A Hybrid Global Optimization Algorithm Based on Wind Driven Optimization and Differential Evolution

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.

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 natureinspired 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  Quartic function, that is, noise 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 highdimensional 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.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]

A Brief Introduction on WDO and DE Algorithm
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 by  where  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 ( ⃗  PG ), 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, Ω 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 in where the acceleration ⃗  in (1) is rewritten as ⃗  = Δ ⃗ /Δ; for simplicity, set Δ = 1; for an infinitesimal air parcel, set  = 1, which simplifies (4) to On the basis of (2), the density  can be written in terms of the pressure; thus (5) can be rewritten as where  cur 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, Δ ⃗  can be written as Δ ⃗  = ⃗  new − ⃗  cur , where ⃗  new represents the velocity in next iteration and ⃗  cur is the velocity at the current iteration.⃗  and ∇ are vectors, they can be broken down in direction and magnitude as ⃗  = ||(0 −  cur ), −∇ = | opt −  cur |( opt −  cur ),  opt is the optimum pressure point that has been found so far,  opt is the optimum location that has been found so far, and  cur 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 (Ω × ⃗ ) is replaced by the velocity influence from another dimension ⃗  other dim cur .Secondly, all the coefficients are combined together; that is,  = −2.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 ⃗  new 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.

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  , ( = 1, 2, . . ., NP) = {  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.
(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.solution.Based on the above description, the main procedure of WDO-DE is as shown in Algorithm 1.

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.

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.0.397887 Goldstein and Price 3   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].

Algorithm Performance Comparison.
In this section, in order to test the performance of WDO-DE algorithm,    results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs for  1 ∼  15 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  5 ,  13 , and  15 .For  5 and  15 , the DE algorithm gives the better results.Although the result of  13 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 ( 1 ∼  4 ,  7 ∼  14 ) 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  11 , the solution of WDO-DE algorithm is the closest to the theoretical optimal solution; for the benchmark function  12 ∼  14 , 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  11 and  13 , the WDO-DE algorithm has converged within 100 generations.And in benchmark functions  12 and  14 , 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      minimum.As can be seen from Figures 1-15, WDO-DE algorithm can converge within the maximum number of iterations except  5 and  15 , 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,  3 and  5 were set to 100 dimensions,  4 and  7 set to 300 dimensions,  1 and  8 set  to 500 dimensions, and  2 ,  6 , and  9 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  1 ∼  9 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  5 .For  5 , the results of  5 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 ( 1 ∼  4 ,  7 ∼  9 ) 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 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.

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 discretevalued 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.

4. 3 .
Parameters Setting.In this section, the parameters setting are presented.Bayraktar et al. did a lot of research

Table 1 :
Description of the benchmark function used in our experiment.
are the ANOVA tests of the global
Generate one initial population with /2 air particles, each air particle assign random location  wdo , and velocity  wdo , , evaluation the population and identify the best solution of WDO algorithm  wdo best, ; Step 2.2.Generate one initial population  de , with /2 individuals, evaluation the population and identify the best solution  de best, .Step 3. Identify the best solution  best among all particles in WDO and DE.Step 4. While stopping criterion is not satisfied Step 4.1.Running process of the WDO algorithm for  = 1 to the /2 do  wdo best, =  best Generate the trial velocity according to (8) Generate the trial location  , by (9) Evaluate the trial location  , If ( , ) ≤ ( wdo , )  wdo ,+1 =  , ,  ( wdo ,+1 ) =  ( , ) If ( , ) ≤ ( wdo best, )  wdo best, =  , ,  ( wdo best, ) =  ( , ) Running process of the DE algorithm for  = 1 to the /2 do  de best, =  best Generate  , using (11) Generate the trial vector  , by (15) Evaluate the trial vector  , [16]le 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 Step 1. Initialize parameters. (Population size);  (Max number of generations); Parameters of WDO: RT (RT coefficient);  (The friction coefficient); max  (Maximum allowed speed);  (Gravitational constant);  (Constant in the update equation).Parameters of DE:  (Mutation scale factor); Pc (Crossover probability).

Table 2 :
The parameters set of WDO.

Table 3 :
The parameters set of DE.
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.)

Table 4 :
Comparison of performance of algorithms in low dimension.

Table 5 :
Comparison of optimal solution of algorithms in low dimension.

Table 6 :
Comparison of performance of algorithms in high dimension.