CN106502963A - Nonlinear function method for solving based on Step-varied back propagation chaos wolf pack optimizing algorithm - Google Patents

Abstract

The present invention discloses a kind of nonlinear function method for solving based on Step-varied back propagation chaos wolf pack optimizing algorithm, and its content includes：Initialization wolf pack, using Chaos Variable as wolf pack initialized location；The optimum election contest wolf of chosen position is used as leader wolf；Position long-range raid movement of other artificial wolves towards leader wolf；Leader wolf searches prey, and other wolves execute jointly attack behavior centered on leader wolf；Colony's renewal is carried out according to " survival of the fittest " principle；Iteration said process, until reaching maximum iteration time, exports optimal solution.The present invention is to being improved based on the wolf pack searching algorithm (LWPS) of leader's strategy, and use it for solving complex nonlinear function Solve problems, it compensate for the deficiency of LWPS algorithms, preferable superiority is shown in terms of low optimization accuracy and convergence rate, there is preferable optimizing ability, the optimal solution of complex nonlinear function can more accurate, be quickly searched out, the theoretical method of nonlinear function solution is enriched.

Description

Nonlinear function method for solving based on Step-varied back propagation chaos wolf pack optimizing algorithm

Technical field

The present invention relates to complex nonlinear function solves field, more particularly to a kind of based on Step-varied back propagation chaos wolf pack The nonlinear function method for solving of optimizing algorithm.

Background technology

Through very long natural selection and biological evolution, many marvellous swarm intelligence phenomenons of nature are created, is made us Also endless the scientific revelation is brought to us while acclaiming as the acme of perfection.For solving complexity nonlinear function problem, Ren Menti Many bionical swarm intelligence optimizing algorithms, such as genetic algorithm (genetic algorithm, GA), particle cluster algorithm are gone out (particle swarm optimization, PSO), artificial bee colony algorithm (artificial bee colony, ABC) etc.. Compared with traditional optimization method, Swarm Intelligence Algorithm realize simple, can not be restricted by search space and object function form, This provides all more options for the solution of large amount of complex nonlinear function.However, going deep into research, researchers gradually send out These swarm intelligence optimizing algorithms existing all have some shortcomings to some extent, and such as algorithm computational accuracy is high, later stage convergence Speed is relatively slow or is easily absorbed in local optimum etc..The problem of these presence also excites researchers progressively to explore new swarm intelligence Method, solves complex nonlinear function Solve problems for the mankind and provides many new thinkings.

Yang et al. in 2007 bionical wolf pack predation in document (Yang Chenguang, Tu Xuyan, Chen Jie.Algorithm of marriage in honey bees optimization based on the wolf pack search[C]//Proceedings of IEEE Computer Society International Conference on Intelligent Pervasive Computing,Jeju Island,2007:462-467.) in propose a kind of swarm intelligence Optimized algorithm wolf pack searching algorithm (wolf pack search algorithm, WPS), and by its successfully with ABC algorithms Fusion, is applied in robot path planning's optimization.Subsequently WPS algorithms workflow optimization (Ye Y, Yin J, Feng Z, et al.Wolf-pack algorithm for business process model syntactic and semantic structure verification in the workflow management environment[C]//IEEE Asia- Pacific Services Computing Conference.IEEE Computer Society,2010:694-699.), electricity Pond charging Strengthening Management system optimization (Hung T C, Huang S J, Pai F S, et al.Design of lithium-ion battery charging system enhanced with wolf pack algorithm[C]//International Conference on Innovations in Bio-Inspired Computing&Applications.IEEE,2012: 195-200.), pid parameter optimization (Wu H S, Zhang F M.A uncultivated wolf pack algorithm for high-dimensional functions and its application in parameters optimization of PID controller[C]//IEEE Congress on Evolutionary Computation,2014:1477-1482.) Etc. aspect obtained applied research.Although WPS algorithms have good convergence rate, it is not high, easy to there is solving precision in which It is absorbed in local optimum and search is crossed the border and reduces the deficiency of convergence rate.For make up this defect, Zhou Qiang et al. in 2013 with Based on WPS algorithms, leader's strategy is introduced,《Computer utility is studied》The 2629-2632 page of periodical of the 9th phase of volume 30 in 2013 A kind of wolf pack search based on leader's strategy is proposed in " a kind of wolf pack searching algorithm based on the leader's strategy " document for carrying Algorithm (leader wolf pack search algorithm, referred to as LWPS algorithms), and table is studied by numerical experiment Bright, LWPS algorithms have larger in terms of convergence rate and solving precision than WPS algorithm, ABC algorithms, PSO algorithms and GA algorithms Raising.

But, find that LWPS algorithms still have the following disadvantages in further investigation：(1) when leader wolf is competed, each competing Select wolf being continuously increased with searching times, current optimal solution also more levels off to locally optimal solution, now the search step of election contest wolf Length also should be finer, to ensure the search precision of algorithm locally optimal solution, and the step-size in search in LWPS algorithms be one often Value, which does not possess adaptive adjustment capability, easily affects the search precision of locally optimal solution；(2) besiege behavior and require artificial wolf tool There is stronger local optimal searching ability, this requires that besieging step-length can be adaptively adjusted with the increase of iterationses, and Jointly attack step-length in LWPS algorithms levels off to 0 in the algorithm iteration later stage, causes optimizing unchanged, showing for Premature Convergence easily occurs As.Therefore, the performance of LWPS algorithms needs further to improve.

Content of the invention

For above-mentioned existing problems, the present invention is improved to LWPS algorithms, and uses it for solving complex nonlinear letter Number Solve problems, propose a kind of based on Step-varied back propagation chaos wolf pack optimizing algorithm (chaos wolf optimization Algorithm with adaptive variable step, CWOA) nonlinear function method for solving, it is intended to make up LWPS The deficiency of algorithm, enriches the theoretical method that nonlinear function is solved.

In order to solve above-mentioned technical problem, the present invention is realized by following technical proposals：

A kind of nonlinear function method for solving based on Step-varied back propagation chaos wolf pack optimizing algorithm (CWOA), which is concrete Implementation steps include following content：

Step one：Initialization wolf pack

The number N of artificial wolf, search space dimension D, search space span [w in initialization wolf pack_dmax,w_dmin], most Big iterationses n_max, campaign for the number q, maximum search number of times H of head wolf_max, direction of search h, step-size in search initial value stepa₀, moving step length stepb, jointly attack threshold value r₀, besiege step-length initial value stepc₀, eliminate number m of wolf；Utilize Logistic chaotic maps expression formulas (1) produces N number of Chaos Variable, and Chaos Variable is projected to optimizing change by expression formula (2) The interval of amount, used as the initialized location of wolf pack；

Chaos_n+1=μ × Chaos_n(1-Chaos_n) μ=4, Chaos_n∈[0,1] (1)

In formula, and Chaos (0,1) it is equally distributed Chaos Variable in interval [0,1]；

Step 2：Competition leader wolf

The preferably artificial wolf of q fitness value is chosen as campaigner, its h direction around oneself is allowed by expression formula (3) Step-varied back propagation search is constantly carried out, if the position p that election contest wolf searches_kjdIt is better than current location w_jd, then enter line position Movement is put, is not otherwise moved；When election contest wolf searching times H reaches maximum search number of times H_max, then terminate search behavior, choose position The election contest wolf of optimum is put as leader wolf；

Jth (j=1,2 ..., q) election contest wolf kth (k=1,2 ..., h) the searching position p that individual direction produces around which_kjd For：

p_kjd=w_jd+Chaos(-1,1)×α×stepa₀(3)

Chaos (- 1,1)=- 1+2 × Chaos (0,1) (4)

In formula, w_jdCurrent location tie up in d of jth election contest wolf, Chaos (- 1,1) be in interval [- 1,1] uniformly The Chaos Variable of distribution, stepa₀For step-size in search initial value, α is the step-size in search Automatic adjusument factor, 0<α<1, α adopts table Adaptive should determine that is carried out up to formula (5)：

Step 3：Leader wolf calls long-range raid

Position long-range raid movement of other artificial wolves towards leader wolf, and prey is continued search for during long-range raid simultaneously Location updating is carried out by expression formula (6)；If the position g after artificial wolf renewal_idIt is better than current location w_id, then position movement is carried out, Otherwise do not move；

Position g after i-th artificial wolf renewal_idFor：

g_id=w_id+Chaos(-1,1)×stepb×(w_ld-w_id) (6)

In formula, w_idIt is current location that i-th wolf is tieed up in d, stepb is moving step length, w_ldIt is leader wolf in d The position of dimension；

Step 4：Surround prey

Leader wolf searches prey, and by yelping, calling companion surrounds prey, during other artificial wolves with leader wolf are The heart launch surround, when meet default jointly attack threshold value r₀During condition, jointly attack behavior is executed, position is carried out more by expression formula (7) Newly；After artificial wolf is besieged to prey, only position movement is just carried out when the position after renewal is better than origin-location, otherwise Holding position is constant；Finally, according to expression formula (9) to renewal after position carry out process of crossing the border；

In formula,For the current location that i-th wolf of the n-th generation is tieed up in d,Tie up in d for i-th wolf of the (n+1)th generation Current location, and rand (0, it is 1) random number that produces in interval [0,1], r₀For default jointly attack threshold value, stepc is to besiege Step-length, its value adaptively reduce with the increase of iterationses, and expression formula is formula (8)：

In formula, stepc₀For besieging step-length initial value；N is current iteration number of times；

After wolf pack besieges prey, position can change, if not in search space, needing according to expression formula (9) to updating Position afterwards carries out process of crossing the border：

Step 5：Distribution food updates wolf pack

Principle according to " survival of the fittest " carries out colony's renewal, removes worst m artificial wolf in wolf pack, while passing through Logistic chaotic maps expression formulas (1) produces m Chaos Variable, and Chaos Variable is projected to optimizing change by expression formula (2) The interval of amount, substitutes m worst artificial wolf, to keep the multiformity of population；

Step 6：End condition judges

Judge whether to reach maximum iteration time, circulation is exited if condition is met, export optimal solution；Otherwise, step is gone to Rapid two.

Due to adopting above-mentioned technical proposal, one kind that the present invention is provided to be based on Step-varied back propagation chaos wolf pack optimizing algorithm (CWOA) nonlinear function method for solving has such beneficial effect：

(1) in order to improve the search precision of artificial wolf, the present invention is improved to step-size in search；Improve in formula and add Automatic adjusument factor-alpha, makes algorithm that there is near excellent solution domain the ability for opening up new solution space, improves the essence of algorithm Fine searching ability；

(2) local optimal searching in order to better adapt to artificial wolf is required, the present invention is improved to besieging stepsize formula； Improve add in formula random regulatory factor rand (0,1), it is to avoid jointly attack step-length is easy due to leveling off to 0 in the iteration later stage There is the phenomenon of Premature Convergence, the ability for making jointly attack behavior possess whole Automatic adjusument；

(3) present invention is optimized search using Chaos Variable, improves Searching efficiency for random search.

Description of the drawings

Fig. 1 is the flow chart of the inventive method；

Fig. 2 is the convergence curve comparison diagram of the embodiment of the present invention；Wherein a) be Matyas convergence curve, b) be Easom Convergence curve, c) be Sumsquares convergence curve, d) be Sphere convergence curve, e) be Eggcrate convergence bent Line, f) be Six Hump Camel Back convergence curve, g) be Bohachevsky3 convergence curve, h) be Bridge Convergence curve, i) be Booth convergence curve, j) be Bohachevsky1 convergence curve, k) be Ackley convergence curve, L) be Quadric convergence curve.

Specific embodiment

Below in conjunction with the accompanying drawings and embodiment the present invention will be described in further detail.

Referring to Fig. 1, it is the flow chart of the inventive method, a kind of based on Step-varied back propagation chaos wolf pack optimizing algorithm (CWOA) nonlinear function method for solving, its specific implementation step include following content：

Step one：Initialization wolf pack

The number N of artificial wolf, search space dimension D, search space span [w in initialization wolf pack_dmax,w_dmin], most Big iterationses n_max, campaign for the number q, maximum search number of times H of head wolf_max, direction of search h, step-size in search initial value stepa₀, moving step length stepb, jointly attack threshold value r₀, besiege step-length initial value stepc₀, eliminate number m of wolf；Utilize Logistic chaotic maps expression formulas (1) produces N number of Chaos Variable, and Chaos Variable is projected to optimizing change by expression formula (2) The interval of amount, used as the initialized location of wolf pack；

Chaos_n+1=μ × Chaos_n(1-Chaos_n) μ=4, Chaos_n∈[0,1] (1)

In formula, and Chaos (0,1) it is equally distributed Chaos Variable in interval [0,1]；

Step 2：Competition leader wolf

The preferably artificial wolf of q fitness value is chosen as campaigner, its h direction around oneself is allowed by expression formula (3) Step-varied back propagation search is constantly carried out, if the position p that election contest wolf searches_kjdIt is better than current location w_jd, then enter line position Movement is put, is not otherwise moved；When election contest wolf searching times H reaches maximum search number of times H_max, then terminate search behavior, choose position The election contest wolf of optimum is put as leader wolf；

Jth (j=1,2 ..., q) election contest wolf kth (k=1,2 ..., h) the searching position p that individual direction produces around which_kjd For：

p_kjd=w_jd+Chaos(-1,1)×α×stepa₀(3)

Chaos (- 1,1)=- 1+2 × Chaos (0,1) (4)

In formula, w_jdCurrent location tie up in d of jth election contest wolf, Chaos (- 1,1) be in interval [- 1,1] uniformly The Chaos Variable of distribution, stepa₀For step-size in search initial value, α is the step-size in search Automatic adjusument factor, 0<α<1, α adopts table Adaptive should determine that is carried out up to formula (5)：

Step 3：Leader wolf calls long-range raid

Position long-range raid movement of other artificial wolves towards leader wolf, and prey is continued search for during long-range raid simultaneously Location updating is carried out by expression formula (6)；If the position g after artificial wolf renewal_idIt is better than current location w_id, then position movement is carried out, Otherwise do not move；

Position g after i-th artificial wolf renewal_idFor：

g_id=w_id+Chaos(-1,1)×stepb×(w_ld-w_id) (6)

In formula, w_idIt is current location that i-th wolf is tieed up in d, stepb is moving step length, w_ldIt is leader wolf in d The position of dimension；

Step 4：Surround prey

Leader wolf searches prey, and by yelping, calling companion surrounds prey, during other artificial wolves with leader wolf are The heart launch surround, when meet default jointly attack threshold value r₀During condition, jointly attack behavior is executed, position is carried out more by expression formula (7) Newly；After artificial wolf is besieged to prey, only position movement is just carried out when the position after renewal is better than origin-location, otherwise Holding position is constant；Finally, according to expression formula (9) to renewal after position carry out process of crossing the border；

In formula,For the current location that i-th wolf of the n-th generation is tieed up in d,Tie up in d for i-th wolf of the (n+1)th generation Current location, and rand (0, it is 1) random number that produces in interval [0,1], r₀For default jointly attack threshold value, stepc is to besiege Step-length, its value adaptively reduce with the increase of iterationses, shown in expression formula such as formula (8)：

In formula, stepc₀For besieging step-length initial value；N is current iteration number of times；

After wolf pack besieges prey, position can change, if not in search space, needing according to expression formula (9) to updating Position afterwards carries out process of crossing the border：

Step 5：Distribution food updates wolf pack

Principle according to " survival of the fittest " carries out colony's renewal, removes worst m artificial wolf in wolf pack, while passing through Logistic chaotic maps expression formulas (1) produces m Chaos Variable, and Chaos Variable is projected to optimizing change by expression formula (2) The interval of amount, substitutes m worst artificial wolf, to keep the multiformity of population；

Step 6：End condition judges

Judge whether to reach maximum iteration time, circulation is exited if condition is met, export optimal solution；Otherwise, step is gone to Rapid two.

In order to verify the effectiveness of the inventive method, choosing the 12 standard nonlinear test functions that commonly uses in the world is carried out Test, and will solve effect with based on leader strategy wolf pack searching algorithm (LWPS) and classics genetic algorithm (GA), Particle cluster algorithm (PSO) is compared.Standard test functions are as shown in table 1.

1 standard test functions of table

In table 1 selected function be related to unimodal, multimodal, can divide, can not grade manifold complex nonlinear function.Unimodal Function only has global optimum in domain of definition, without local extremum；And Solving Multimodal Function has multiple local extremums in domain of definition, More complicated with respect to for unimodal function.General optimizing algorithm is easily ensnared into local most when optimizing is carried out to Solving Multimodal Function Excellent, it is more difficult to find the global optimum of Solving Multimodal Function, so Solving Multimodal Function is often used to the global search for checking optimizing algorithm The ability of Premature Convergence can be avoided and.In addition, this function is for can if a compound function can be represented with one-variable function sum Point, otherwise it is inseparable function.Due to inseparable function relationship between variables complicated, therefore to this class function optimizing relatively more Difficult.Additionally, the dimension of search space is also a key factor, much for the good algorithm of low-dimensional function effect is for higher-dimension The optimizing effect of complicated function is poor.In table 1, the dimension of function is tieed up to 30 dimensions from 2, is all larger complicated non-of difficulty Linear function optimization problem, with good testability, more can comprehensively react the performance of optimizing algorithm.

The inventive method realizes with M Programming with Pascal Language that based on Matlab R2013b GA algorithms adopt Univ Sheffield UK The algorithmic tool case that (University of Sheffield) is developed, PSO algorithms adopt North Carolina, USA university The workbox of professor's Brian exploitation of (North Carolina A＆T State University), LWPS algorithms are according to Zhou Qiang Deng《Computer utility is studied》A kind of " wolf based on leader's strategy of the 2629-2632 page of publication of the 9th phase of volume 30 in 2013 The program thread provided in group hunting algorithm " document is realized.The initialization scale of each optimizing algorithm is all set to N=50, maximum Iterationses are set to n_max=600, other parameters follow Sriniva M etc. respectively and exist《Computer》1994 volume 27 the 6th " the Genetic algorithms of the 17-26 page publication of phase:A survey " documents, Kennedy J etc. exist《IEEE International Conference on Neural Networks》The 1942-1948 page of volume 4 publication of nineteen ninety-five " Particle swarm optimization " document and Zhou Qiang etc. exists《Computer utility is studied》The 9th phase of volume 30 in 2013 A kind of basic principle of " wolf pack searching algorithm based on leader's strategy " document of 2629-2632 page of publication, is arranged such as 2 institute of table Show.

2 algorithm parameter table of table

In order to intuitively compare low optimization accuracy and the convergence rate of each optimized algorithm, respectively each test function is entered at random Row optimizing, respectively obtains its convergence curve, as shown in Fig. 2 wherein a) be Matyas convergence curve, b) be Easom Convergence curve, c) be Sumsquares convergence curve, d) be Sphere convergence curve, e) be Eggcrate convergence bent Line, f) be Six Hump Camel Back convergence curve, g) be Bohachevsky3 convergence curve, h) be Bridge Convergence curve, i) be Booth convergence curve, j) be Bohachevsky1 convergence curve, k) be Ackley convergence curve, L) be Quadric convergence curve.As seen from Figure 2, compared with GA, PSO and LWPS algorithm, CWOA proposed by the present invention is calculated Method has relatively higher low optimization accuracy；And target function value can be converged to faster, with convergence rate faster, Preferable superiority is shown in terms of low optimization accuracy and convergence rate, with preferable optimizing ability, can be more accurate, quickly Search out the optimal solution of complex nonlinear function.

Claims (1)

1. a kind of nonlinear function method for solving based on Step-varied back propagation chaos wolf pack optimizing algorithm, it is characterised in that：Should Method specific implementation step includes following content：

Step one：Initialization wolf pack

The number N of artificial wolf, search space dimension D, search space span [w in initialization wolf pack_dmax,w_dmin], maximum changes For frequency n_max, campaign for the number q, maximum search number of times H of head wolf_max, direction of search h, step-size in search initial value stepa₀, move Dynamic step-length stepb, besieges threshold value r₀, besiege step-length initial value stepc₀, eliminate number m of wolf；Using Logistic chaotic maps The N number of Chaos Variable of expression formula (1) generation, and Chaos Variable is projected to by expression formula (2) interval of optimizing variable, as The initialized location of wolf pack；

Chaos_n+1=μ × Chaos_n(1-Chaos_n) μ=4, Chaos_n∈[0,1] (1)

$w_{id}^0=w_{dmin}+Chaos(0,1)\&times;(w_{dmax}-w_{dmin}),(i=1,2,...,N;d=1,2,...,D)---\left(2\right)$

In formula, and Chaos (0,1) it is equally distributed Chaos Variable in interval [0,1]；

Step 2：Competition leader wolf

The preferably artificial wolf of q fitness value is chosen as campaigner, its h direction around oneself is allowed by expression formula (3) Step-varied back propagation search is constantly carried out, if the position p that election contest wolf searches_kjdIt is better than current location w_jd, then carry out position shifting Dynamic, otherwise do not move；When election contest wolf searching times H reaches maximum search number of times H_max, then terminating search behavior, chosen position is most Excellent election contest wolf is used as leader wolf；

Jth (j=1,2 ..., q) election contest wolf kth (k=1,2 ..., h) the searching position p that individual direction produces around which_kjdFor：

p_kjd=w_jd+Chaos(-1,1)×α×stepa₀(3)

Chaos (- 1,1)=- 1+2 × Chaos (0,1) (4)

In formula, w_jdIt is current location tie up in d of jth election contest wolf, and Chaos (- 1, it is 1) to be uniformly distributed interval [- 1,1] is interior Chaos Variable, stepa₀For step-size in search initial value, α is the step-size in search Automatic adjusument factor, 0<α<1, α adopts expression formula (5) adaptive should determine that is carried out：

$\mathrm{\&alpha;}=1-{\left(\frac{H-1}{H_{max}}\right)}^2;---\left(5\right)$

Step 3：Leader wolf calls long-range raid

Position long-range raid movement of other artificial wolves towards leader wolf, and prey is continued search for during long-range raid and by table Location updating is carried out up to formula (6)；If the position g after artificial wolf renewal_idIt is better than current location w_id, then position movement is carried out, otherwise Do not move；

Position g after i-th artificial wolf renewal_idFor：

g_id=w_id+Chaos(-1,1)×stepb×(w_ld-w_id) (6)

In formula, w_idIt is current location that i-th wolf is tieed up in d, stepb is moving step length, w_ldFor the position that leader wolf is tieed up in d Put；

Step 4：Surround prey

Leader wolf searches prey, and by yelping, calling companion surrounds prey, and other artificial wolves are opened up centered on leader wolf Open encirclement, when meet default jointly attack threshold value r₀During condition, jointly attack behavior is executed, location updating is carried out by expression formula (7)；People After work wolf is besieged to prey, only just position movement is carried out when the position after renewal is better than origin-location, otherwise kept Position is constant；Finally, according to expression formula (9) to renewal after position carry out process of crossing the border；

$w_{id}^{n+1}=\left\{\begin{array}{cc}{w}_{id}^n& rand(0,1)\&le;r_0\\ w_{ld}+Chaos(-1,1)\&times;stepc& rand(0,1)>r_0\end{array}\right.---\left(7\right)$

In formula,For the current location that i-th wolf of the n-th generation is tieed up in d,For the present bit that i-th wolf of the (n+1)th generation is tieed up in d Put, and rand (0, it is 1) random number that produces in interval [0,1], r₀For default jointly attack threshold value, stepc is to besiege step-length, its Value adaptively reduces with the increase of iterationses, and expression formula is formula (8)：

$stepc={\mathrm{stepc}}_0\&times;rand(0,1)\&times;\&lsqb;1-{\left(\frac{n-1}{n}\right)}^2\&rsqb;---\left(8\right)$

In formula, stepc₀For besieging step-length initial value；N is current iteration number of times；

Wolf pack besiege prey after position can change, if not in search space, need according to expression formula (9) to renewal after Position carries out process of crossing the border：

$w_{id}^{n+1}=\left\{\begin{array}{cc}{w}_{d\max }& w_{id}^{n+1}>w_{dmax}\\ w_{d\min }& w_{id}^{n+1}<w_{d\min }\end{array}\right.;---\left(9\right)$

Step 5：Distribution food updates wolf pack

Principle according to " survival of the fittest " carries out colony's renewal, removes worst m artificial wolf in wolf pack, while passing through Logistic chaotic maps expression formulas (1) produces m Chaos Variable, and Chaos Variable is projected to optimizing change by expression formula (2) The interval of amount, substitutes m worst artificial wolf, to keep the multiformity of population；

Step 6：End condition judges

Judge whether to reach maximum iteration time, circulation is exited if condition is met, export optimal solution；Otherwise, step 2 is gone to.