CN106529666A - Difference evolution algorithm for controlling parameter adaptive and strategy adaptive - Google Patents

Abstract

The invention discloses a difference evolution algorithm for controlling parameter adaptive and strategy adaptive. According to the method, through simulation tests of 13 100-dimensional standard test functions and actual optimization problem applications, the algorithm is proved to have relatively strong global excellence searching performance and a relatively rapid convergence rate, and prematurity phenomena existing in multiple traditional intelligent optimization methods can be well avoided; adaptive of control parameters can not only be realized, but also adaptive of variation strategy selection is further realized; as proved by test simulation results, the DE-CPASA algorithm has higher solution precision and a relatively faster convergence speed; the DE-CPASA algorithm can be applied to Hg oxidation dynamics parameter estimation, and a relatively excellent optimization result is acquired.

Description

Differential evolution algorithm for control parameter adaptation and strategy adaptation

Technical Field

The invention relates to the field of control algorithms, in particular to a differential evolution algorithm with adaptive control parameters and adaptive strategies.

Background

Since the differential evolution algorithm is proposed by storm and Price, compared with other intelligent optimization algorithms, the algorithm has better optimization performance and faster convergence speed, so that the method rapidly gets the attention and research of numerous scholars [2,3,4 and 5], and a great deal of research work is carried out on control parameters (a variation factor F and a cross probability CR) and strategies of the algorithm.

Although the differential evolution algorithm has a satisfactory optimization effect, in the face of an increasingly complex optimization problem, the fixed control parameter setting and the mutation strategy are difficult to adapt to the needs of algorithm evolution, and therefore, the existing algorithm defects are not negligible.

Disclosure of Invention

The present invention aims to solve the above problems and provide a differential evolution algorithm for controlling parameter adaptation and policy adaptation.

The invention realizes the purpose through the following technical scheme:

the invention aims at the optimization problem

Where f (x) is an optimized objective function, x is a D-dimensional optimization vector,andare respectively j variationQuantity x_jLower and upper limits of (d); the method specifically comprises the following steps:

(1) initialization: generating original population in respective feasible domainsAnd control parameter populationSetting the maximum number of iterations G_mAnd NP of the population size, and simultaneously, N_choice＝0，F_strategy1＝0，F_strategy2＝0；

(2) Original population S₁The evolution of (2): each original individualBy respective use ofAs control parameters, realizing differential evolution operation and generating new individuals

Selection of NP/2+ N_choiceIndividual mutation operation of the formula (2)

Wherein i_strategy1＝1,2，···，NP/2+N_choice

Selecting the remaining NP/2-N_choiceCarrying out individual mutation operations of the formulas (3) and (4) respectively according to the principle that rand is larger than 0.4, if so, carrying out the mutation operation by using the formula (3), and if not, carrying out the mutation operation by using the formula (4);

wherein i_strategy2＝NP/2+N_choice+1，NP/2+N_choice+1, NP, and i_strategy1+i_strategy2＝NP；

Individual boundary processing: if it isOrThenRandomly selecting from the feasible region;

individual crossover operation:

whereinIs a new individualThe j gene of (3);

operation of the adaptation of the policy: to obtainWhile at the same time causingTo obtainWhile at the same time causingIf F_strategy1≤F_strategy2Then N_choice＝N_choice+1, otherwise N_choice＝N_choice-1; at the same time, to N_choiceIf N is processed_choice> NP/2, then N_choiceIf N is NP/2-1_choice< -NP/2, then N_choice＝-NP/2+1；

Selection of individuals:

(3) control parameter populationThe evolution of (2):

F_i ^G+1＝N(0.5,σ)，(7)

wherein, sigma is 1.2-G/G_m；

Setting of control parameter boundaries: if F_i ^G+1> 1 or F_i ^G+1< 0, then F_i ^G+11 or F_i ^G+1＝0.

If it is notOrThenor

(4) Repeating the steps 2 to 3 until the evolution algebra exceeds the maximum evolution algebra G_m。

The invention has the beneficial effects that:

compared with the prior art, the differential evolution algorithm of the invention has stronger global search and optimization performance and faster convergence rate through simulation tests of 13 100-dimensional standard test functions and application to practical optimization problems, and can better avoid the phenomenon of early maturity of a plurality of traditional intelligent optimization algorithms. Not only the self-adaptation of the control parameters is realized, but also the self-adaptation of the selection of the variation strategy is realized. Test simulation results show that the DE-CPASA algorithm has higher solving precision and faster convergence speed. And finally, the DE-CPASA algorithm is applied to Hg oxidation kinetic parameter estimation, and a better optimization result is obtained.

Detailed Description

The invention is further illustrated below:

for optimization problems

Where f (x) is an optimized objective function, x is a D-dimensional optimization vector,andare respectively the jth variable x_jLower and upper limits of (d);

in the DE-CPASA algorithm, the structure and the strategy of individuals are used as shown in a table 0, and for the structure of the individuals, a mode that control parameters and the individuals are coded together is adopted, namely, each individual has one control parameter corresponding to the individual, so that the algorithm can realize dynamic change of the control parameters on the individual level while population evolution is realized, and a mode that the control parameters in the traditional DE algorithm are fixed is changed; for the use of the strategy, the algorithm makes N at the initial stage_choice0, namely, at the beginning of algorithm iteration, dividing the population into two sub-populations with the number of individuals NP/2, then carrying out mutation operation on the individuals by using mutation strategies such as DE/rand/1, DE/rand-to-best/1 and DE/rand/2, and evaluating the individuals, and finally using N_choiceThe strategy is self-adaptive, so that the overall optimization performance of the algorithm is improved.

TABLE 0 original populationControl parameter populationAnd mutation strategies

Fig.1.Primary populationthe population ofcontrol parametersandmutation strategy

The method specifically comprises the following steps:

(1) initialization: generating original population in respective feasible domainsAnd control parameter populationSetting the maximum number of iterations G_mAnd NP of the population size, and simultaneously, N_choice＝0，F_strategy1＝0，F_strategy2＝0；

(2) Original population S₁The evolution of (2): each original individualBy respective use ofAs control parameters, realizing differential evolution operation and generating new individuals

Selection of NP/2+ N_choiceIndividual mutation operation of the formula (2)

Wherein i_strategy1＝1,2，···，NP/2+N_choice

Selecting the remaining NP/2-N_choiceCarrying out individual mutation operations of the formulas (3) and (4) respectively according to the principle that rand is larger than 0.4, if so, carrying out the mutation operation by using the formula (3), and if not, carrying out the mutation operation by using the formula (4);

wherein i_strategy2＝NP/2+N_choice+1，NP/2+N_choice+1, …, NP, and i_strategy1+i_strategy2＝NP；

Individual boundary processing: if it isOrThenRandomly selecting from the feasible region;

individual crossover operation:

whereinIs a new individualThe j gene of (3);

operation of the adaptation of the policy: to obtainWhile at the same time causingTo obtainAverage value of whileMake itIf F_strategy1≤F_strategy2Then N_choice＝N_choice+1, otherwise N_choice＝N_choice-1; at the same time, to N_choiceIf N is processed_choice> NP/2, then N_choiceIf N is NP/2-1_choice< -NP/2, then N_choice＝-NP/2+1；

Selection of individuals:

(3) control parameter populationThe evolution of (2):

F_i ^G+1＝N(0.5,σ)， (7)

wherein, sigma is 1.2-G/G_m；

Setting of control parameter boundaries: if F_i ^G+1> 1 or F_i ^G+1< 0, then F_i ^G+11 or F_i ^G+1＝0.

If it is notOrThenor

(4) Repeating the steps 2 to 3 until the evolution algebra exceeds the maximum evolution algebra G_m。

Simulation test

In order to illustrate the performance of the DE-CPASA algorithm (differential evolution algorithm with control parameter adaptation and strategy adaptation), the performance of the algorithm is verified by testing 13 standard test functions with 100 dimensions, and the optimized result is tracked by the known algorithm JADE^[6]And a traditional differential evolution algorithm DE/rand/1 for comparison.

In order to embody the fairness of comparison, NP is 400 in the DE-CPASA algorithm, but the DE-CPASA algorithm adopts different parameter settings at the setting of maximum iteration algebra, see table 1. Wherein, the control parameter setting of the traditional differential evolution algorithm adopts Price and storm^[7]Recommended parameters are as follows: f-0.5 and CR-0.9. For each 100-dimensional standard test function, the DE-CPASA algorithm, the JADE algorithm and the traditional differential evolution algorithm are independently operated for 50 times respectively, and then the average value and the standard deviation of each test function are respectively obtained. As can be seen from table 2, for the two tests of Penalized1 and Penalized2, the optimization effect of DE-CPASA is slightly different from that of the JADE algorithm than that of the optimization result of JADE, especially the test function of Penalized2, but is better than that of the traditional differential evolution algorithm; for the Rosenbrock test function, it can be found that when the iteration algebra is 6000, the optimization result of the DE-CPASA is not good as JADE, and when the number of generations reaches 20000, the optimization result far exceeds the optimization result of JADE, and from the result of JADE optimization, the algorithm has premature convergence, but the DE-CPASA algorithm does not have the condition, and meanwhile, the optimization result of the DE-CPASA is better than that of the traditional differential evolution algorithm; comparing the Step function, the optimization result of the DE-CPASA is better than that of JADE and the traditional differential evolution algorithm, except that the difference is not very large when the algebra of JADE with archive is 1500; for theFor other residual test functions, the optimization result of the DE-CPASA algorithm is better than that reported in the literature, and the maximum iteration algebra is obviously much smaller than that reported in the literature, so that the optimization performance of the DE-CPASA for high-dimensional test functions generally exceeds that of JADE and the traditional differential evolution algorithm, and the DE-CPASA has excellent performance in high-dimensional complex test functions.

TABLE 1 Standard test function and parameter settings

Table 1.benchmark functions and parameters setting

Table 2: optimization results comparison of DE-CPASA with literature

Table 2 comprise of the DE-CPASA and the reports for optimizationresults

Application of DE-CPASA algorithm in Hg oxidation kinetic parameter estimation

The mercury directly causes huge damage and serious pollution to the ecological environment, and therefore, the mercury is very important for capturing the mercury element in the flue gas. Studies have shown that mercury is oxidized to Hg²⁺Thereafter, the capture of elemental mercury is a very effective method, for which Agarwal^[8]The mercury oxidation mechanism has been studied intensively, and at the same time, the following kinetic models of the mercury oxidation process are proposed:

wherein, [ Hg ] is]: concentration of elemental mercury [ Cl ]₂]: concentration of chlorine gas

[HgCl₂]: concentration of mercuric chloride [ H ]₂O]: concentration of water

[HCl]: concentration of hydrochloric acid [ O ]₂]: concentration of oxygen

[SO₂]: concentration of sulfur dioxide [ NO]: concentration of nitric oxide

r_i: reaction rate, i ═ 1,2,3,4, 5A_i: antecedent factor, i ═ 1,2,3,4,5

E_i: activation energy, i ═ 1,2,3,4, 5R: gas constant

T: absolute temperature

In the formula, A_iAnd E_i(i ═ 1,2,3,4,5) are ten unknown parameters, and the objective of optimizing these ten unknown parameters is to minimize the difference between the conversion of mercury oxide calculated from equation 14 and the measured value.Then, the objective function is expressed as:

wherein,expressing the Hg conversion by kinetic equation fitting; c. C_iIndicating the experimentally measured Hg conversion; EQS represents the sum of the squares of errors of Hg conversion fit; m represents the number of experimental points.

Parameter setting of DE-CPASA and document [ 5]]Same, i.e. NP is 50, G_m1000, and the data used in the experiments are from document [9 ]]. As can be seen from Table 3, the optimization results obtained by the DE-CPASA algorithm are better than those obtained by Agarwal et al and Huchun Ping et al, which fully shows that the DE-CPASA algorithm can also obtain good optimization effects in practical application.

Table 3: comparison of optimization results of DE-CPASA with those reported in the literature

Table 3 comprise of the DE-CPASA and reported data(Agarwal etal.2007b，Chunping Hu et al.2009)for the results of optimization

	A1	A2	A3	A4	A5	E1	E2	E3	E4	E5	EQS
												Agarwaletal.	62.271	0.37688	98.682	138.85	36.113	8.8668	0.089337	23.509	17.638	15.108	88.134
ISDE	3.6024	0.44288	77.06	8.3261	10.866	3.9994	0.27544	17.442	13.134	13.233	73.746
												DE-CPASA	1.6601	0.46399	60.29	13.8569	95.1584	3.0001	0.34532	17.0017	14.0217	17.8262	70.147

In the algorithm implementation, the control parameters are involved in the dynamic evolution of the algorithm evolution process, and the variation strategy is also selected autonomously along with the evolution of the population, so that the DE-CPASA algorithm can automatically adjust the control parameters and the variation strategy to achieve a better optimization effect under various complex optimization conditions, and the influence of fixed settings in the traditional differential evolution algorithm on different optimization problems is reduced. In the test of 13 100-dimensional test functions, the DE-CPASA algorithm has stronger searching capability, can better avoid the condition of early maturity in the presence of a complex optimization problem, and simultaneously embodies the better self-adaptive capability of the DE-CPASA algorithm. Finally, the DE-CPASA algorithm is applied to Hg oxidation kinetic parameter estimation, and the algorithm obtains a better optimization result from the optimization result of an experiment, which shows that the algorithm also has strong competitiveness and better optimization capability in practical application.

The foregoing shows and describes the general principles and features of the present invention, together with the advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (1)

1. A differential evolution algorithm for controlling parameter adaptation and strategy adaptation is characterized in that: for optimization problems

$\begin{array}{ccc}\underset{x}{min}f(x_1,x_2,...,x_D)& x_j\&Element;(x_j^L,x_j^U)& j=1,2,...,D\end{array}---\left(1\right)$

Where f (x) is an optimized objective function, x is a D-dimensional optimization vector,andare respectively the jth variable x_jLower and upper limits of (d); the method specifically comprises the following steps:

(1) initialization: generating original population in respective feasible domainsAnd control parameter populationSetting the maximum number of iterations G_mAnd NP of the population size, and simultaneously, N_choice＝0，F_strategy1＝0，F_strategy2＝0；

(2) Original population S₁The evolution of (2): each original individualBy respective use ofAs control parameters, realizing differential evolution operation and generating new individualsi＝1,2...,NP；

Selection of NP/2+ N_choiceIndividual mutation operation of the formula (2)

${\hat{x}}_{i_{strategy1}}^{G+1}=x_{r_1}^G+F_{i_{strategy1}}^G\&CenterDot;(x_{r_2}^G-x_{r_3}^G)---\left(2\right)$

Wherein i_strategy1＝1,2，...，NP/2+N_choice

Selecting the remaining NP/2-N_choiceCarrying out individual mutation operations of the formulas (3) and (4) respectively according to the principle that rand is larger than 0.4, if so, carrying out the mutation operation by using the formula (3), and if not, carrying out the mutation operation by using the formula (4);

${\hat{x}}_{i_{strategy2}}^{G+1}=x_{r_1}^G+F_{i_{strategy2}}^G\&CenterDot;(x_{r_2}^G-x_{r_3}^G)+F_{i_{strategy2}}^G\&CenterDot;(x_{r_4}^G-x_{r_5}^G)---\left(3\right)$

${\hat{x}}_{i_{strategy2}}^{G+1}=x_i^G+F_{i_{strategy2}}^G\&CenterDot;(x_{best}^G-x_i^G)+F_{i_{strategy2}}^G\&CenterDot;(x_{r_1}^G-x_{r_2}^G)---\left(4\right)$

wherein i_strategy2＝NP/2+N_choice+1，NP/2+N_choice+ 1., NP, and i_strategy1+i_strategy2＝NP；

Individual boundary processing: if it isOrThenRandomly selecting from the feasible region;

individual crossover operation:

whereinIs a new individualThe j gene of (3);

operation of the adaptation of the policy: to obtainWhile at the same time causingTo obtainWhile at the same time causingIf F_strategy1≤F_strategy2Then N_choice＝N_choice+1, otherwise N_choice＝N_choice-1; at the same time, to N_choiceIf N is processed_choice> NP/2, then N_choiceIf N is NP/2-1_choice< -NP/2, then N_choice＝-NP/2+1；

Selection of individuals:

(3) control parameter populationThe evolution of (2):

F_i ^G+1＝N(0.5,σ)， (7)

${\mathrm{CR}}_i^{G+1}=N(0.9,\mathrm{\&sigma;});---\left(8\right)$

wherein, sigma is 1.2-G/G_m；

Setting of control parameter boundaries: if F_i ^G+1> 1 or F_i ^G+1< 0, then F_i ^G+11 or F_i ^G+1If not equal to 0OrThen

(4) Repeating the steps 2 to 3 until the evolution algebra exceeds the maximum evolution algebra G_m。