optimization algorithm called WDO-DE, which is based on the wind driven optimization (WDO) and differential evolution (DE) for solving optimization problems. We use 15 benchmark functions which contain unimodal, multimodal, low-dimensional, and high-dimensional unconstrained test functions to test the performance of WDO-DE algorithm. The WDO-DE algorithm can converge within the maximum number of iterations in most functions. In comparison with the GA-PSO, WDO, DE, BA, and PSO, the WDO-DE algorithm is more effective in finding better solutions and the convergence speed and precision of WDO-DE are higher. It is an effective and reliable global optimization algorithm.
Although in this paper the hybrid WDO-DE algorithm was implemented only for function optimization, in the field of optimization, there are still many aspects worthy of our study. Firstly, the hybrid algorithm proposed in this paper is based on the continuous space optimization. The future research may concentrate on discrete WDO algorithm. We can utilize many discretized strategies to discretize WDO algorithm to solve a problem characterized by discrete-valued design variables. Secondly, in engineering application, production management, and national defense construction, many optimization problems are multiobjective optimization problems, which are widely used in practical engineering. We would apply our proposed hybrid approach to solve multiobjective optimization problem. Lastly, we will learn more algorithms which have better optimization performance and analyze their characteristics. We would develop new hybrid approaches to solve the optimization problems raised above.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Science Foundation of China under Grants nos. 61165015 and 61463007 and the Innovation Project of Guangxi Graduate Education.