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

Control parameter self adaptation and tactful adaptive differential evolution algorithm

Technical field

The present invention relates to control algolithm field, more particularly to a kind of control parameter self adaptation and tactful adaptive difference are entered Change algorithm.

Background technology

After differential evolution algorithm is proposed by Storn and Price, for other intelligent optimization algorithms, due to Which has better optimizing performance and convergence rate faster, therefore, obtain rapidly the concern and research of numerous scholars [2,3,4,5], and the aspect such as the control parameter (mutagenic factor F and crossover probability CR) to the algorithm, strategy carried out in a large number Research work, simultaneously, be also applied to various actual complicated optimum problems, achieve gratifying optimization effect Really.

Although differential evolution algorithm has gratifying effect of optimization, in the face of the optimization problem for becoming increasingly complex, Control parameter setting and Mutation Strategy which is fixed all are difficult in adapt to the needs of algorithm evolution, therefore, its algorithm errors for existing It is very important.

The content of the invention

The purpose of the present invention to be that and provide a kind of control parameter self adaptation to solve the above problems and strategy is adaptive The differential evolution algorithm answered.

The present invention is achieved through the following technical solutions above-mentioned purpose：

The present invention is for optimization problem

Wherein, f (x) is the object function of optimization, and x is D dimension optimization vectors,WithRespectively j-th variable x_jUnder Limit and the upper limit；Specifically include following steps：

(1) initialize：In respective feasible zone, initial population is generatedWith control parameter populationMaximum is set Iterationses G_mWith scale NP of population, meanwhile, make N_choice=0, F_strategy1=0, F_strategy2=0；

(2) initial population S₁Evolution：Each original individualityUsing respectiveAs control ginseng Number, realizes differential evolution operator, generates new individual

Select NP/2+N_choiceThe individual variation operation of individual progressive form (2)

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

Select remaining NP/2-N_choiceIt is individual according to the principle of rand ＞ 0.4 distinguish the individual of progressive form (3) and (4) Body mutation operation, if it is carries out mutation operation using formula (3), if it is not, then carrying out mutation operation using formula (4)；

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

Individual BORDER PROCESSING：IfOrSoRandomly select from feasible zone；

Individual intersection is operated：

WhereinIt is new individualJ-th gene；

The adaptive operation of strategy：Try to achieveMeansigma methodss, while making Try to achieveMeansigma methodss, while makingIf F_strategy1≤F_strategy2, then N_choice=N_choice+ 1, otherwise N_choice=N_choice-1；Meanwhile, to N_choiceBorder processed, if N_choice＞ NP/2, So N_choice=NP/2-1, if N_choice＜-NP/2, then N_choice=-NP/2+1；

Individual selection:

(3) control parameter populationEvolution：

F_i ^G+1=N (0.5, σ), (7)

Wherein, σ=1.2-G/G_m；

The setting on control parameter border：If F_i ^G+1＞ 1 or F_i ^G+1＜ 0, then F_i ^G+1=1 or F_i ^G+1=0.

IfOrSoor

(4) repeat the 2nd～the 3rd step, until evolutionary generation exceedes maximum evolutionary generation G_m。

The beneficial effects of the present invention is：

The present invention is a kind of control parameter self adaptation and tactful adaptive differential evolution algorithm, compared with prior art, The present invention shown by the emulation testings of 13 100 dimension standard test functions and in the application of actual optimization problem, the algorithm Dominance is searched with the stronger overall situation with rate of convergence faster, and can preferably avoid many traditional intelligent optimization algorithms from depositing " precocity " phenomenon.The self adaptation of control parameter is not only realized, and self adaptation is also realized in the selection to Mutation Strategy.Survey Examination simulation result shows that DE-CPASA algorithms have higher solving precision and convergence rate faster.Finally, by DE-CPASA Algorithm is applied to the estimation of Hg kinetic parameters, has obtained preferable optimum results.

Specific embodiment

The invention will be further described below：

For optimization problem

Wherein, f (x) is the object function of optimization, and x is D dimension optimization vectors,WithRespectively j-th variable x_jUnder Limit and the upper limit；

In DE-CPASA algorithms, the use of individual construction and strategy is shown in Table shown in 0, for individual construction, is adopted Be control parameter and the mode that encodes together of individuality, i.e. each individuality has a corresponding control parameter, so as to Algorithm is enough realized while Evolution of Population, control parameter realizes dynamic change in individual level, change traditional DE and calculate The mode that control parameter is fixed in method；For the use of strategy, algorithm makes N at the starting stage_choice=0, i.e. change in algorithm When generation starts, population is divided into the sub- population that two individual amounts are NP/2, then using DE/rand/1, DE/rand-to- The Mutation Strategies such as best/1 and DE/rand/2 are carrying out mutation operation to individuality, and they are evaluated, and finally utilize N_choiceChange come the self adaptation of implementation strategy, the optimization performance overall so as to improve algorithm.

0. initial population of tableControl parameter populationAnd Mutation Strategy

Fig.1.Primary populationthe population ofcontrol parametersand mutation strategy

Specifically include following steps：

(1) initialize：In respective feasible zone, initial population is generatedWith control parameter populationMaximum is set Iterationses G_mWith scale NP of population, meanwhile, make N_choice=0, F_strategy1=0, F_strategy2=0；

(2) initial population S₁Evolution：Each original individualityUsing respectiveAs control ginseng Number, realizes differential evolution operator, generates new individual

Select NP/2+N_choiceThe individual variation operation of individual progressive form (2)

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

Select remaining NP/2-N_choiceIt is individual according to the principle of rand ＞ 0.4 distinguish the individual of progressive form (3) and (4) Body mutation operation, if it is carries out mutation operation using formula (3), if it is not, then carrying out mutation operation using formula (4)；

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

Individual BORDER PROCESSING：IfOrSoRandomly select from feasible zone；

Individual intersection is operated：

WhereinIt is new individualJ-th gene；

The adaptive operation of strategy：Try to achieveMeansigma methodss, while making Try to achieveMeansigma methodss, while makingIf F_strategy1≤F_strategy2, then N_choice=N_choice+ 1, otherwise N_choice=N_choice-1；Meanwhile, to N_choiceBorder processed, if N_choice＞ NP/2, So N_choice=NP/2-1, if N_choice＜-NP/2, then N_choice=-NP/2+1；

Individual selection:

(3) control parameter populationEvolution：

F_i ^G+1=N (0.5, σ), (7)

Wherein, σ=1.2-G/G_m；

The setting on control parameter border：If F_i ^G+1＞ 1 or F_i ^G+1＜ 0, then F_i ^G+1=1 or F_i ^G+1=0.

IfOrSoor

(4) repeat the 2nd～the 3rd step, until evolutionary generation exceedes maximum evolutionary generation G_m。

Emulation testing

In order to illustrate DE-CPASA algorithms (control parameter self adaptation and tactful adaptive differential evolution algorithm) performance Quality, herein by testing come the performance of verification algorithm by the standard test functions to 13 100 dimensions, and optimized As a result with well-known algorithm JADE^[6]And a kind of conventional differential evolution algorithm DE/rand/1 is compared.

In order to embody the fairness for comparing, NP=400 in DE-CPASA algorithms, but setting in greatest iteration algebraically Put, DE-CPASA algorithms take different parameter settings, are shown in Table 1.Wherein, the control parameter of conventional differential evolution algorithm sets Put using Price and Storn^[7]The parameter of recommendation：F=0.5 and CR=0.9.For the standard test functions of each 100 dimension, DE-CPASA algorithms, JADE algorithms and conventional differential evolution algorithm all independently run 50 times, then try to achieve each survey respectively The meansigma methodss of trial function and standard variance.From Table 2, it can be seen that for the two surveys of Penalized1 and Penalized2 Examination, the effect of optimization of DE-CPASA are more not good enough than the optimum results of JADE, especially this test function of Penalized2, with JADE algorithms have some gaps, but are better than the optimum results of conventional differential evolution algorithm；And for Rosenbrock This test function, it is found that when iterative algebra is 6000, the optimum results of DE-CPASA do not have the good of JADE, and When algebraically reached for 20000 generation, the optimum results of the result of optimization considerably beyond JADE, and the result from JADE optimizations comes See, algorithm all occurs in that too early convergence, and DE-CPASA algorithms do not occur such case, meanwhile, DE-CPASA's is excellent Change result and will be better than conventional differential evolution algorithm；Again comparing this function of Step, the optimum results of DE-CPASA will It is better than JADE and conventional differential evolution algorithm, except JADE with archive are when algebraically is 1500, but gap is simultaneously It is not very big；For other remaining test functions, the optimum results of DE-CPASA algorithms are not only better than document report Optimum results, and the iterative algebra of maximum is substantially less than document more, it can thus be seen that for higher-dimension test function For, DE-CPASA's searches dominance and can be above JADE and traditional differential evolution algorithm on the whole, it is shown that DE- Outstanding performances of the CPASA in the complicated test function of higher-dimension.

1 standard test functions of table and parameter setting

Table 1.benchmark functions and parameters setting

Table 2：DE-CPASA is compared with the optimum results of document

Table 2 comprise of the DE-CPASA and the reports for optimization results

Application of the DE-CPASA algorithms in the estimation of Hg kinetic parameters

Hydrargyrum directly can cause huge destruction and serious pollution to ecological environment, therefore, for mercury in flue gas element Capture is most important.Research shows, hydrargyrum is oxidized to Hg²⁺Afterwards, then to capture mercury element, it is an effectively method, is This, Agarwal^[8]Et al. in-depth study has been made to hydrargyrum oxidation mechanism, meanwhile, propose the dynamic of following hydrargyrum oxidation course Mechanical model：

Wherein, [Hg]：Concentration [the Cl of element mercury₂]：The concentration of chlorine

[HgCl₂]：Concentration [the H of mercuric chloride₂O]：The concentration of water

[HCl]：Concentration [the O of hydrochloric acid₂]：The concentration of oxygen

[SO₂]：The concentration [NO] of sulfur dioxide：Nitric oxide production concentration

r_i：Reaction rate, i=1,2,3,4,5 A_i：The preceding paragraph factor, i=1,2,3,4,5

E_i：Activation energy, i=1,2,3,4,5 R：Gas constant

T：Absolute temperature

In formula, A_iAnd E_i(i=1,2,3,4,5) are ten unknown parameters, and the target for optimizing this ten unknown parameters is to make It is minimum by the conversion ratio of 14 calculated mercury oxide of formula and the difference of measured value.So, object function is expressed as：

Wherein,Represent the Hg conversion ratios obtained by kinetics equation fitting；c_iRepresent the Hg conversions that experiment measurement is obtained Rate；EQS represents the error sum of squares of Hg conversion ratios fitting；M represents experimental point number.

The parameter setting of DE-CPASA is identical with document [5], i.e. NP=50, G_m=1000, meanwhile, data used by experiment are equal From document [9].From table 3 it is observed that the optimum results obtained by DE-CPASA algorithms are better than Agarwal et al. and Hu Optimum results obtained by spring equality people, this also fully indicates DE-CPASA algorithms in actual application, can also obtain very Good effect of optimization.

Table 3：DE-CPASA is compared with the optimum results of document report

Table 3 comprise of the DE-CPASA and reported data(Agarwal et Al.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

Control parameter self adaptation and tactful adaptive differential evolution algorithm, on algorithm is realized, not only make control parameter Dynamic evolution during participation algorithm evolution, and its Mutation Strategy also carries out autonomous choosing in the evolution with population Select, so that DE-CPASA algorithms can automatically adjust its control parameter and change under various complicated optimization situations Different strategy reduces fixed setting in conventional differential evolution algorithm to Different Optimization problem institute reaching more preferable effect of optimization The impact for bringing.In the test of the test function to 13 100 dimensions, it can be seen that DE-CPASA algorithms have stronger search Ability, and in face of complicated optimization problem, preferably can avoid " precocity " and situation, meanwhile, embody DE-CPASA The preferable adaptive ability of algorithm.Finally, DE-CPASA algorithms are applied to into the estimation of Hg kinetic parameters, from the excellent of experiment From the point of view of changing result, this algorithm has obtained more preferable optimum results, and this explanation, this algorithm similarly have in actual application Very strong competitiveness and preferable optimizing ability.

The ultimate principle and principal character and advantages of the present invention of the present invention has been shown and described above.The technology of the industry Personnel it should be appreciated that the present invention is not restricted to the described embodiments, the simply explanation described in above-described embodiment and description this The principle of invention, without departing from the spirit and scope of the present invention, the present invention also has various changes and modifications, these changes Change and improvement is both fallen within scope of the claimed invention.The claimed scope of the invention by appending claims and its Equivalent thereof.

Claims (1)

1. a kind of control parameter self adaptation and tactful adaptive differential evolution algorithm, it is characterised in that：For optimization problem

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

Wherein, f (x) is the object function of optimization, and x is D dimension optimization vectors,WithRespectively j-th variable x_jLower limit and upper Limit；Specifically include following steps：

(1) initialize：In respective feasible zone, initial population is generatedWith control parameter populationMaximum iteration is set Number of times G_mWith scale NP of population, meanwhile, make N_choice=0, F_strategy1=0, F_strategy2=0；

(2) initial population S₁Evolution：Each original individualityUsing respectiveIt is as control parameter, real Existing differential evolution operator, generates new individualI=1,2..., NP；

Select NP/2+N_choiceThe individual variation operation of individual progressive form (2)

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

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

Select remaining NP/2-N_choiceIt is individual to become come the individuality for distinguishing progressive form (3) and (4) according to the principle of rand ＞ 0.4 ETTHER-OR operation, if it is carries out mutation operation using formula (3), if it is not, then carrying out mutation operation using formula (4)；

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

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

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

Individual BORDER PROCESSING：IfOrSoRandomly select from feasible zone；

Individual intersection is operated：

WhereinIt is new individualJ-th gene；

The adaptive operation of strategy：Try to achieveMeansigma methodss, while makingTry to achieveMeansigma methodss, while makingIf F_strategy1≤F_strategy2, then N_choice= N_choice+ 1, otherwise N_choice=N_choice-1；Meanwhile, to N_choiceBorder processed, if N_choice＞ NP/2, then N_choice=NP/2-1, if N_choice＜-NP/2, then N_choice=-NP/2+1；

Individual selection:

(3) control parameter populationEvolution：

F_i ^G+1=N (0.5, σ), (7)

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

Wherein, σ=1.2-G/G_m；

The setting on control parameter border：If F_i ^G+1＞ 1 or F_i ^G+1＜ 0, then F_i ^G+1=1 or F_i ^G+1If=0.OrSo

(4) repeat the 2nd～the 3rd step, until evolutionary generation exceedes maximum evolutionary generation G_m。