Improved Alopex-based evolutionary algorithm (AEA) by quadratic interpolation and its application to kinetic parameter estimations

Accepted Manuscript Title: Improved Alopex-based evolutionary algorithm (AEA) by quadratic interpolation and its application to kinetic parameter estimations Author: Yihang Yang XuepengZong Dacheng Yao Shaojun Li PII: DOI: Reference:

S1568-4946(16)30606-8 http://dx.doi.org/doi:10.1016/j.asoc.2016.11.037 ASOC 3927

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

1-4-2016 8-11-2016 19-11-2016

Please cite this article as: Yihang Yang, XuepengZong, Dacheng Yao, Shaojun Li, Improved Alopex-based evolutionary algorithm (AEA) by quadratic interpolation and its application to kinetic parameter estimations, Applied Soft Computing Journal http://dx.doi.org/10.1016/j.asoc.2016.11.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights: 1. The quadratic interpolation method was employed to improve the performance of the AEA. 2. QIAEA was compared with the AEA and LAEA and some state-of-the-art algorithms. 3. Two applications for estimate parameters were used to test the performance of the QIAEA.

Alopex-based algorithm (AEA)

Application QIAEA

Parameter estimation

QI method

Improved Alopex-based evolutionary algorithm (AEA) by quadratic interpolation and its application to kinetic parameter estimations Yihang Yang, XuepengZong, Dacheng Yao, Shaojun Li Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, (East China University of Science and Technology), Shanghai 200237, China Abstract: The Algorithm of Pattern Extraction (Alopex)-based evolutionary algorithm (AEA) is an intelligent optimization algorithm that combines the Alopex with the evolution algorithm. It has high exploratory ability due to its especial searching mechanism. However, the AEA is poor at exploitation to some extent. To improve the performance of AEA, quadratic interpolation (QI) method is introduced to the AEA in this paper. The improved algorithm chooses three individuals to fit a quadratic function to approximate the objective function in each iterative of the AEA and uses the extreme point of the quadratic function to generate new individuals. The QI method can help the algorithm to converge rapidly near optimal solutions, which can greatly improve the exploitation ability of AEA. The traditional and CEC 2013 benchmark functions are both used to test the performance of the improved algorithm (QIAEA); the experimental results show that the QIAEA has better convergent speed and higher accuracy than the AEA. The QIAEA is also compared with several state-of-the-art algorithms and the results show that the QIAEA is an effective and efficient algorithm. Furthermore, its applications to parameter estimation for a model of sulfur dioxide (SO2) oxidation with a cesium–rubidium–vanadium (Cs–Rb–V) sulfuric acid catalyst and a heavy oil thermal cracking three-lump model demonstrate that QIAEA can forecast more accurately than other methods tested. Keywords: Alopex; Evolutionary algorithm; Quadratic interpolation; Kinetic parameter estimation *Corresponding author: Shaojun Li School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China e-mail: [email protected]

1. Introduction In engineering applications, many complex optimization problems need to be solved. Traditional methods such as Newton method, gradient descent method, etc. can hardly to be applied when solving the problems with characteristics like non-convex, discontinuous, and non-differentiable. Inspired by natural evolution, researchers developed evolutionary algorithms (EAs), which mimic the natural evolution or social behavior of species. Without making any assumptions about the underlying fitness landscape, EAs often perform well in approximating solutions to many types of problems. Genetic algorithms (GAs) [1] are the first family of EA computational models to be developed. GAs simulate the adaptive process of natural systems and develop artificial systems while retaining the features of natural systems. In addition, GAs have been proved to perform well in many areas, such as structure of clusters [2], production and operation management [3], and so on. There are also other evolutionary algorithms that have been developed in last two decades, such as particle swarm optimization (PSO), differential evolution (DE), cultural algorithms (CA), the ant colony optimization algorithm (ACO), and the artificial bee colony algorithm (ABC). To improve optimization speed and efficiency, these EAs apply various methods, such as improved selection techniques [4], recombination (crossover) [5], or mutation progress [6]. Another kind of algorithm that can extricate solution vectors from a local optimum is simulated annealing (SA) [7]. SA imitates the process of metal annealing. The algorithm accepts an iterative solution with a certain probability depending on the difference between the function solutions before and after as well as a global parameter “temperature” whose value decreases with the increase of iterations. In recent years, SA has received significant attention for extremal optimization problems, where a global optimum is hidden among many local optima. It has been employed by researchers to combine with other EAs to monitor the population diversity and avoid

being trapped into local optima [8-10]. First proposed by Harth and Tzanakou in 1974 [11], the algorithm of pattern extraction (Alopex) is similar to the SA. This algorithm uses the influence of previous changes of independent variables on the objective function value as heuristic information, and it also utilizes “temperature” to calculate the probability of searching direction to find a competitive solution. As a result, the algorithm can jump out of local optima with a high probability. However, Alopex is a single-point iterative algorithm, causing it to have low efficiency. Combining the Alopex with the concept of EAs, Li and Li proposed the Alopex-based evolutionary algorithm (AEA) in 2011 [12]. According to literature [12], AEA has better performance than an improved hybrid GA and PSO. Nevertheless, due to the probability of searching in the direction of increasing the value of the objective function (for a minimization problem), the AEA is in a way good at exploration but poor at exploitation. Usually, exploration and exploitation of an evolutionary algorithm are contradictory to each other. Proper methods should be used to keep these two abilities well balanced. Some researchers combined the EAs with traditional methods to make an improvement. For example, Gao et al. proposed a novel ABC with Powell‟s method [13]. Davoodi et al. combined the Quantum-behaved Particle Swarm Optimization with Nelder Mead Simplex method to improve the optimizing efficiency [14]. Quadratic interpolation (QI) is a one-dimensional optimization method and uses a parabola to fit the shape of the objective function near the optima. It is simple and can find local/global optima quickly. However, it is inappropriate to apply this method when the problems become complex as it may fall into local optima. Since the AEA has good exploration ability but is poor at exploitation, we introduce the QI method to improve the exploitation ability of the AEA. The outstanding individuals in the population are chosen to execute the QI operation, which helps to enhance the convergence speed. Besides, quadratic interpolation among different individuals

can increase population diversity. As a result, the introduction of the QI method is beneficial to both the exploration and exploitation of the AEA. The rest of the paper is organized as follows. In Section 2, the basic principle of AEA is introduced. Section 3 is devoted to the description and implementation of AEA with QI (QIAEA), and the performance of QIAEA as demonstrated by comparing test results for benchmark functions with other algorithms is presented in Section 4. Section 5 includes two application examples on reaction kinetic parameters estimation. Conclusions are presented in Section 6.

2 Alopex-based evolutionary algorithm (AEA) The Alopex (Algorithm of Pattern Extraction)-based evolutionary algorithm [12] is an intelligent optimization algorithm that merges Alopex with the swarm intelligence. In the AEA, a new individual is generated by adding (or subtracting) the weighted difference vector between two chosen individuals to (or from) the first individual according to a probability determined by the objective values and variables of the two individuals. If the objective function value is improved, the new individual will replace the old one. Taking a minimization problem as an example (without loss of generality, illustrations and test functions in this paper will consider the optimum as the minimum), suppose that there are two D-dimensional

populations

X i  ( xi1 , xi 2 ,...,xiD )

F( X i )

and

F (Yi )

and

G1 and

G2

that

Yi  ( yi1 , yi 2 ,..., yiD )

have

the

same

number

of

individuals.

are the ith individuals in G1 and G2 , respectively.

are the objective function values of the optimization problem, respectively. The

update process of the individuals can be depicted as follows：

Cij  xij  yij  [F ( X i )  F (Yi )]

Pij 

1 1 e

(Cij / T )

,   1

(1)

(2)

xij  ( xij  yij )  r1 if Pij  r2 ( xij )'    xij  ( xij  yij )  r1 otherwise where

(3)

i  1, 2,  , N ( N is the number of individuals in a population) and j  1, 2,  , D ( D is the

number of dimension of the variable), F (.) is the objective function values of the optimization problem,

Cij is the correlation coefficient between the variables and the corresponding function value. T is the annealing temperature. ith individual.

r1

and

Pij is the probability to determine the changing direction of the jth variable in the

r2

the trial variable. In Eq.2,

are uniform distributions of a random number between 0 and 1 and

 1

for a minimization problem and

  1

( xij )' is

for a maximization problem.

The AEA uses Eq.2 and Eq.3 to generate new individuals. For convenience of analyzing the optimization principle of the AEA, we focus on a one-dimensional optimization problem to describe the optimization process.

Fig. 1

As shown in Figure 1, suppose there is a one-dimensional function individuals and are on the indicated locations in the figure, where Through Eq.3 we can see that possibility

x1

will “walk” towards

x2

f (x) , x1 and x2 are two

x1  x2

and

f (x1 )  f (x2 ) .

(the direction of local optimum) with

(1  P) and “walk” in the opposite direction with possibility P . According to Eq.1 and Eq.2,

we can calculate that P  0.5 , so

x1

has a larger possibility to “walk” towards

possibility to climb the hill and move close to the global optimum. If the value of C in Eq.1 is small, at this point, the valve of move towards

x2

x1

and

x2

x2

but it also has the

are close to each other,

P is close to 0.5. So the possibilities of x1 to

and to move conversely are approximately equal, which means

x1

“walks” in the way

that is similar to random search. If

f ( x1 ) 0 and

and

x1

f (x2 )

x1

and

x2

are far away from each other or the difference between

is large, the value of the C in Eq.1 is relatively large, so the value of

P is close to

will “walk” in the direction of decreasing function value with a large possibility. Searching in

this way, the algorithm has strong ability to jump out of local optima when the individuals are trapped in local optima. In other cases when

f (x1 )  f (x2 ) ,

x1  x2 , f (x1 )  f (x2 )

or

x1  x2 , f (x1 )  f (x2 )

or

x1  x2 ,

the individual will also move in the direction of decreasing the function value with a

larger probability. In short, the variables always move in the direction of decreasing function value with a possibility that are more than 50 percent, and they also have the possibility less than 50 percent to move conversely. The temperature

T is another important parameter in this algorithm. To make adaptive adjustment, it

is set as follows:

1 D N T  C ND j 1 i 1 ij

(4)

Thus, the calculation process of AEA can be summarized below: through Eq.1, we can obtain the correlation coefficient between the variables and the corresponding function values. Then the annealing temperature

T and the probability in walking direction can be gotten via Eq.4 and Eq.2. The trial

individual ( X i )' can be determined via Eq.3. If the trial individual has smaller objective function value than the old individual X i , the latter will be replaced by the former to update the population. The progress is repeated until the termination criterion is met. The more detail steps of the algorithm can be referenced in [12].

3. The improved Alopex-based evolutionary algorithm (AEA) by quadratic

interpolation, QIAEA We have shown in the previous section that the AEA has strong global search capability. However, the exploration and exploitation of an EA in a way contradict with each other. Due to the existence of randomness in the search direction, the AEA has the ability to escape from local optima, which, at the meantime, reduces the convergence speed of the algorithm. In other words, the AEA is good at exploration but poor at exploitation. QI is a simple method that can find the minimum in the given initial space. The basic idea of the QI method is to use three known points to form a quadratic curve and use the curve to approximate the shape of the objective function, so that the optima of the objective function can be approximated by the extreme point of the quadratic function. The extreme point can be easily obtained through the vertex formula of the quadratic curve. In the proposed algorithm, the QI method is applied right after the population is updated by AEA in each generation. Suppose that x a  ( x1a , x2a ,...,xDa ) , x b  ( x1b , x2b ,...,xDb ) , x c  ( x1c , x2c ,...,xDc ) are the three individuals that use QI method to generate a new individual

x , then it can be calculated as

follows:

x  (x1,...,xD ) xj 

[(xcj )2  ( xbj )2 ]F ( xa )  [(xaj )2  ( xcj )2 ]F ( xb )  [(xbj )2  ( xaj )2 ]F ( xc ) 2[(xcj  xbj )F ( xa )  ( xaj  xcj )F ( xb )  ( xbj  xaj )F ( xc )]

(5)

,

j 1   D (6)

where F ( x a ) , F ( x b ) and F ( x c ) are the fitnesses of the three selected individuals, respectively. In the optimizing process of the QIAEA, after the population is updated, the individuals are sorted according to their fitnesses. Choose three individuals X k , X k 1 , X k 2 ( k increases from 1 to N  2 ,

N is population size) from the sorted population to generate a new individual via Eq.5 and Eq.6. The

interpolation among those outstanding individuals can help to greatly improve the convergence speed of AEA. Because when these individuals locate in the vicinity of the possible optimal solution, this operation takes advantage of the strong exploitation ability of QI method. And the quadratic interpolation among other individuals act as crossover operators, which in a way increase the diversity of population. The old individuals are then replaced by the new ones if the latter performs better and the population is updated. This process continues until the termination condition is met. Searching in this way, QIAEA is expected to have higher convergence speed and higher solutions quality. The flowchart of QIAEA is shown in Fig.2 and the implementation steps of QIAEA are described in detail as follows: Step 1. Set the population size ( N ), the variable dimensions ( D ), the function optimization goal, the current function evaluations FEs  0 , and the termination condition. Step 2. Randomly generate the initial population G1 within the search space. Step 3. Calculate the objective function values of population G1. FEs  FEs  N . Step 4. Generate another population G2 . The generation process follows the rules: supposing the individuals in population G1 have a sequence number from 1 to N, randomly choose an integer

I within [2, N ] , and then rearrange the order of individuals in population G1 according to the numeric sequence

[I , I  1 ,..., N , 1, 2 ,..., I  1] to form population G2 . Actually, G2 is

just a reordering of the individuals in G1. Step 5. Calculate the cross-correlation annealing temperature

Cij according to Eq.1. According to Eq.4 and Eq.2, calculate the

T and the probability Pij . Then, create the interim population G3

based on Eq.3. Note that, if the variables in the new generated individuals exceed the limit ranges, they will be replaced by their nearest boundary values. FEs  FEs  N .

Step 6. Calculate the objective function values of the interim population G3 . According to the greedy law, the individuals in G1 are updated via Eq.7 as follows:

 G1 , if F (G1i ) F (G3i ) G1i   i , i  1,2,...,N G3i , if F (G1i )  F (G3i )

(7)

Step 7. Rearrange the updated G1 according to their ascending objective function values to form the population G4 . Set k  1 . Step 8. Choose three individuals X k , X k 1 , X k 2 from the population G4 . The three individuals are used to generate a new individual (denoted as Z k ) according to Eq.5 and Eq.6. Again, correct the variables that beyond the search region. FEs  FEs  1 . Step 9. Similarly, use the greedy law and compare the objective function value of the new individual Z k with that of the old individual X k ; the one with the smaller value will be the remainder. Set

k  k  1 and if k  N  1, go to Step 8. The population G1 is updated and if the termination criterion is met (reaching the maximum number of function evaluations, Max_FEs), output the result; otherwise, go to Step 4. Fig. 2

The pseudocode of QIAEA can be written as follows: Initialize all relevant parameters and the population G1 While (terminal condition is not met) Generate the comparison population G2 Calculate the cross-correlation

Cij , the annealing temperature T , and the probability Pij .

Generate the intermediate population G3 Update the population G1 Generate population G4 as an ascending sorted population of G1

Set k  1 While ( k < the number of individuals - 1) Choose three individuals X k , X k 1 , X k 2 from G4 Generate a new individual Z k according to Eq.5 and Eq.6. Compare the objective function value of the new individual with that of X k , the one with the smaller value will be the remainder.

k  k 1 End while Update the population G1 with the updated new individuals Record the optimal value End while In order to illustrate the effect of the combination, a numerical example is presented here. For the sake of simplicity, a one-dimensional test function is used to explain the process:



f ( x)  1  exp  0.5x 2



(8)

Here, the population size is assumed as 4. The evolution process of the population is shown in Table 1. First, randomly initialize 4 individuals: X 1 ,

X 2 , X 3 and X 4 . The values outside and inside the

brackets are the individuals and their corresponding objective function values, respectively. After the Alopex operation, four new individuals (0.2129, -0.837, 0.0287, and 0.0112) are generated and their function values are 0.0224, 0.0035, 4.12E-4, and 6.32E-5, respectively. According to the greedy law, the initial population G1 is updated and the temporary population G1' is generated. Then, the individuals are sorted according to their fitness values and the QI method is executed. generate a new individual

X 4 , X 3 , X 2 are used to

Z1 ( Z1 =2.83e-5, f (Z1 ) =4.00e-10), and X 3 , X 2 , X 1 are used to

generate another new individual Z2 ( Z2 =3.13e-4, f (Z2 ) =4.89e-8) via Eq.5 and Eq.6, respectively. After the greedy selection, a new generation is produced. This procedure goes on until reaching the termination condition. In this case, the objective function value (4.00e-10) is gotten just in one iteration.

Table 1.

4. Simulation results and analysis In this section, the traditional benchmark functions are utilized to test the performance of the proposed algorithm. The mathematical formulas of these benchmark functions as well as their dimensions and their value ranges are shown in Table 2. The benchmark functions have different characteristics: f1 - f8 are unimodal functions, while f6 is discontinuous step function and f7 is a noisy quartic function. The ninth function, Rosenbrock, is unimodal for dimensions of 2 and 3 but turns out to be multimodal in high dimensions [15]. The rest functions (f10 - f22) are multimodal and their local optima increases exponentially with the increase of dimension. Further, the CEC 2013 benchmark functions [16] are also used to verify whether the proposed algorithm can deal effectively with real parameter single objective optimization. The results on CEC 2013 benchmark functions are presented in Section 4.4. Considering the improvement in this paper is based on the AEA, hence, it is necessary to compare the QIAEA，AEA and an improved AEA by using local searching (LAEA) [17] to observe the effectiveness of the improvement. In addition, to give the overall performance of the proposed algorithm, we also conduct comparisons between QIAEA and several state-of-the-art algorithms. To avoid randomness, the Wilcoxon signed-rank test [18], a nonparametric test for pairwise comparisons, is employed.

Table 2.

4.1 Comparison of QIAEA with AEA and an improved AEA

A series of tests with AEA, LAEA, and QIAEA are conducted in this section. Parameters about this test part are set as follows: The number of individuals in the population are all fixed at 100. For most functions, a run is regarded as successful if the difference between the value obtained by the algorithm and the known optimal value is less than 0.01. However, for the two functions Rosenbrock and Schwefel2.26, for which it is difficult to find an optimal solution, the accuracy threshold is set to 15 and 0.4, respectively; and for the function Himmelblau, whose optimal value is less than -75, the accuracy threshold is set as -75. In LAEA, the number of the individuals using local search, the number of searches using local search and the number of consecutive generations are set to 50, 100, and 5, respectively according to literature [17]. First, we test the algorithms on 10-dimentional problems. A total of 100 runs are conducted for each function. Here, the function evaluations (FEs) is used as the termination condition for the algorithms. In order to compare the dynamic performance, three FEs, 50,000, 100,000, and 200,000, are used to evaluate the three algorithms. Tables 3–5 list the rates of success (RS), the average best function values (ABFV), and the standard deviations (SD) of 100 independent runs of the three algorithms. It should be noted that ABFV and SD are calculated using successful runs only, while unsuccessful runs are not considered and if all the runs are failed, the corresponding indicators are denoted as (-) instead. The best results among these algorithms are marked in boldface for clarity. To investigate the overall performance, for each algorithm the number of no worse results (NNWR) compared with the other two algorithms is calculated; these are listed at the bottom of Tables.

Tables 3–5

On observing the results given in Tables 3–5, it can be seen that the NNWRs of RS obtained by

QIAEA are 22/22/22, while those for AEA are 17/19/20, and those for LAEA are 19/21/22. These results show that QIAEA has the highest success rate in finding solutions in the three algorithms. The NNWRs of ABFV, which represent the quality of solutions obtained by the algorithms, are 19/19/21 for QIAEA, 4/11/9 for AEA, and 7/11/13 for LAEA. The NNWRs of SD, representing the stability of the algorithms, are 17/18/21 for QIAEA, 4/9/9 for AEA, and 8/11/14 for LAEA. These results show that the QIAEA is more efficient than AEA and LAEA. Moreover, its stability is better than those of AEA and LAEA. We then further conduct the comparison on 30 and 60 dimension problems. Table 6 and 7 summarize the mean result of 30 runs with 100,000 and 200,000 FEs of all benchmarks with dimension 30 and 60, respectively. From the Tables we can note that, for the 30 dimension, QIAEA has best performance on 18 benchmark functions in terms of solutions. LAEA obtains the best results on function 14, 15 and 17. AEA wins the first place on one function (f19). It should be also noted that only QIAEA can find the optimum results on function 4, 6, 16 and 21. For the 60 dimension, the results are similar with the 30-dimensional case except on function 17. QIAEA obtains better result than LAEA while it loses to LAEA when the dimension is 30. That may because the local search strategy in LAEA is executed on only one dimension, when the dimension of the problem increases, this strategy may become inefficient to some degree. In order to make a more intuitive comparison, we also plot the convergence curves for some 10-dimensional functions (most of them are multimodal) in Fig.3. The termination condition of the algorithms is a maximum of 200,000 function evaluations. The abscissa represents the number of function evaluations, and the ordinate represents the logarithm of the absolute objective function value.

Table 6.

Table 7.

Fig. 3

We can observe from Table 6, 7 and Fig.3 that QIAEA is faster than AEA and LAEA on almost all the test functions. Thus, it can be concluded that the proposed algorithm can obtain the most effective and successful results for both low and high dimensional problems among the three algorithms. To confirm that the results are not due to randomly factors, the Wilcoxon signed-rank test is implemented between QIAEA and the other two algorithms. In the relevant positions of Tables 3–7, we report the results of the Wilcoxon signed-rank test with significance level 0.05. It should be noted that the results are produced with respect to the function values obtained by the algorithms. In the charts, „+‟ indicates that the difference is significant at a 0.05 level of significance, while „-‟ indicates that the difference is not significant. And if the two algorithms achieve the same function value, „.‟ is used. It can been seen from Tables 3–7 that, for the 10-dimensional cases, QIAEA is significantly better than AEA and LAEA on most of the functions except the functions that they achieve the same ABFVs. For 30-dimensinal cases, QIAEA is significantly better than AEA and LAEA on 18 and 17 functions, respectively. For 60-dimensional cases, the numbers of functions on which QIAEA is significantly better than AEA and LAEA are both 19. This shows that the differences between QIAEA and other two algorithms are statistically significant.

4.2 Comparison of QIAEA with state-of-the-art PSOs To further evaluate the performance of the algorithm, QIAEA are compared with seven algorithms that are improvements based on PSO: CLPSO [19], HPSO-TVAC [20], FIPSO [21], SPSO-40, LPSO [22], DMS-PSO [23], and LFPSO [24]. The number of particles or population size for all algorithms is set to 40, the dimension of all

benchmark functions is set to 30, and the maximum number of function evaluations is set to 200,000 for each run for all algorithms. For the purpose of reducing statistical errors, each algorithm is tested 25 times independently for each function, and the mean results are used in the comparison. Table 8 shows the results of the comparison. The best mean results and the best standard deviations of the benchmark functions obtained by the algorithms are shown in bold. The algorithm finding the best result for the function is ranked 1, the worst algorithm is ranked 8, and the other algorithms are ranked between 1 and 8 according to their respective results.

Table 8.

It can be seen from Table 8 that, SPSO-40 can find the best results on Sphere and Schwefel2.22 unimodal functions (f1, f5), while LFPSO wins first on Noise unimodal function (f7). For the remaining multimodal functions, CLPSO finds the best solution on Rosenbrock (f9), Rastrigin (f14) and Schwefel2.26 (f15), but it does not show same success on Ackley (f10), Penalized1 (f17) and Penalized2 (f18). In spite of ranking sixth on Rosenbrock, the proposed algorithm is successful on Ackley, Griewank, Penalized1, and Penalized2 and is ranked second on Schwefel2.22, Noise and Schwefel2.26. Other comparing algorithms are failed to achieve best solution for any function. The bottom of the table shows the final rank of all the competitors. We can note that QIAEA offers the best overall performance, followed by LFPSO. CLPSO is the third, while SPSO-40 is in the fourth place, followed by HPSO-TVAC, FIPSO, and DMS-PSO. LPSO is the final. Therefore, QIAEA has the best performance among these competitors. In order to further characterize the significant difference between QIAEA and the other seven algorithms in the quality of their solutions, the Wilcoxon signed-rank test is also employed and the

comparison results are displayed in Table 9.

Table 9.

From Table 9, the number of P_values less than 0.1 is five, which means that QIAEA has an advantage of more than 90% over five algorithms. The P_value for LFPSO is 0.1875, which means that a nearly 82% advantage over LFPSO is obtained by QIAEA. For CLPSO, however, the P_value is 0.461, which means that QIAEA has a performance roughly equivalent to that of CLPSO.

4.3 Comparison of QIAEA with state-of-the-art DEs DEs are a class of highly efficient parallel search algorithms with strong global search capability and robustness. In Table 10, we conduct a comparison between QIAEA with several state-of-the-art DEs, namely ODE [25], SaDE [26], MoDE [27], CoDE [28] and MDE_pBX [29]. The results of these DEs are derived directly from literature [30]. To make a fair comparison, the algorithms are all tested on 30 dimensional problems and the population sizes are all set at 100. The parameter settings of DEs refer to corresponding literatures. To be consistent with the references, the FEs are set differently among the benchmark functions. Each algorithm is repeated 30 times independently for each function and the best mean results and the best standard deviations results are recorded in Table 10. The best results obtained by the algorithms are shown in bold.

Table 10.

From Table 10, we can see that the proposed algorithm outperforms the other algorithms on f6, f9, f11

and f18. It wins the second place on f5, f14, f15, f17 and ranks third on the remaining functions. SaDE wins the first place on two unimodal functions (f1, f5) and one multimodal function (f10), but has poor performance on the other functions. CoDE can obtain the best results on f14 and f15, while ODE only wins one function. The other two competitors, MDE_pBX and MoDE cannot yield best results on the whole functions. QIAEA is located first three rank on the whole test functions, which demonstrates that it has good robustness when solving the optimization problems. It can be observed that the average rank of QIAEA is much better than other competitive DEs. The Wilcoxon signed-rank test is also employed to further test the significant difference between the QIAEA and other algorithms. Table 11 displays comparisons among the algorithms. We can see that the number of P_values less than 0.05 is 3, which means that the difference between QIAEA and three DE variants is significant. The P_values for the other two DEs, ODE and CoDE, are 0.0527 and 0.138, which means QIAEA has an advantage of 94.73% and 86.2% over ODE and CoDE, respectively. Thus it can be concluded that QIAEA has the best performance among the competitive algorithms.

Table 11.

4.4 Comparison with ARFO on CEC 2013 benchmark functions The CEC‟13 test suite includes 5 unimodal functions, 15 multimodal functions and 8 composition functions, which are specifically designed and are much more challenging than the traditional ones. The properties of these functions can be referred to [16]. In this experiment, the QIAEA is tested on CEC 2013 benchmark functions and is compared with a state-of-the-art algorithm, namely artificial root foraging optimization (ARFO) algorithm [31]. In the literature [31], the ARFO algorithm has been comprehensively

tested and analyzed on all the CEC 2013 benchmarks and has been proved to be more efficient than several other state-of-the-art algorithms. To compare with ARFO, experiments are conducted for two dimensions: D = 10 and D = 50 on each test function, with a maximum number of FEs equal to 10,000 * D. Each function is run for 30 times and the search space is set to [-100, 100] for all the 28 functions. Here, the best, worst and mean results of the two algorithms, as well as the SD throughout the whole runs are recorded in Table 12 and 13, where F1 to F28 are the functions in the CEC‟13 test suite and F(x*) is the theoretical optimum.

Table 12

Table 13

From the tables we can observe that, for the 10-dimensional problems, ARFO can always find the global minimum on the whole unimodal functions (F1-F5), while QIAEA can find the global optimum only on two functions (F1, F5). However, for the multimodal functions, it is clear that, QIAEA can achieve more excellent results than ARFO on almost all the multimodal problems except F6, F10, F16 and F20. For the composition functions, it seems that the ARFO algorithm can sometimes find better results but is also easy to fall into local optima that are far from the global optimum. It outperforms QIAEA only on 3 functions (F21, F23, F27) out of 8. The results of the 50-dimensional cases are listed in Table 13. We can observe that ARFO still gives good results on unimodal functions except F4. For the multimodal functions, the average results indicate little difference between the performances of the two algorithms. Nevertheless, QIAEA has more chance of finding better solutions than ARFO. It is also apparent that, QIAEA performs

much better than ARFO on the composition functions F22, F24, F25, F26 and F28. In Table 14, the Wilcoxon signed-rank test is executed for QIAEA vs. ARFO for all the CEC 2013 benchmark functions. It can be seen that, QIAEA obtains higher R+ values than R- values in all the cases, which means QIAEA performs better than ARFO, although the differences are not significant at a 0.05 level significance (except for 10-dimentional test functions in regard to the worst results and SD).

Table 14

The results for the CEC 2013 functions with the proposed algorithm further demonstrate the effectiveness of the improvement and we believe that QIAEA has the potential of solving complex optimization problems.

4.5 Algorithm complexity The algorithm was conducted in a Windows 7 Professional OS environment using an Intel i5, 3.2 GHz, 12 GB RAM and the codes were implemented in Matlab 7.11.0. Table 15 shows the algorithm complexity for the QIAEA.

Table 15

According to [16], T0 is calculated by running the following test problem: for i = 1: 1000000 x = 0.55 + (double) i; x = x + x; x = x./2; x = x*x;

x = sqrt(x); x = log(x); x = exp(x); y = x/x; end In Table 15, T1 is the computing time on F14 for 200,000 evaluations and for the proposed algorithm on F14 for 200,000 evaluations.

T2 is the computing time

T2 is executed five times and the mean for

T2 is denoted as Tˆ2 . The algorithm complexities are calculated on 10, 30 and 50 dimensions. According to Table 15, the algorithm complexity grows with the increase of the dimensions, as shown by the growth of (Tˆ2  T1 ) / T0 .

5. Application to parameter estimation for kinetic models In chemical production, the rate of a chemical reaction is a very important parameter. The reaction rate model can be used to predict and optimize the parameters in actual production processes. Typically, the reaction rate is related to the concentration, pressure and temperature of reactants and also the catalyst. Building reaction rate model can also help to study connotation of chemical reactions deeply. Lots of scholars have set up kinetic models for chemical processes. For example, Chen et al. [32] have set up a low-temperature sulfur dioxide (SO2) oxidation with Cs–Rb–V sulfuric acid catalyst model to provide the basis for the design of the reactor; Song et al. [33] have established a heavy oil thermal cracking three lumps model to predict the yield of product. These kinetic models are usually derived from the associated mechanism and the relevant parameters need to be estimated through the sample data, which can be measured under different conditions (for example, different temperature, pressure and so on). Through these sample data, we can build an objective function, that is, the deviation between the model output and actual output. Thus, algorithms can be utilized to solve this estimation problem. In this section, we apply the QIAEA to solve two parameter estimation problems and evaluate the algorithm at the meantime.

5.1 Parameter estimation for low-temperature SO2 oxidation with Cs–Rb–V sulfuric acid catalyst model Used as a low-temperature sulfuric acid catalyst, Cs–Rb–V has lower initiation temperature, simpler technology, and lower cost than other catalysts. This catalyst can improve the conversion rate of SO2 for the smelter acid production and thus it is necessary to study the reaction mechanism and kinetics of SO2 with Cs–Rb–V sulfuric acid catalyst. The oxidation reaction of SO2 on Cs–Rb–V at 380~520 degrees Celsius can be divided into three catalytic reaction steps [32]: First, the SO2 dissolves in molten salt and is combined with the complexes of pentavalent vanadium (denoted as X) to produce X·SO2. X·SO2 is oxidized to Y·SO3; Then Y·SO3 releases SO3 and produces tetravalent vanadium compound (denoted as Z); Finally Z is oxidized to pentavalent vanadium compound (X) by oxygen that dissolved in the molten salt, while SO3 removes to the gas phase from the molten salt. When the molar ratio of alkali metal element and vanadium in the catalyst is in the range of 3 to 5, then X is mainly complexes of pentavalent vanadium; Y is the conjugates of X and SO2 and Z is mainly complexes of tetravalent vanadium. The reaction mechanism of the above process can be written as follows: X + SO2 Y·SO3

⇋

⇋

Z + O2

Y·SO3

Z+SO3

⇋

X

(9) (10) (11)

Through research on the physicochemical properties and reaction kinetics of the Cs–Rb–V catalyst molten salt system, Chen et al. have given the model for low-temperature SO2 oxidation with Cs–Rb–V sulfuric acid [32], which is shown below:

r

k 1PO12/ 2 k2  k3 PSO3

PSO  3 PSO2

(1 

PSO3 ) PSO2 PO12/ 2 K P

(12)

where

r

is total reaction rate of the three catalytic reaction steps.

pressure of the corresponding gas component.

k0

and activation energy

k

PSO3 , PSO2

and

PO2

are partial

is rate constant, defined by the pre-exponential factor

E . KP is equilibrium constant of the oxidation reaction. k and KP

can be defined by:

k  k0 exp(

E ) (  1,2,3) RT

(13)

 4812.3  K P exp[2.3026   2.8254lg T   2.284103T   T  7 2 10 3 7.01210 T  1.19710 T  2.23] where

(14)

T is reaction temperature. E is activation energy. R is molar gas constant and equals to

8.314J/(mol K). The reaction rate r , partial pressure

PSO3 , PSO2 , PO2

and reaction temperature

this model can be determined by experiments, while the remaining six parameters,

k0

and

T in

E , are to

be estimated. The reaction is a Multiple-Input Multiple-Output system with nonlinearity, which is difficult to optimize using traditional methods. To work with the equation in a realistic manner, QIAEA and other algorithms are utilized to estimate parameters. Data provided by Chen et al. [32] are used as modeling samples and test samples. The object function is set as:

min Arer 

1 m (ri  rˆi ) / ri 2  m i 1

(15)

Where rˆi is obtained by using Eq.12, ri is given by the experimental data, and m is the number of modeling samples.

Arer is average self-checking relative error.

We employ AEA, LAEA and QIAEA to implement on parameter estimation of this model. To make a fair competition, the population size of the AEAs is set at 100 and the terminal condition is set at 100,000

FEs. The results obtained by various algorithms, including the Powell method and a eugenic evolution genetic algorithm (EGA) [34], are listed in Table 16. From Table 16, it can be seen that, the average self-checking error obtained by QIAEA is 3.919%, while those for Powell method, EGA, AEA, and LAEA are 6.677%, 3.926%, 3.934% and 3.928%, respectively. QIAEA obtained the smallest

Arer , indicating

that it can offer a fairly high fitting precision of the model.

Table 16.

5.2 Parameter estimation for a heavy oil thermal cracking three-lump model Cracking of heavy oil is a very complex chemical process. Using lumped reaction model for the heavy oil thermal cracking can simplify the reaction system that has variety of components. In other words, the simplified model can be built by merging a large number of single components into several virtual components, according to the principle of similar kinetic characteristics. The heavy oil thermal cracking three-lump model has already been given by Song et al. [33]. The material transformation process of heavy oil thermal cracking three lumps model is shown in Fig. 4. Assume x is the sum of the yield of various fractions, pyrolysis gas and toluene insoluble.

xW

is the yield of the heavy distillate.

xL

is the sum of

the yield of pyrolysis gas, light fraction and condensation products. There are two hypothesizes as follows: Fig. 4

a) Pyrolysis reaction raw material H is toluene solved in residual oil hotter than 510 degrees Celsius. The middle of the heavy fraction W can further degenerate into cracking gas, light fraction, and condensation compound L at the same time. b) The secondary reaction that transfers heavy fraction W into lumped fraction L is a first-order

reaction. Given the conditions of constant temperature and an overall reaction order of 1, the equation for the process can be expressed as follows:

xL 





K LP0e  ELP / T K K e ( EWPEWLP) / T 1  (1  x) nL  WP 0 WLP0 EWLP / T nL nW  KWLP0e









 EWLP / T   1 1  1  (1  x) KWLP0e  (1  x) nW 1   EWLP / T nW  KWLP0e 

where x and

(16)

T are the input variables and xL is the system output. Eight parameters are to be

determined: KLPO , KWPO , KWLPO ,

ELP , EWP , EWLP , nL , and nW .

As with Eq.16, this equation is a nonlinear multi-input equation; thus, it is difficult to use traditional methods to perform the optimization. Therefore, QIAEA and other algorithms are employed to find the best parameters to apply the equation in practice. For modeling samples and test samples, 56 groups of data provided by Song et al. [33] are used. The object function is set as:

min Arer  where

1 m  x (i)  xˆ L (i) / xL (i) m i 1 L

(17)

xˆL (i) is obtained by using Eq.16, xL (i) is given by the 56 groups of data, and m is the number

of modeling samples.

Arer is average self-checking relative error.

Results of parameter estimation using various algorithms are given in Table 17. We can see that the average self-checking error for QAEA is 5.49%, while the errors are 6.28%, 5.85%, 5.69%, 5.73%, and 5.53% for EGA, a composite particle swarm optimization (CPSO) [35], an adaptive particle swarm optimization (APSO) [36], AEA and LAEA, respectively. Hence, QIAEA is found to be superior to other competitive algorithms.

Table 17.

6. Conclusions To overcome the weakness of AEA, we have proposed QIAEA, which combines AEA with the QI method. QIAEA can find optima more quickly than does AEA, and the results are of higher quality. According to the results of experiments using benchmark functions, QIAEA is significantly superior to AEA and LAEA. In addition, it has better performance than some state-of-the-art algorithms in some situations, which demonstrate that the modified method is effective. Applications of the algorithm to estimate the kinetic parameters on low-temperature SO2 oxidation with a Cs–Rb–V sulfuric acid catalyst and on heavy oil thermal cracking indicate that QIAEA can also predict models well. Although QIAEA is efficient, there is still room for improvement. For example, how to choose three individuals to execute the quadratic interpolation operation is worth considering. In our study, the individuals are sorted according to their fitnesses and three adjacent individuals are selected. However, this is not the most efficient way when the algorithm is doing the local search. In fact, if every three individuals contain the best individual, it will have faster convergence but is not conductive to exploration. Therefore, in future work, we may introduce a selection mechanism that can either choose adjacent individuals or the best individual to be one of them adaptively, so as to further improve the performance of the algorithm.

Conflict of interest There is no conflict of interest.

Acknowledgments

The authors of this paper appreciate the National Natural Science Foundation of China (under Project No.21406064 and No.21676086) and the Fundamental Research Funds for the Central Universities for their financial support.

References [1] J. H. Holland, Adaptation in natural and artificial systems, The University of Michigan Press, Ann Arbor, 1975.

[2] R. L. Johnston, Evolving better nanoparticles: Genetic algorithms for optimizing cluster geometries, Dalton Transactions

22 (2003) 4193-4207.

[3] S. S. Chaudhry, W. Luo, Application of genetic algorithms in production and operations management: a review,

International Journal of Production Research 43 (2005) 4083-4101. [4] E. Nabil, A modified flower pollination algorithm for global optimization, Expert Systems with Applications 57 (2016)

192-203.

[5] Q. Q. Fan, Y. L. Zhang, Self-adaptive differential evolution algorithm with crossover strategies adaptation and its

application in parameter estimation, Chemometrics and Intelligent Laboratory Systems 151 (2016) 164-171.

[6] M. Mohammadi, M. Nastaran, A. Sahebgharani, Development, application, and comparison of hybrid meta-heuristics for

urban land-use allocation optimization: Tabu search, genetic, GRASP, and simulated annealing algorithms, Computers,

Environment and Urban Systems 60 (2016) 23-36.

[7] S. Kirkpatrick, C. D. Gelatt, M. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671-680.

[8] C. H. Mu, J. Xie, Y. Liu, F. Chen, Y. Liu, L. C. Jiao, Memetic algorithm with simulated annealing strategy and tightness

greedy optimization for community detection in networks, Applied Soft Computing 34 (2015) 485-501.

[9] H. F. Zhang, D. Yue, X. P. Xie, S. L. Hu, S. X. Weng, Multi-elite guide hybrid differential evolution with simulated

annealing technique for dynamic economic emission dispatch, Applied Soft Computing 34 (2015) 312-323.

[10] M. T. Assadi, M. Bagheri, Differential evolution and Population-based simulated annealing for truck scheduling problem

in multiple door cross-docking systems, Computers & Industrial Engineering 96 (2016) 149-161.

[11] E. Harth, E. Tzanakou, Alopex: a stochastic method for determining visual receptive fields, Vision Research 14 (1974)

1475-1482.

[12] S. J. Li, F. Li, Alopex-based evolutionary optimization algorithms and its application to reaction kinetic parameter

estimation, Computers & Industrial Engineering 60 (2011) 341-348. [13] W. F. Gao, S. Y. Liu, L. L. Huang, A novel artificial bee colony algorithm with Powell‟s method, Applied Soft

Computing 13 (2013) 3763-3775.

[14]E. Davoodi, M. T. Hagh, S. G. Zadeh, A hybrid improved quantum-behaved particle swarm optimization-simplex method

(IQPSOS) to solve power system load flow problems, Applied Soft Computing 21 (2014) 171-179.

[15] Y. W. Shang, Y. H. Qiu, A note on the extended rosenbrock function, Evolutionary Computation 14 (2006) 119-126.

[16] J. J. Liang, B. Y. Qu, P. N. Suganthan, A. G. Hernndez-Diaz, Problem Definitions and Evaluation Criteria for the CEC

2013 Special Session on Real-Parameter Optimization, Zhengzhou University/Nanyang Technological University,

Zhengzhou, China/Singapore, 2013, Technical Report 201212.

[17] J. R. Cheng, J. Y. Zhao, S. J. Li, An improved AEA using local search and its application to parameter estimation for

homogeneous mercury oxidation, Chemometrics and Intelligent Laboratory Systems 141 (2015) 68-78.

[18] Y. Xue, L. P. Chen, R statistical modeling and R software, Tsinghua University press, Beijing, China, 2012.

[19] J. J. Liang, A. K. Qin, P. N. Suganthan, S. Baskar, Comprehensive learning particle swarm optimizer for global

optimization of multimodal functions, IEEE Transactions on Evolutionary Computation 10 (2006) 281-295.

[20] A. Ratnaweera, S. K. Halgamuge, H. C. Watson, Self-organizing hierarchical particle swarm optimizer with

time-varying acceleration coefficients, IEEE Transactions on Evolutionary Computation 8 (3) (2004) 240-255.

[21] R. Mendes, J. Kennedy, J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on

Evolutionary Computation 8 (3) (2004) 204-210.

[22] J. Kennedy, R. Mendes, Population structure and particle swarm performance, In: Proceedings of IEEE international

conference on evolutionary computation, 2002, pp. 1671-1676.

[23] J. J. Liang, P. N. Suganthan, Dynamic multi-swarm particle swarm optimizer, Swarm Intelligence Symposium,

California, 2005, pp. 124-129. [24] H. Haklı, H. Ûguz, A novel particle swarm optimization algorithm with Levy flight, Applied Soft Computing 23 (2014)

333-345.

[25] S. Rahnamayan, H. Tizhoosh, M. Salama, Opposition-based differential evolution, IEEE Transactions on Evolutionary

Computation 12 (1) (2008) 64-79.

[26] A. Qin, V. Huang, P. Suganthan, Differential evolution algorithm with strategy adaptation for global numerical

optimization, IEEE Transactions on Evolutionary Computation 13 (2) (2009) 398-417. [27] U. Maulik, I. Saha, Automatic fuzzy clustering using modified differential evolution for image classification, IEEE

Transactions on Geoscience and Remote Sensing 48 (9) (2010) 3503-3510.

[28] Y. Wang, Z. Cai, Q. Zhang, Differential evolution with composite trial vector generation strategies and control

parameters, IEEE Transactions on Evolutionary Computation 15 (1) (2011) 55-66.

[29] S. Islam, S. Das, S. Ghosh, S. Roy, P. Suganthan, An adaptive differential evolution algorithm with novel mutation and

crossover strategies for global numerical optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part B:

Cybernetics 42 (2) (2012) 482-500.

[30] Y. Q. Cai, J. H. Wang, Differential evolution with hybrid linkage crossover, Information Sciences 320 (2015) 244-287.

[31] L. B. Ma, Y. L. Zhu, Y. Liu, L. W. Tian, H. N. Chen, A novel bionic algorithm inspired by plant root foraging behaviors,

Applied Soft Computing 37 (2015) 95-113.

[32] Z. X. Chen, H. Ye, J. Liu, Study on the mechanism for the low temperature SO 2 oxidation with Cs-Rb-V sulfuric acid

catalyst, Chemical Reaction Engineering and Technology 17 (2001) 119-124.

[33] X. F. Song, D. Z. Chen, S. X. Hu, J. Z. Xiao, F. Z. Liu, Eugenic evolution strategy genetic algorithms for estimating

parameters of heavy oil thermal cracking model, Journal of Chemical Engineering of Chinese University 17 (2003) 411-417.

[34] Q. F. Zheng, D. Z. Chen, Eugenic evolution genetic algorithm and its application for chemical engineering data

processing, Journal of Zhejiang University (Engineering Science) 37 (2003) 303-306.

[35] H. J. Yu, L. P. Zhang, D. Z. Chen, S. X. Hu, Estimation of model parameters using composite particle swarm

optimization, Journal of Chemical Engineering of Chinese University 19 (2005) 675-680.

[36] D. X. Zhang, Z. H. Guan, R. Q. Liao, Parameter estimation of the thermal cracking model for heavy oil based on particle

swain optimization, Journal of Xi'an Shiyou University (Natural Science Edition) 23 (2008) 94-98.

f 1-P

x1

P

x2 local global

x Fig. 1. A one-dimensional function

Set all relevant parameters and generate the initial population. Set FEs = N

Generate the comparison population

Perform the Alopex operation and update the population. FEs = FEs + N and set k = 1

Choose three individuals Xk, Xk+1, Xk+1 and generate a new individual Zk through QI method. FEs = FEs + 1

No

Yes F(Zk) < F(Xk)

k=k+1

Replace Xk with the new individual, and k = k + 1

No k = the number of individuals - 1 1 Yes FEs = Max_FEs

Yes Output the best solution

Fig. 2. Flowchart of QIAEA

No

Schwefel2.22

Noise

10

Zakharov's

3

AEA LAEA QAEA

0 -10

10

AEA LAEA QAEA

2

AEA LAEA QAEA

0 -10

-20 -30 -40 -50

log10(|f(x)|)

log10(|f(x)|)

log10(|f(x)|)

1

0

-20 -30 -40

-1 -60 -50 -70

-2 -60

-80 -90

0

0.5

1

1.5

2

FEs

-3

2.5 x 10

0

0.5

1

1.5

2

FEs

5

Rosenbrock

-70

2.5 x 10

0

0.5

7

2.5 x 10

5

5

AEA LAEA QAEA

2 0

6

2

Levy and Montalvo-1

4

AEA LAEA QAEA

1.5

FEs

Griewank

8

1

5

AEA LAEA QAEA

0 -5

-2

4 3

log10(|f(x)|)

log10(|f(x)|)

log10(|f(x)|)

5 -4 -6 -8

-10 -15 -20

2 -10 -25

1

-12

0 -1

-30

-14

0

0.5

1

1.5

2

FEs

-16

2.5 x 10

0

0.5

1

1.5

2

FEs

5

Sinusoidal

-35

2.5 x 10

0

0.5

1

Penalized1

2 0 -2

2

2.5 x 10

5

Alpine

10

AEA LAEA QAEA

1.5

FEs

5

5

AEA LAEA QAEA

5

AEA LAEA QAEA

0

0 -5

-4 -6 -8 -10

log10(|f(x)|)

log10(|f(x)|)

log10(|f(x)|)

-5 -10 -15

-10

-15

-20 -12

-20 -25

-14

-18

-25

-30

-16

0

0.5

1

1.5

FEs

2

2.5 x 10

5

-35

0

0.5

1

1.5

FEs

2

2.5 x 10

5

-30

0

0.5

1

1.5

2

FEs

Fig. 3. Convergence performance of difference AEAs on the 9 test functions

2.5 x 10

5

W

xW

KW KP H

M

(1-x)

x

KWL KL

L xL

Fig. 4. Heavy oil thermal cracking 3 lumps reaction model

The list of tables

Table 1. A numerical example of combining QI method with AEA. Table 2. Benchmark functions. Table 3. Comparison of the optimization results on 10-dimentioanl problems after 50,000 FEs. Table 4. Comparison of the optimization results on 10-dimentioanl problems after 100,000 FEs. Table 5. Comparison of the optimization results on 10-dimentioanl problems after 200,000 FEs. Table 6. Experimental results of AEAs on 30-dimensional functions with 100,000 FEs. Table 7. Experimental results of AEAs on 60-dimensional functions with 200,000 FEs. Table 8. Comparison between QIAEA and the state-of-the art PSOs. Table 9. The difference of ABFVs between QIAEA and other seven PSOs. Table 10. Comparison between QIAEA and the state-of-the art DEs. Table 11. The difference of ABFVs between QIAEA and other five DEs. Table 12. Results obtained by ARFO and QIAEA through 30 independent runs on 10-dimensional CEC2013 functions. Table 13. Results obtained by ARFO and QIAEA through 30 independent runs on 50-dimensional CEC2013 functions. Table 14. Results of the Wilcoxon signed-rank test for QIAEA vs. ARFO for all CEC 2013 functions. Table 15. Computational complexity. Table 16. Results of parameter estimation of Cs–Rb–V sulfuric acid catalyst model given by various algorithms. Table 17. Results of parameter estimation of heavy oil thermal cracking three lumps model given by various algorithms.

Table 1. A numerical example of combining QI method with AEA.

Generation

X1（f(X1)）

X2（f(X2)）

X3（f(X3)）

X4（f(X4)）

G1

0.5468(0.1388)

-0.0699(0.0024)

-0.0216(2.35E-4)

-0.0095(4.50E-5)

Initial Population

0.2129(0.0224)

-0.0699(0.0024)

-0.0216(2.35E-4)

-0.0095(4.50E-5)

Temporary population G1'

3.13E-4(4.89E-8)

2.83E-5(4.00E-10)

-0.0216(2.35E-4)

-0.0095(4.50E-5)

New Population

0

1

Table 2. Benchmark functions. No.

Name

f1

Sphere

Function

Bounds

n

x

2 i

i 1

i 1

10  n

f2

6

Elliptic

n1 x 2

n

SumPower

f4

Exponential

f5

Schwefel2.22

f6

x i 1

n

n

i 1

i 1

n

ix

4 i

 random[0,1) 2

4

2 i

100x   20 exp  0.02  

n

1 n

x i 1



2

i 1

i 1

Ackley

[-1.28,1.28]

 n i   n i  x    xi     xi   i 1  i 1 2   i 1 2  n

n1

f10

[-100,100]

2

i 1

Rosenbrock

[-10,10]

i

n

f9

[-1.28,1.28]

 floorx  0.5

Noise Zakharov‟s

[-1,1]

 xi   xi

Step

f8

i 1

i

n   1  exp  0.5 xi2  i 1  

i 1

f7

[-100,100]

i

i 1

f3

[-5.12,5.12]

2 i

 xi2  ( xi 1)2



n    exp 1n  cos2xi   20  e   i 1  

[-5.12,5.12]

[-30,30]

[-32,32]

f11

Griewank

f12

Levy and Montalvo-1

1

1 n 2 n x  xi  cos i   4000 i 1  i i 1

n1  2 2 10 sin 2 y1     yi  1 1  10 sin 2 yi 1    y n  1  n i 1 

, yi  1  f13

Levy and Montalvo-2

f14

Rastrigin

1 x  1 4 i







n

418.9829n   xi sin

Schwefel2.26

i 1

Sinusoidal

No.

Name

f17

Penalized1





[-5,5]



[-5.12,5.12]

 x

[-500,500]

10n   xi2 10 cos2xi  i 1

f16

[-10,10]

n1   0.1  sin 2 3x1   ( xi 1)2 1  sin 2 3xi 1   ( xn 1)2 1  sin 2 2xn    i 1  

n

f15

[-600,600]

i

n     n     3.5  2.5sin xi    sin 5 xi    6  i 1   6    i 1 

[0,π]

Table 2. (Continued) Function





Bounds



n1 n 2 2 10sin 2 y1     yi 1 1  10sin 2 yi 1    yi 1   uxi ,10,100,4 n i 1  i 1

yi  1  14 xi  1

[-50,50]

k xi  am , xi  a  u( xi , a, k , m)  0 ,a  xi  a  k  x  am , x  a i i  f18

Penalized2

f19

Alpine









n1   n 0.1sin 2 3x1   xi 12 1  sin 2 3xi1   xn 12 1  sin 2 2xn    uxi ,5,100,4 i 1   i1

[-50,50]

n

 x sinx   0.1x i 1

Levy

i 1

2

Weierstrass

n

Himmelblau

[-10,10]

i

2

i 1

1





n

20





1 1  sin 2 3xn 

[-10,10]

 

[-0.5,0.5]

0.5k cos 2  3k  xi  0.5  n0.5k cos 3k  k 0 i 1

f22

2

i

20

f21

i

x 1 1 sin 3x  sin 3x   x n1

f20

i



k 0

1 n 4  x 16xi2  5xi n i 1 i



[-5,5]

R

Table 3. Comparison of the optimization results on 10-dimentioanl problems after 50,000 FEs. AEA

LAEA

QIAEA

RS

ABFV

SD

Significant

RS

ABFV

SD

Significant

RS

ABFV

SD

100

5.76E-29

3.18E-28

+

100

8.63E-29

1.60E-28

+

100

4.25E-28

1.30E-2

100

7.00E-14

5.27E-14

+

100

6.27E-23

1.74E-22

+

100

8.12E-21

1.76E-2

100

1.98E-35

2.51E-35

+

100

1.87E-52

8.58E-52

+

100

2.83E-109

1.31E-1

100

4.44E-18

2.19E-17

+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

5.05E-17

4.39E-17

+

100

8.56E-17

5.62E-17

+

100

9.71E-21

1.22E-2

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

8.26E-03

3.07E-03

+

100

4.47E-03

2.01E-03

+

100

3.17E-03

1.11E-0

100

2.80E-10

3.59E-10

+

100

1.35E-09

1.63E-09

+

100

8.96E-14

3.88E-1

100

3.75E+00

8.27E-01

-

98

3.96E+00

9.15E-01

-

100

3.65E+00

2.22E+

31

2.27E-03

2.35E-03

+

100

3.41E-06

2.08E-05

+

100

1.40E-03

1.73E-0

73

1.59E-03

2.68E-03

+

100

1.52E-03

3.01E-03

+

100

9.38E-05

7.55E-0

100

2.48E-28

6.36E-28

+

100

1.96E-28

1.02E-27

+

100

4.71E-32

4.95E-4

100

1.22E-28

3.05E-28

+

100

1.36E-28

5.51E-28

+

100

1.35E-32

2.48E-4

0

-

-

+

100

3.40E-03

2.85E-03

+

100

2.80E-09

2.78E-0

81

5.21E-02

9.50E-02

+

100

1.42E-02

2.74E-02

+

100

1.27E-04

0.00E+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

1.25E-26

3.99E-26

+

100

3.21E-27

6.88E-27

+

100

4.71E-32

4.95E-4

100

3.00E-24

2.90E-23

+

100

6.46E-26

2.43E-25

+

100

1.35E-32

2.48E-4

100

1.51E-04

5.20E-04

+

100

1.75E-05

1.93E-05

+

100

1.68E-12

9.27E-1

100

4.04E-25

8.55E-25

+

100

2.74E-27

5.60E-27

+

100

1.35E-31

2.64E-4

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

-7.69E+01

1.05E+00

+

100

-7.83E+01

2.07E-14

.

100

-7.83E+01

6.03E-0

17

4

4

19

7

8

22

19

17

R

Table 4. Comparison of the optimization results on 10-dimentioanl problems after 100,000 FEs. AEA

LAEA

QIAEA

RS

ABFV

SD

Significant

RS

ABFV

SD

Significant

RS

ABFV

SD

100

4.73E-58

2.26E-57

+

100

2.06E-57

6.95E-57

+

100

1.73E-54

6.22E-5

100

1.01E-33

9.44E-34

+

100

2.46E-51

1.47E-50

+

100

2.27E-45

9.22E-4

100

1.65E-60

8.22E-60

+

100

4.47E-100

2.13E-99

+

100

0.00E+00

0.00E+

100

5.55E-18

2.43E-17

+

100

1.11E-18

1.11E-17

+

100

0.00E+00

0.00E+

100

1.26E-34

1.29E-34

+

100

3.35E-34

4.19E-34

+

100

2.98E-35

3.79E-3

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

7.06E-03

2.36E-03

+

100

3.68E-03

1.61E-03

+

100

2.09E-03

8.22E-0

100

9.09E-22

2.06E-21

+

100

5.11E-20

9.82E-20

+

100

2.09E-30

1.05E-2

100

1.27E+00

7.93E-01

+

98

1.87E+00

3.22E-01

+

100

1.34E+00

2.18E+0

95

5.07E-05

3.07E-04

+

100

3.24E-07

9.10E-07

+

100

8.88E-16

0.00E+

95

1.26E-03

2.95E-03

+

100

5.88E-04

2.09E-03

+

100

0.00E+00

0.00E+

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-4

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-4

51

9.39E-04

2.04E-03

+

100

4.71E-08

1.41E-07

+

100

0.00E+00

0.00E+

100

1.27E-04

3.83E-08

.

100

1.27E-04

4.19E-11

.

100

1.27E-04

0.00E+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-4

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-4

100

1.72E-08

7.72E-08

+

100

3.93E-09

8.33E-09

+

100

1.58E-16

1.42E-1

100

1.35E-31

2.64E-46

.

100

1.35E-31

2.64E-46

.

100

1.35E-31

2.64E-4

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

-7.70E+01

9.61E-01

+

100

-7.83E+01

3.71E-14

.

100

-7.83E+01

2.11E-1

19

11

9

21

11

11

22

19

18

R

Table 5. Comparison of the optimization results on 10-dimentioanl problems after 200,000 FEs. AEA

LAEA

QIAEA

RS

ABFV

SD

Significant

RS

ABFV

SD

Significant

RS

ABFV

SD

100

8.57E-101

2.96E-101

+

100

4.75E-102

3.25E-102

+

100

1.29E-108

2.14E-1

100

1.41E-73

2.04E-73

+

100

1.24E-101

8.42E-102

+

100

3.37E-95

2.83E-9

100

7.83E-103

5.65E-103

+

100

3.68E-112

1.02E-111

+

100

0.00E+00

0.00E+

100

2.22E-18

1.56E-17

+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

7.24E-70

1.35E-69

+

100

1.10E-68

2.56E-68

+

100

8.54E-84

1.33E-8

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

5.26E-03

1.74E-03

+

100

2.51E-03

1.28E-03

+

100

1.51E-03

6.72E-0

100

1.20E-44

2.27E-44

+

100

1.38E-40

5.26E-40

+

100

6.72E-62

2.41E-6

100

2.40E-01

7.85E-01

+

100

2.30E-01

4.78E-01

+

100

1.35E-01

7.28E-0

100

1.70E-06

1.70E-05

+

100

4.44E-15

0.00E+00

+

100

8.88E-16

0.00E+

98

1.33E-03

3.05E-03

+

100

7.43E-04

2.31E-03

+

100

0.00E+00

0.00E+

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-4

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-4

99

5.69E-06

5.50E-05

+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

1.27E-04

0.00E+00

.

100

1.27E-04

0.00E+00

.

100

1.27E-04

0.00E+

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-47

.

100

4.71E-32

4.95E-4

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-47

.

100

1.35E-32

2.48E-4

100

1.70E-15

9.30E-15

+

100

4.96E-15

2.03E-14

+

100

1.42E-26

4.64E-2

100

1.35E-31

2.64E-46

.

100

1.35E-31

2.64E-46

.

100

1.35E-31

2.64E-4

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+00

.

100

0.00E+00

0.00E+

100

-7.71E+01

9.30E-01

+

100

-7.83E+01

1.33E-13

.

100

-7.83E+01

1.33E-1

20

9

9

22

13

14

22

21

21

Table 6. Experimental results of AEAs on 30-dimensional functions with 100,000 FEs. AEA

LAEA

QIAEA

f ABFV

SD

Significant

ABFV

SD

Significant

ABFV

SD

1

2.30E-17

1.54E-17

+

3.32E-17

2.24E-17

+

8.81E-26

1.29E-25

2

1.08E-04

6.79E-05

+

2.57E-11

2.58E-11

+

3.72E-20

5.03E-20

3

1.32E-32

1.89E-32

+

1.60E-37

6.35E-37

+

3.88E-86

3.88E-85

4

2.46E-16

4.62E-17

+

2.63E-16

5.44E-17

+

0.00E+00

0.00E+00

5

1.58E-11

5.62E-12

+

8.73E-12

3.45E-12

+

2.25E-15

1.33E-15

6

5.00E-02

2.61E-01

+

2.00E-01

4.84E-01

+

0.00E+00

0.00E+00

7

5.85E-02

6.77E-03

+

1.15E-01

2.46E-02

+

1.39E-02

3.40E-03

8

1.62E-01

6.26E-02

+

4.49E-01

1.62E-01

+

4.85E-05

7.07E-05

9

3.41E+01

1.97E+01

+

4.12E+01

2.47E+01

-

2.81E+01

1.85E+01

10

1.28E-08

4.01E-09

+

1.49E-08

6.66E-09

+

5.96E-14

9.35E-14

11

4.09E-03

5.96E-03

+

6.67E-03

7.31E-03

+

9.86E-05

9.86E-04

12

2.94E-18

2.07E-18

+

1.92E-18

1.51E-18

+

1.57E-32

3.58E-47

13

2.53E-03

4.65E-03

+

3.30E-03

5.12E-03

+

1.94E-06

1.94E-05

14

7.22E+01

2.49E+01

+

4.23E+00

1.33E+00

+

8.09E+01

2.91E+01

15

2.63E+03

4.64E+02

-

3.86E+02

1.39E+02

+

3.07E+03

7.77E+02

16

1.04E-15

5.98E-16

+

1.48E-15

6.21E-16

+

0.00E+00

0.00E+00

17

1.10E-03

1.10E-02

+

1.09E-16

1.02E-16

+

2.21E-03

1.51E-02

18

4.83E-03

5.48E-03

+

2.93E-03

4.94E-03

+

2.96E-10

2.92E-09

19

4.39E-12

8.82E-12

+

8.72E-12

3.11E-11

+

8.08E-10

2.27E-10

20

2.53E-02

4.65E-02

+

1.83E-02

4.16E-02

+

1.60E-09

6.00E-09

21

2.35E-03

2.31E-02

+

1.73E-02

9.47E-02

+

0.00E+00

0.00E+00

22

-7.81E+01

5.08E-01

+

-7.80E+01

5.04E-01

+

-7.82E+01

3.76E-01

Table 7. Experimental results of AEAs on 60-dimensional functions with 200,000 FEs. AEA

LAEA

QIAEA

f ABFV

SD

Significant

ABFV

SD

Significant

ABFV

SD

1

1.48E-16

7.11E-17

+

3.11E-16

1.88E-16

+

2.82E-26

2.79E-26

2

9.78E-01

3.92E-01

+

2.00E-10

9.48E-11

+

4.05E-21

4.15E-21

3

2.30E-37

2.92E-37

+

1.11E-41

3.06E-41

+

1.56E-193

0.00E+00

4

6.70E-16

8.98E-17

+

7.25E-16

1.54E-16

+

0.00E+00

0.00E+00

5

1.44E-11

3.24E-12

+

2.41E-12

1.34E-12

+

2.83E-16

1.40E-16

6

1.63E+00

1.59E+00

+

8.43E+00

4.73E+00

+

0.00E+00

0.00E+00

7

1.27E-01

1.44E-02

+

3.20E-01

6.32E-02

+

3.76E-02

6.60E-03

8

4.96E+00

1.59E+00

+

1.01E+01

1.72E+00

+

4.24E-02

4.48E-02

9

1.11E+02

4.24E+01

+

1.37E+02

4.54E+01

+

6.08E+01

4.57E+01

10

1.31E+00

7.91E-01

+

1.91E+00

3.50E-01

+

3.55E-15

0.00E+00

11

2.14E-03

4.13E-03

+

3.12E-03

5.89E-03

+

4.57E-13

2.07E-12

12

2.65E-17

3.88E-17

+

1.39E-17

2.21E-17

+

7.85E-33

2.78E-48

13

8.73E-03

9.21E-03

+

1.05E-02

8.76E-03

+

1.46E-03

8.02E-03

14

5.45E+01

6.07E+01

-

1.21E+01

2.33E+00

+

1.49E+02

8.02E+01

15

9.44E+03

5.79E+02

+

1.42E+03

3.02E+02

+

1.08E+04

1.46E+03

16

6.42E-15

1.58E-15

+

1.44E-14

8.00E-15

+

2.96E-17

1.62E-16

17

1.43E-02

2.41E-02

+

2.07E-02

4.39E-02

+

7.68E-03

2.00E-02

18

8.69E-03

6.52E-03

+

3.56E-02

1.08E-01

+

1.98E-12

9.94E-12

19

1.33E-12

4.37E-13

+

1.24E-07

6.78E-07

+

1.34E-03

5.04E-03

20

5.84E-02

1.28E-01

+

8.67E-02

1.33E-01

+

1.74E-09

4.09E-09

21

1.32E+00

6.18E-01

+

3.68E-01

3.69E-01

+

0.00E+00

0.00E+00

22

-7.61E+01

1.01E+00

+

-7.44E+01

2.05E+00

+

-7.79E+01

1.50E+00

Table 8. Comparison between QIAEA and the state-of-the art PSOs. f

1

5

7

9

10

11

14

15

CLPSO

HPSOTVAC

FIPSO

SPSO-40

LPSO

DMS-PSO

LFPSO

QIAEA

Mean

1.58E-12

2.83E-33

2.42E-13

2.29E-96

3.34E-14

2.65E-31

4.69E-31

2.45E-30

SD

7.70E-13

3.19E-33

1.73E-13

9.48E-96

5.39E-14

6.25E-31

2.50E-30

2.20E-30

Rank

8

2

7

1

6

3

4

5

Mean

2.51E-08

9.03E-20

2.76E-08

1.74E-53

1.70E-10

1.57E-18

2.64E-17

6.94E-23

SD

5.84E-09

9.58E-20

9.04E-09

1.58E-53

1.39E-10

3.79E-18

6.92E-17

3.10E-23

Rank

7

3

8

1

6

4

5

2

Mean

5.85E-03

9.82E-02

4.24E-03

4.02E-03

2.81E-02

1.45E-02

2.41E-03

3.92E-03

SD

1.11E-03

3.26E-02

1.28E-03

1.66E-03

5.60E-03

5.05E-03

8.07E-04

9.51E-04

Rank

5

8

4

3

7

6

1

2

Mean

1.14E+01

2.39E+01

2.51E+01

1.35E+01

2.81E+01

4.16E+01

2.38E+01

2.59E+01

SD

9.85

2.65E+01

5.10E-01

1.46E+01

2.18E+01

3.03E+01

3.17E-01

1.75E+00

Rank

1

4

5

2

7

8

3

6

Mean

3.66E-07

7.29E-14

2.33E-07

3.73E-02

8.20E-08

1.84E-14

1.68E-14

3.55E-15

SD

7.57E-08

3.00E-14

7.19E-08

1.90E-01

6.73E-08

4.35E-15

4.84E-15

0.00E+00

Rank

7

4

6

8

5

3

2

1

Mean

9.02E-09

9.75E-03

9.01E-12

7.48E-03

1.53E-03

6.21E-03

8.14E-17

0.00E+00

SD

8.57E-09

8.33E-03

1.84E-11

1.25E-02

4.32E-03

8.14E-03

4.46E-16

0.00E+00

Rank

4

8

3

7

5

6

2

1

Mean

9.09E-05

9.43E+00

6.51E+01

4.10E+01

3.51E+01

2.72E+01

4.54E+00

4.69E+00

SD

1.25E-04

3.48E+00

1.34E+01

1.11E+01

6.89E+00

6.02E+00

1.03E+01

1.11E+01

Rank

1

4

8

7

6

5

2

3

Mean

3.82E-04

1.59E+03

9.93E+02

3.14E+03

3.16E+03

3.21E+03

1.37E+03

7.90E+00

17

18

SD

1.28E-07

3.26E+02

5.09E+02

7.81E+02

4.06E+02

6.51E+02

6.36E+02

3.00E+01

Rank

1

5

3

6

7

8

4

2

Mean

1.25E-12

2.79E-28

2.70E-14

7.47E-02

3.26E-13

2.64E-03

1.51E-28

1.35E-32

SD

9.45E-13

1.88E-29

1.57E-14

3.11E+00

3.70E-13

4.79E-03

8.00E-28

5.59E-48

Rank

6

3

4

8

5

7

2

1

Mean

5.85E-03

9.82E-02

4.24E-03

4.02E-03

2.81E-02

1.45E-02

2.41E-03

5.70E-13

SD

1.11E-03

3.26E-02

1.28E-03

1.66E-03

5.60E-03

5.05E-03

8.07E-04

2.92E-12

Rank

5

8

4

3

7

6

2

1

Mean Rank

4.5

4.9

5.2

4.6

6.1

5.6

2.7

2.4

Final Rank

3

5

6

4

8

7

2

1

Table 9. The difference of ABFVs between QIAEA and other seven PSOs. Algorithm

CLPSO

HPSO-TVAC

FIPSO

SPSO-40

LPSO

DMS-PSO

LFPSO

P_value

0.461

0.032

0.0244

0.0527

0.0010

0.0020

0.1875

Table 10. Comparison between QIAEA and the state-of-the art DEs. f

1

5

6

9

10

11

14

15

FEs

150,000

150,000

150,000

500,000

150,000

200,000

300,000

ODE

SaDE

MoDE

CoDE

MDE_pBX

QIAEA

Mean

2.24E-27

1.62E-48

6.51E-04

2.28E-07

1.24E-03

7.25E-23

SD

2.72E-27

4.46E-48

2.81E-03

8.29E-08

4.83E-03

5.63E-23

Rank

2

1

5

4

6

3

Mean

1.52E-11 1.52E-11

6.11E-45 6.11E-45

2.89E-03 2.89E-03

1.16E-07 1.16E-07

4.17E-08 4.17E-08

4.63E-17 4.63E-17

SD

1.06E-11

1.54E-44

1.23E-02

3.79E-08

1.85E-07

1.93E-17

Rank

3

1

6

5

4

2

Mean

1.7+03 2.32E+04

1.65E+03 1.00E+04

5.43E+02 6.05E+02

1.91E+03 6.25E+04

1.18E+04 8.12E+03

0.00E+00 0.00E+00

SD

1.76E+03

1.65E+03

5.43E+02

1.91E+03

1.18E+04

0.00E+00

Rank

5

4

2

6

3

1

Mean

2.52E+01 2.52E+01

1.28E+01 1.28E+01

4.51E+00 4.51E+00

3.33E+00 3.33E+00

4.70E+01 4.70E+01

2.71E+00 2.71E+00

SD

1.10E+00

5.86E+00

5.30E+00

1.96E+00

3.15E+01

7.26E+00

Rank

5

4

3

2

6

1

Mean

3.53E-14

4.02E-15

7.70E+00

1.31E-04

1.77E-02

2.15E-11

SD

2.04E-14

6.49E-16

1.82E+00

3.74E-05

9.30E-02

9.20E-12

Rank

2

1

6

4

5

3

Mean

2.47E-04

2.79E-03

2.49E-01

4.19E-09

4.92E-03

0.00E+00

SD

1.35E-03

6.61E-03

2.33E-01

3.72E-09

1.24E-02

0.00E+00

Rank

3

4

6

2

5

1

Mean

3.60E+01

1.43E+00

6.01E+01

1.02E-11

7.80E+00

8.39E-01

SD

1.91E+01

1.13E+00

1.41E+01

1.73E-11

2.25E+00

4.41E+00

Rank

5

3

6

1

4

2

Mean

6.99E+03

3.55E+01

4.14E+03

1.21E-12

1.22E+03

3.82E-04

300,000

17

18

SD

3.08E+02

6.34E+01

6.51E+02

8.72E-13

4.34E+02

0.00E+00

Rank

6

3

5

1

4

2

Mean

3.49E-28

1.38E-02

5.00E+00

8.21E-08

1.98E-02

6.21E-11

SD

5.11E-28

7.57E-02

6.30E+00

3.76E-08

6.57E-02

3.40E-10

Rank

1

4

6

3

5

2

Mean

3.49E-28 2.49E-28

1.38E-02 1.10E-03

5.00E+00 5.57E+00

8.21E-08 1.92E-07

1.98E-02 2.22E-02

6.21E-11 1.35E-32

SD

3.62E-28

3.35E-03

6.70E+00

7.83E-08

4.92E-02

5.57E-48

Rank

2

4

6

3

5

1

Mean Rank

3.4

2.9

5.1

3.1

4.7

1.8

Final Rank

4

2

6

3

5

1

150,000

150,000

Table 11. The difference of ABFVs between QIAEA and other five DEs. Algorithm

ODE

SaDE

MoDE

CoDE

MDE_pBX

P_value

0.0527

0.0137

0.00098

0.138

0.00098

Table 12. Results obtained by ARFO and QIAEA through 30 independent runs on 10-dimensional CEC2013 functions. ARFO

Fun

QIAEA

F(x*) Best

Worst

Mean

SD

Best

Worst

Mean

SD

F1

-1400

−1.400E+03

−1.400E+03

−1.400E+03

0.000E+00

-1.400E+03

-1.400E+03

-1.400E+03

0.000E+00

F2

-1300

−1.300E+03

−1.300E+03

−1.300E+03

0.000E+00

-1.222E+03

5.975E+04

2.036E+04

1.312E+04

F3

-1200

−1.2000E+03

−1.200E+03

−1.200E+03

1.411E+00

-1.197E+03

4.250E+08

3.299E+07

7.013E+07

F4

-1100

−1.100E+03

−1.100E+03

−1.100E+03

0.000E+00

-7.039E+02

1.210E+04

3.906E+03

2.816E+03

F5

-1000

−1.000E+03

−1.000E+03

−1.000E+03

2.356E−11

-1.000E+03

-1.000E+03

-1.000E+03

0.000E+00

F6

-900

−9.000E+02

−8.960E+02

−8.998E+02

8.914E−01

-9.000E+02

-8.900E+02

-8.915E+02

3.350E+00

F7

-800

−7.345E+02

−6.723E+02

−7.144E+02

1.342E+01

-7.994E+02

-7.450E+02

-7.801E+02

1.645E+01

F8

-700

−6.796E+02

−6.791E+02

−6.794E+02

6.603E−02

-6.799E+02

-6.795E+02

-6.796E+02

7.868E-02

F9

-600

−5.964E+02

−5.832E+02

−5.905E+02

3.680E+00

-5.985E+02

-5.942E+02

-5.969E+02

8.497E-01

F10

-500

−5.000E+02

−5.000E+02

−5.000E+02

2.036E−02

-5.000E+02

-4.988E+02

-4.996E+02

2.800E-01

F11

-400

−3.930E+02

2.884E+02

−2.884E+02

2.022E+02

-4.000E+02

-3.871E+02

-3.961E+02

2.291E+00

F12

-300

−2.920E+02

5.079E+02

−3.700E+01

3.308E+02

-2.960E+02

-2.692E+02

-2.883E+02

4.814E+00

F13

-200

−1.761E+02

1.204E+03

1.416E+02

3.834E+02

-1.947E+02

-1.500E+02

-1.771E+02

9.614E+00

F14

-100

1.165E+03

2.330E+03

1.741E+03

3.167E+02

-5.824E+01

9.492E+02

4.993E+02

2.841E+02

F15

100

7.050E+03

1.012E+04

8.604E+03

5.563E+02

2.257E+02

1.535E+03

8.229E+02

3.665E+02

F16

200

2.000E+02

2.003E+02

2.002E+02

8.771E−02

2.006E+02

2.019E+02

2.013E+02

2.377E-01

F17

300

3.148E+02

1.469E+03

4.363E+02

3.419E+02

3.106E+02

3.289E+02

3.194E+02

5.782E+00

F18

400

4.167E+02

1.069E+03

4.570E+02

1.441E+02

4.132E+02

4.411E+02

4.291E+02

7.839E+00

F19

500

5.005E+02

5.022E+02

5.012E+02

5.0548E−01

5.003E+02

5.017E+02

5.009E+02

3.141E-01

F20

600

6.000E+02

6.010E+02

6.005E+02

1.235E+00

6.020E+02

6.036E+02

6.029E+02

3.997E-01

F21

700

8.000E+02

1.100E+03

1.085E+03

6.713E+01

1.100E+03

1.100E+03

1.100E+03

0.000E+00

F22

800

1.906E+03

3.689E+03

3.096E+03

4.420E+02

8.206E+02

2.113E+03

1.068E+03

3.296E+02

F23

900

9.000E+02

8.543E+03

1.400E+03

2.234E+03

1.129E+03

2.226E+03

1.502E+03

2.610E+02

F24

1000

1.113E+03

1.628E+03

1.214E+03

1.635E+03

1.200E+03

1.221E+03

1.211E+03

5.294E+00

F25

1100

1.300E+03

1.334E+03

1.305E+03

5.212E+01

1.214E+03

1.317E+03

1.303E+03

1.328E+01

F26

1200

1.304E+03

1.561E+03

1.424E+03

8.254E+01

1.303E+03

1.517E+03

1.372E+03

6.004E+01

F27

1300

1.558E+03

1.871E+03

1.581E+03

7.414E+01

1.601E+03

1.700E+03

1.633E+03

2.665E+01

F28

1400

1.800E+03

2.203E+03

1.902E+03

9.323E+02

1.500E+03

2.116E+03

1.784E+03

1.582E+02

Table 13. Results obtained by ARFO and QIAEA through 30 independent runs on 50-dimensional CEC2013 functions. ARFO Fun

QIAEA

F(x*) Best

Worst

Mean

SD

Best

Worst

Mean

SD

F1

-1400

-1.400E+03

-1.400E+03

-1.400E+03

2.655E-13

-1.400E+03

-1.400E+03

-1.400E+03

1.110E-13

F2

-1300

8.466E+03

7.272E+04

3.726E+04

1.829E+04

3.439E+05

9.689E+05

6.232E+05

1.575E+05

F3

-1200

8.918E+05

7.541E+07

2.651E+07

2.501E+07

2.268E+06

7.486E+08

2.035E+08

1.699E+08

F4

-1100

4.250E+03

2.345E+05

6.452E+04

7.494E+04

1.629E+03

1.808E+04

9.914E+03

3.325E+03

F5

-1000

-1.000E+03

−1.000E+03

−1.000E+03

8.262E−06

-1.000E+03

-1.000E+03

-1.000E+03

2.831E-11

F6

-900

-9.000E+02

−8.524E+02

−8.875E+02

3.696E+00

-8.782E+02

-8.037E+02

-8.333E+02

2.390E+01

F7

-800

-7.292E+02

−6.609E+02

−6.998E+02

2.289E+01

-7.584E+02

-6.979E+02

-7.281E+02

1.373E+01

F8

-700

-6.791E+02

−6.789E+02

−6.790E+02

3.603E−02

-6.790E+02

-6.788E+02

-6.788E+02

4.078E-02

F9

-600

-5.549E+02

−5.218E+02

−5.479E+02

9.038E+00

-5.683E+02

-5.522E+02

-5.603E+02

3.496E+00

F10

-500

-5.000E+02

−4.999E+02

−5.000E+02

2.543E−02

-4.999E+02

-4.987E+02

-4.997E+02

2.601E-01

F11

-400

-3.920E+02

2.864E+02

−2.534E+02

3.032E+02

-3.692E+02

-3.284E+02

-3.564E+02

9.046E+00

F12

-300

-1.429E+02

3.264E+03

1.620E+03

1.369E+03

-2.105E+02

-4.927E+01

-1.403E+02

3.371E+01

F13

-200

1.361E+02

5.996E+03

3.302E+03

1.336E+03

3.228E+01

2.696E+02

1.229E+02

4.809E+01

F14

-100

7.254E+03

9.961E+03

8.492E+03

7.928E+02

7.510E+02

1.254E+04

4.480E+03

4.566E+03

F15

100

7.091E+03

1.054E+04

8.007E+03

5.294E+03

6.297E+03

1.452E+04

1.131E+04

3.123E+03

F16

200

2.023E+02

2.029E+02

2.003E+02

6.150E−01

2.015E+02

2.039E+02

2.033E+02

4.589E-01

F17

300

4.674E+02

6.572E+02

5.366E+02

4.140E+01

3.775E+02

7.211E+02

4.974E+02

1.309E+02

F18

400

5.775E+02

7.157E+02

6.407E+02

3.700E+01

5.269E+02

8.991E+02

7.804E+02

1.121E+02

F19

500

5.059E+02

5.143E+02

5.088E+02

2.164E+00

5.036E+02

5.118E+02

5.072E+02

1.842E+00

F20

600

6.201E+02

6.250E+02

6.219E+02

4.424E+02

6.183E+02

6.227E+02

6.209E+02

8.744E-01

F21

700

9.000E+02

1.888E+03

1.491E+03

3.726E+02

9.000E+02

1.822E+03

1.643E+03

2.578E+02

F22

800

9.624E+03

1.455E+04

1.303E+04

1.283E+03

1.386E+03

1.298E+04

5.104E+03

4.719E+03

F23

900

9.165E+03

1.394E+04

1.422E+03

1.151E+03

6.884E+03

1.552E+04

1.027E+04

2.592E+03

F24

1000

1.265E+03

4.603E+03

2.835E+03

1.052E+03

1.274E+03

1.342E+03

1.312E+03

1.510E+01

F25

1100

1.378E+03

2.421E+03

1.527E+03

2.966E+02

1.438E+03

1.486E+03

1.463E+03

1.062E+01

F26

1200

1.400E+03

3.539E+03

1.747E+03

6.814E+02

1.400E+03

1.627E+03

1.591E+03

5.773E+01

F27

1300

2.236E+03

3.186E+03

2.474E+03

2.245E+02

2.555E+03

2.968E+03

2.745E+03

1.063E+02

F28

1400

1.800E+03

1.955E+04

5.826E+03

5.596E+03

1.800E+03

5.808E+03

2.443E+03

1.406E+03

Table 14. Results of the Wilcoxon signed-rank test for QIAEA vs. ARFO for all CEC 2013 functions. Dim

10

50

Criteria

R+

R-

P_value

Best

181.5

118.5

0.1879

Worst

235

90

0.02635

Mean

217.5

133.5

0.1459

SD

274

132

0.02117

Best

177

99

0.1208

Worst

211

140

0.187

Mean

201

150

0.2627

SD

245

161

0.1751

Table 15. Computational complexity.

T0 D = 10 D = 30 D = 50

0.1680

T1

Tˆ2

(Tˆ2  T1 ) / T0

1.5968

2.8427

7.4142

2.4113

5.8200

20.0034

3.6712

8.4561

28.4740

Table 16. Results of parameter estimation of Cs–Rb–V sulfuric acid catalyst model given by various algorithms. Method

K01

K02

K03

E1

E2

E3

Arer(%)

Powell

0.152

8.18

0.221

62073

2384

18949

6.677

EGA

0.320

10.255

6.945

62610

17348

55694

3.926

AEA

0.340

14..642

5.339

62010

2775

57665

3.934

LAEA

0.263

13.529

19.578

59802

1474

98152

3.928

QIAEA

0.263

9.918

14.968

60900

1000

50490

3.919

Table 17. Results of parameter estimation of heavy oil thermal cracking three lumps model given by various algorithms. Method

KLP0

KWP0

KWLP0

ELP

EWP

EWLP

nL

nW

Arer(%)

EGA

4.032

5.058

4.228

1166

3676

2382

1.57

1.357

6.28

CPSO

0.151

4.032

4.303

-1013

1770

-400

2.97

1.605

5.85

APSO

5.303

2.9616

1.209

2388.3

493.88

-240.34

5.545

8.867

5.69

AEA

1.402

0.1533

6.567

596.40

-973.70

623.76

1.583

4.189

5.73

LAEA

1.317

0.174

5.756

501.79

-503.47

561.90

0.507

3.484

5.53

QIAEA

4.197

0.605

3.240

1294.58

-699.26

240.14

6.601

3.916

5.49