CN103218657A - Population survival dynamics optimization method under environment pollution - Google Patents

Abstract

The invention provides a population survival dynamics optimization method under environment pollution, namely a PSDO-EP algorithm. A population survival dynamics theory under environment pollution is used, an environment system and the solution space of an optimization problem correspond to each other, pollution phenomena exist in the environment system, a plurality of populations live in the environment system, each population corresponds to a trial solution of the optimization problem, one feature of the population corresponds to a variable in the trial solution, the populations change all the time under the effect of environment pollution, strong populations which can resist environment pollution grow, weak populations stop growing, a population survival dynamics model under environment pollution is used for constructing evolution operators and achieving information interchange between environment and the populations and among the populations, during a population evolution process, the populations convert from one growing state to another growing state, searching on the optimal solution of the optimization problem of the populations is achieved, the PSDO-EP algorithm has the advantages of being strong in searching capacity, global convergence is achieved, and a solution is provided for the complex function optimization problem.

Description

The environmental pollution all living creatures that sows deposits the Dynamics Optimization method

Technical field

The present invention relates to intelligent optimization algorithm, be specifically related to a kind of environmental pollution all living creatures that sows and deposit Dynamics Optimization method-PSDO-EP algorithm.

Background technology

Consider the function optimization problem:

minf(X)

In the formula: R ⁿIt is n dimension Euclidean space; X=(x ₁, x ₂..., x _n) be a n dimension decision vector, variable x _i(i=1,2 ..., n) be nonnegative real number; S is non-negative search volume, claims solution space again; F (X) is an objective function; g _i(X) 〉=0 be i constraint condition, i=1,2 ..., I, I are inequality constrain condition number; h _i(X)=0 be i equality constraint, i=1,2 ..., E, E are the equality constraint number.Objective function f (X) and constraint condition g _i(X), h _i(X) do not need special restrictive condition.Therefore, traditional mathematical optimization method based on function continuity and the property led can't address this problem.

In order to solve this type of optimization problem, people have been developed and intelligent optimization algorithm, and this class algorithm does not generally need special restrictive condition to objective function and constraint condition, has than extensive applicability.Existing intelligent optimization algorithm has: (1) genetic algorithm: the proposition of this algorithm monograph by the Holland of Univ Chicago USA " Adaptation in Natural and Artificial Systems " in 1975, the technical scheme that is adopted is to utilize Heredity theory to construct individual evolvement method, thereby optimization problem is found the solution; (2) ant group algorithm: this algorithm by people such as Colorni A and Dorigo M at document " Distributed optimization by ant colonies, Proceedings of the1 ^StEurope Conference on Artificial Life, 1991,134-142 " the middle proposition, the technical scheme that is adopted is that simulation ant colony foraging behavior is optimized finding the solution of problem; (3) particle cluster algorithm: this algorithm by Eberhart R and Kennedy J at document " New optimizer using particle swarm theory, MHS ' 95Proceedings of the Sixth International Symposium on Micro Machine and Human Science, IEEE, Piscataway, NJ, USA, 1995:38-43 " the middle proposition, the technical scheme that is adopted is to utilize the group behavior of imitation birds to be optimized finding the solution of problem; (4) fish-swarm algorithm: this algorithm is " a kind of based on the communal optimizing chess of animal formula: fish-swarm algorithm at document by people such as Li Xiaolei, Shao Zhijiang River and Qian Jixin, the system engineering theory and practice, 2002,22 (11): 32-38 " propose in, the technical scheme that is adopted be utilize fish looking for food in water, knock into the back, behavior such as clustering searches for the optimization problem solution space, thereby obtain the globally optimal solution of optimization problem; (5) biogeography algorithm: this algorithm 2008 has proposed the biogeography optimized Algorithm by Dan Simon with the method for biogeography, document is " Simon D.Biogeography-based Optimization[J] .IEEE Transactions.Evolutionary Computation, 2008,12 (6): 702-713 ".This algorithm has been realized search to the optimization problem optimum solution by the migration of population between the habitat; (6) bat algorithm: this algorithm 2010 by Yang X S at document " A new metaheuristic bat-inspired algorithm, Nature Inspired Cooperative Strategies for Optimization (NICSO2010), Studies in Computational Intelligence284, Springer-Verlag, Berlin Eidelberg, 2010,65-74 " the middle proposition, a kind of new intelligent optimization algorithm that this algorithm proposes by simulation bat echolocation behavior, it also is a kind of random search optimizing algorithm based on population, the bat individuality is the elementary cell of bat algorithm, the motion of whole colony produces the evolutionary process from disorder to order in the problem solving space, thereby obtains optimum solution.

Environmental pollution is very attractive problem in the real society to the problem that influences of biotic population.In the time of the modern industry prosperity, a large amount of pollutants that are harmful to biotic population are discharged in the environment, and are remarkable day by day to the influence that biotic population brought.Pollution brings great influence present can for the growth of biotic population, and people have proposed multiple mathematical model and studied this problem.Although there is contamination phenomenon in the environmental system, live in wherein some biotic population and but can resist to such an extent that live to pollute the influence that brings and obtain survival, the other biotic population is but withered away gradually because of resisting the influence that pollution brings.Here it is the survival of the fittest, the reason of the survival of the fittest.The new algorithm that this paper proposes has utilized this thought just.

Utilize environmental pollution sow all living creatures deposit kinetic theory construct can the solved function optimization problem algorithm yet there are no report.

Summary of the invention

The object of the present invention is to provide a kind of environmental pollution all living creatures that sows to deposit Dynamics Optimization method-PSDO-EP algorithm, this algorithm has the characteristics of the strong and global convergence of search capability, be that finding the solution of complex function optimization problem, particularly higher-dimension optimization problem provides a solution.

In order to achieve the above object, the present invention adopts following technical scheme:

A kind of environmental pollution all living creatures that sows deposits Dynamics Optimization method-PSDO-EP algorithm, and it is characterized in that: establishing the function optimization problem that will solve is:

minf(X)

In the formula: R ⁿIt is n dimension Euclidean space; X=(x ₁, x ₂..., x _n) be a n dimension decision vector, variable x _i(i=1,2 ..., n) be nonnegative real number; S is non-negative search volume, claims solution space again; F (X) is an objective function; g _i(X) 〉=0 be i constraint condition, i=1,2 ..., I, I are inequality constrain condition number; h _i(X)=0 be i equality constraint, i=1,2 ..., E, E are the equality constraint number; Objective function f (X) and constraint condition g _i(X), h _i(X) do not need special restrictive condition;

The PSDO-EP algorithm adopts is that the environmental pollution all living creatures that sows deposits kinetic theory, environmental system is corresponding with the solution space of optimization problem (1), there is contamination phenomenon in this environmental system, several populations are wherein living, each population correspondence a trial solution of optimization problem, and the feature of population is corresponding to a variable in the trial solution; The kinetic model that is applicable to some feature of population also be applicable in the trial solution to dependent variable, population constantly changes under the environmental pollution effect, the strong person that can resist environmental pollution obtains growth, valetudinarian, invalid stops growing; Because population is being shared the resource in the environmental system, so there is the phenomenon that influences each other between the population; These interactions between environment and population and population and the population are used to construct the growth strategy of population, utilize the environmental pollution all living creatures that sows to deposit kinetic model and come the structural evolution operator to realize message exchange between environment and population, population and the population; In the population evolution process, the growth of population or maintain the original state constantly or develops to better direction, and population is transferred to another kind of growth conditions from a kind of growth conditions and realized the search of population to the optimization problem optimum solution;

In order to make the PCDO-HNS algorithm be applicable to various optimization problems, the objective function of optimization problem (1) is rewritten into following formula:

In the formula: F _MaxBe very large real number, be used for the trial solution that does not satisfy constraint condition is punished.

The environmental pollution all living creatures that sows deposits kinetic model

Consider that one exists the environmental system of pollution source, wherein life has N kind biotic population, and the environmental pollution meeting influences the growth of population, and the existence of population can influence the pollution level of environment conversely.For example, population is the pollutant of absorbing environmental not only, and the drainage of population will pollute the environment again conversely, and the mathematical model about the rule of the concentration c time to time change of the Konzentration of environmental contaminants and population internal contamination thing is accordingly

$\left\{\begin{array}{c}\frac{{\mathrm{dx}}_i}{\mathrm{dt}}=x_i(r_i-r_{i}c-f\left(x_i\right))\\ \frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{Ke}-\underset{i=1}{\overset{N}{\mathrm{\Σ}}}(g_i+m_i)c\\ \frac{\mathrm{de}}{\mathrm{dt}}=-K_{1}e\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{x}_i+g_{1}c\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{x}_i-\mathrm{he}+u\end{array}\right.,i=\mathrm{1,2},\·\·\·,N---\left(3\right)$

In the formula: t represents period; U is the intensity of pollution source to environmental emissions; K is the efficient of environmental pollution population, 0＜K＜1; G is the efficient that population drains pollutant to environment, 0＜g＜1; H is the part that pollutant eliminates by other approach in environment, 0＜h＜1; M is the efficient that population absorbs, purifies the pollutant of being eliminated, 0＜m＜1; x _iBe population P _iQuantity, and x _i〉=0; r _iBe population P _iRate of growth, 0＜r _i＜1; r _iC is population P _iThe pollution mortality ratio, the size of polluting mortality ratio is directly proportional 0 with the contaminated degree c of population＜c＜1; F (x _i) be population P _iThe natural death function.Here

With

Be respectively the unit interval environmental system and input to all populations and all populations and drain pollutant levels to environment, this amount not only with the total amounts of all populations

Relevant, and with these populations to absorb the ability of pollutant relevant.How this ability is described? the whole bag of tricks can be arranged, and the average quality of people these populations commonly used is weighed, if establish the average quality of M for each population, M〉0, K then ₁=KM, g ₁=gM.

Modular form (3) is a general mathematical model.In formula (3), get f (x _i)=a _ix _i, a _iBe population P _iNatural mortality rate, neglect the nonlinear terms in formula (3) the 3rd formula, and get g _i=g, m _i=m, i=1,2 ..., N can obtain more succinct model, promptly

$\left\{\begin{array}{c}\frac{{\mathrm{dx}}_i}{\mathrm{dt}}=x_i(r_i-r_{i}c-a_i{x}_i)\\ \frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{Ke}-N(g+m)c,\\ \frac{\mathrm{de}}{\mathrm{dt}}=-\mathrm{he}+u\end{array}\right.i=\mathrm{1,2},\·\·\·,N---\left(4\right)$

Formula (4) is exactly that the environmental pollution that will use of this paper all living creatures that sows deposits kinetic model.

The design of algorithm scene

Consider that one exists the environmental system of pollution source, wherein life has N kind biotic population, i.e. { P ₁, P ₂..., P _N.Pollution source are at random to the contaminant capacity (being called pollution intensity) of environmental system discharging, and promptly the different times pollution source are different to the contaminant capacity of environmental system discharging.The mode of polluting in the environmental system of clearing up has two kinds, has a kind ofly been sponged by environmental system self, and another kind has been sponged by the biotic population in the environmental system.The biotic population that survives in the environmental system is on the one hand also to the environmental system exhaust emission, and biotic population is caused negative effect because of the pollution in the absorbing environmental system to himself growth on the other hand.

Each biotic population P _iWith its character representation is exactly P _i=(p _I1, p _I2..., p _In), p wherein _IjBe exactly biotic population P _iJ feature, n is biotic population P _iThe total characteristic number.Pollution in the environmental system is to biotic population P _iMurder by poisoning just show in the influence to its feature that and this murder by poisoning is time dependent.That is to say, certain the time, polluting can be once to biotic population P _iSome feature impact simultaneously, but be not whole n feature; And in the time of other certain, pollute but once to biotic population P _iThe other feature impact whole n features certainly simultaneously.In a word, certain the time, polluting can be to biotic population P _iWhich feature to impact simultaneously be at random fully.Biotic population P _iAfter poisoning, its growth conditions has just changed, and this variation is at random fully.

Procedure correlation with top argumentation and solving-optimizing problem (1) globally optimal solution gets up below.

In the solution space S of optimization problem (1), select N trial solution, i.e. S={X at random ₁, X ₂..., X _N, X wherein _i=(x _I1, x _I2..., x _In).Obviously, solution space S is corresponding with environmental system, and N population is just corresponding one by one with N trial solution of optimization problem (1) in this environmental system, i.e. P _iWith X _iCorresponding one by one, i=1,2 ..., N.Further, population P _iProper vector and optimization problem trial solution X _iVector is corresponding fully, i.e. population P _iFeature p _IjWith trial solution X _iVariable x _IjCorresponding.

In summary, population and trial solution are no longer distinguished later in conceptive complete equivalence.Population P _iAfter being poisoned, its growth conditions can change, and the solution space S of optimization problem is hinted obliquely in this variation, just is equivalent to trial solution X _iTransfer to the another one locus from a locus.For the sake of simplicity, a locus is called a state, and represents with its subscript.

Suppose population P _iCurrent state be i, be equivalent to promptly that residing position is X in solution space S _iIf population P _iAfter being poisoned, change to new state k, promptly be equivalent in solution space S from present located position X from current state i _iTransfer to reposition X _kCalculate by formula (2), if F (X _k)＜F (X _i), show reposition X _kThan original position X _iMore excellent (because of reposition X _kTarget function value little), then think population P _iEnergy for growth strong.Otherwise, if F (X _k) 〉=F (X _i), show reposition X _kThan original position X _iPoorer (because of reposition X _kTarget function value little), or do not have what difference (because of reposition X _kEquate with the target function value of original position), then think population P _iEnergy for growth a little less than.The population that energy for growth is strong can obtain higher probability continued growth; And the weak population of energy for growth then may stop growing.

In addition, in environmental system, biotic population influences each other because of mutually contention existence resource or mutualism exist, and this influence must be also embodied in the interaction between species characteristic.This influencing each other hinted obliquely at the solution space of optimization problem, is exactly that certain trial solution and some other trial solutions existence influence each other.

The N kind biotic population that generates at random in the environmental system, because of characteristic is different separately, certainly can be not identical to the ability to bear of pollution and toxic hazard yet.Because the pollution intensity that the different times pollution source inject to environmental system is a random variation, this makes the growth conditions of N kind biotic population also ensue variation.Solution space is hinted obliquely in this variation, is moved over time and constantly with regard to the state that is equivalent to N trial solution.

The PSDO-EP algorithm adopts above-mentioned these search strategies to realize the globally optimal solution of optimization problem (1) is searched for exactly.

The design of population evolution operator

The PSDO-EP algorithm utilizes the environmental pollution all living creatures that sows to deposit kinetic model and come the structural evolution operator to realize message exchange between environment and population, population and the population, and then realization is searched for the optimization problem solution space.The described dynamic law of formula (4) is applied on some features of population, just is equivalent to be applied in the trial solution on those corresponding variablees, because these features or claim that variable all is non-negative real number.And, for randomness and the popularity that strengthens search, we in addition can select some features of population to come application formula (4) at random.That is to say that in the optimization problem solution space, we are applied to the described dynamic law of formula (4) from (x _I1, x _I2..., x _In) on some variable of selecting at random.The evolution operator of PSDO-EP algorithm is as described below.

(1) pollutes operator, population P _i(i=1,2 ..., some feature N) causes feature to change because of being subjected to polluting, and makes the population growth change, thereby produces a new generation, is got by formula (4):

In the formula:

With

Be respectively period t+1 and period population i the state value of feature j, and all be nonnegative real number;

Be population P _iRate of growth,

c ^tBe the concentration of population absorption pollutant, 0＜c ^t＜1;

Be population P _iThe pollution mortality ratio, population P _iThe size of pollution mortality ratio and population absorb the degree c that pollutes ^tBe directly proportional;

Be population P _iNatural mortality rate,

g ^tBe the efficient of population to environment drainage pollutant, 0＜g ^t＜1; h ^tThe part that in environment, eliminates for pollutant by other approach, 0＜h ^t＜1; m ^tBe the efficient of the pollutant that population absorbs, purification is eliminated, 0＜m ^t＜1; e ^tBe the concentration of environmental contaminants, 0＜e ^t＜1; K is the efficient of environmental pollution population, 0＜K ^t＜1; get during calculating

c ⁰=Rand (c _l, c _u), e ⁰=Rand (e _l, e _u),

g ^t=Rand (g _l, g _u), m ^t=Rand (m _l, m _u), h ^t=Rand (h _l, h _u); r _lAnd r _uExpression respectively

The lower limit and the upper limit of interval, and 1〉r _uR _l0; c _lAnd c _uRepresent c respectively ^tThe lower limit of interval and the upper limit, and 1〉c _uC _l0; e _lAnd e _uRepresent e respectively ⁰The lower limit of interval and the upper limit, and 1〉e _uE _l0; a _lAnd a _uExpression respectively

The lower limit of interval and the upper limit, and 1〉a _uA _l0; g _lAnd g _uRepresent g respectively ^tThe lower limit of interval and the upper limit, and 1〉g _uG _l0; m _lAnd m _uRepresent m respectively ^tThe lower limit of interval and the upper limit, and 1〉m _uM _l0; h _lAnd h _uRepresent h respectively ^tThe lower limit of interval and the upper limit, and 1〉h _uH _l0; K _lAnd K _uRepresent K respectively ^tThe lower limit of interval and the upper limit, and 1〉K _uK _l0; (a b) is illustrated in uniform random number of [a, b] interval generation to Rand; WR ₀Be high pollution probability, p is the selected probability of feature j; u ^tBe pollutant discharge amount;

Because increase or the minimizing pollutant discharge amount, can reduce or improve the state value of certain feature of population, just reduce or improve the size of trial solution variable.More briefly, be exactly the size that pollutant discharge amount can be controlled the trial solution variable, this is our needed characteristic just.So u ^tCan be calculated as follows

$u^t=\mathrm{Rand}(\min (x_{\mathrm{ij}}^t,x_j^{*t}),\max (x_{\mathrm{ij}}^t,x_j^{*t}))$

In the formula:

Be the current globally optimal solution X of t in period ^*J variable; (a b) is illustrated in uniform random number of [a, b] interval generation to Rand.

(2) disturb operator mutually, what this operator was described is influencing each other between population, population P _i(i=1,2 ..., N) with environmental system in other several populations exist and influence each other, this influence shows on some feature, it is expressed as:

In the formula: HR ₀Be the highest probability of disturbing mutually; s _kFor from 1,2 ..., the A that selects at random among the N} its contamination resistance is higher than population P _iPopulation, represent to be exactly { s with its numbering ₁, s ₂..., s _{| A|};

s _k≠ i; A is and population P _iProduce the good variety population number of disturbing mutually, be called the optimum population number of disturbing mutually, A 〉=1,

Be called the optimum rate of disturbing mutually, and

u _kFor from 1,2 ..., B the population P that selects at random among the N} _iPopulation, represent to be exactly { u with its numbering ₁, u ₂..., u _{| B|}; u _k≠ i; B is and population P _iProduce the ordinary population number of disturbing mutually, be called the ordinary population number of disturbing mutually, B 〉=1,

Be called the ordinary rate of disturbing mutually, and Get during calculating ${\mathrm{\λ}}_{s_k}^0=\mathrm{Rand}({\mathrm{\λ}}_l^0,{\mathrm{\λ}}_u^0),$ ${\mathrm{\λ}}_{u_k}^1=\mathrm{Rand}({\mathrm{\λ}}_l^1,{\mathrm{\λ}}_u^1),$

With

Expression respectively

The lower limit and the upper limit of interval, and

With

Expression respectively The lower limit and the upper limit of interval, and

(3) invade and eliminate operator, what this operator was described is that invading mutually between population eliminated phenomenon, and promptly the feature of some populations is invaded population P _i(i=1,2 ..., in character pair N); The feature of other populations is from population P _iCharacter pair in be eliminated; The feature of some populations is from population P again _iProcess invade mutually eliminate the back implanted population P _iCharacter pair in, it is expressed as:

In the formula: JD ₀For increased resistance invasion is eliminated probability, p ₀=Rand (0,1);

i _s, i _{2 (s-1)}, i _2s-1For from 1,2 ..., the numbering of the population that the N} random choose goes out; If s ≠ i, then i _s≠ i; α _s, β _sEliminate constant, 0 for invading＜α _s, β _s＜1, get α during calculating _s=Rand (α _l, α _u), β _s=Rand (β _l, β _u), α _lAnd α _uRepresent α respectively _sThe lower limit and the upper limit of interval, and 1〉α _uα _l0; β _lAnd β _uRepresent β respectively _sThe lower limit and the upper limit of interval, and 1〉β _uβ _l0;

With

Eliminate constant for invading,

Get during calculating ${\mathrm{\δ}}_s^0=\mathrm{Rand}({\mathrm{\δ}}_l^0,{\mathrm{\δ}}_u^0),$ ${\mathrm{\δ}}_s^1=\mathrm{Rand}({\mathrm{\δ}}_l^1,{\mathrm{\δ}}_u^1),$

With

Expression respectively

The lower limit and the upper limit of interval, and

With

Expression respectively

The lower limit and the upper limit of interval, and

C is for implementing the population number that feature is invaded, C 〉=1; Other population number that D is eliminated for its feature, D 〉=1; E is other implanted population number of feature, E 〉=1;

(4) accretive operatos, what this operator was described is the growth of population, if population P _i(i=1,2 ..., N) can bear the murder by poisoning of environmental pollution to it, then this population goes down continued growth; Otherwise this population will stop growing, and it is expressed as:

Initialization of population

The dimension of supposing the optimization problem search volume is n, and the region of search of each variable is [l _i, u _i], i=1,2 ..., n then utilizes orthogonal Latin square generating algorithm INIT to produce the orthogonal arrage L of N initial solution _N(N ⁿ) construction algorithm.

Algorithm INIT produces the orthogonal arrage L of N initial solution _N(N ⁿ) construction algorithm

Step 1: the discrete point y that calculates each variable _Ij:

y _ij=l _i+(j-1)(u _i-l _i)/(N-1)，i=1，2，…，n；j=1，2，…，N。

Step 2: the generation method according to orthogonal Latin square is calculated initial solution x _Ij:

x _ij=y _jk，i=1，2，…，N，j=1，2，…，n

In the formula: k=(i+j-1) mod N; If k=0, then k=N.

The determined N of an algorithm INIT initial solution X _i=(x _I1, x _I2..., x _In), i=1,2 ..., N has good balanced dispersed and neat comparability.

Described PSDO-EP algorithm comprises the steps:

(1) initialization: a) make t=0 in period; By all parameters that this algorithm of table 1 initialization relates to, wherein t represents current the evolution to t in period; B) the generating algorithm INIT by orthogonal Latin square determines N initial population;

The obtaining value method of table 1 parameter

(2) make period t from 0 to G, following step (3)～(14) are carried out in circulation, and wherein G is the maximum epoch number that develops;

(3) make population i from 1 to N, following step (4)～(13) are carried out in circulation;

(4) the feature j that makes population is from 1 to n, and following step (5)～(9) are carried out in circulation;

(5) calculate p=Rand (0,1); Wherein p is for polluting operator, disturbing operator and invade and eliminate the actual probabilities that operator is performed mutually;

(6) if p≤WR ₀, then carry out and pollute operator by formula (5), obtain

(7) if WR ₀＜p≤HR ₀, then carry out and disturb operator mutually by formula (6), obtain

(8) if HR ₀＜p≤JD ₀, then carry out to invade and eliminate operator by formula (7), obtain

(9) make j=j+1, if j≤n then changes above-mentioned steps (5), otherwise changes following step (10);

(10) carry out accretive operatos by formula (8), obtain

(11) if the error between globally optimal solution that newly obtains and the last globally optimal solution that obtains satisfies minimum requirements ε, then change following step (15);

(12) preserve the globally optimal solution that newly obtains;

(13) make i=i+1, if i≤N then changes above-mentioned steps (4), otherwise changes following step (14);

(14) make t=t+1, if t≤G then changes above-mentioned steps (3), otherwise changes following step (15);

(15) finish.

Beneficial effect

The present invention compares with prior art, has following characteristics:

1, the invention discloses and a kind ofly deposit dynamic (dynamical) optimized Algorithm based on the environmental pollution all living creatures that sows, it is the PSDO-EP algorithm, this algorithm is corresponding with the environmental system that has contamination phenomenon with the solution space of optimization problem to be solved, and several populations are living in this environmental system; Population constantly changes under contamination, can resist the strong person who pollutes and obtain growth, and the yellowbelly stops growing; Each population correspondence a trial solution of optimization problem, and the feature of population is corresponding to a variable component in the trial solution; Be used to construct the growth strategy of population between environment and population and population and the population; Utilize the environmental pollution all living creatures that sows to deposit kinetic model and come the structural evolution operator to realize message exchange between environment and population, population and the population; In the population evolution process, population is transferred to another kind of growth conditions from a kind of growth conditions and has realized the search of population to the optimization problem optimum solution.

2, evolutionary process has the Markov characteristic.From polluting operator, disturbing operator and invade the definition of eliminating operator and know mutually, the generation of any one new trial solution is only relevant with the current state of this trial solution, and with this trial solution be that the course that how to develop current state has nothing to do in the past.

3, evolutionary process has " not poor step by step " characteristic, and promptly develop not can be to than the worse state transitions of current state per step.This characteristic is just known from the definition of accretive operatos.

4, time complexity is lower.The time complexity computation process of PSDO-EP algorithm is as shown in table 2, and its time complexity is relevant with time complexity and other non-productive operations of evolution number of times G, population scale N, variable number n and each operator.

The time complexity reckoner of table 2PCDO-HNS algorithm

Operation	Time complexity	Maximum cycle indexes
			Initialization	O(n+4nN+N(N+1))	1

[0090]?

Pollute operator	O(5n/3)	GN
			Disturb operator mutually	O(4n(A+B)/3)	GN
Invade and eliminate operator	O(4n(C+D+E)/3)	GN
			Objective function calculates	O(n)～O(n 2)	GN
Accretive operatos	O(3n)	GN
			Result's output	O(n)	1

Know that by table 2 time complexity of PSDO-EP method is lower, so algorithm speed is fast.

5, the value of the related parameter of PSDO-EP method need not too high accuracy, and the improper speed of convergence that only influences of value is to the not influence of precision of globally optimal solution.

6, the model parameter value is simple.Adopt random device to determine to pollute in the algorithm operator, disturb operator and invade the correlation parameter of eliminating in the operator mutually, both significantly reduced parameter input number, make model more can express actual conditions again.

7, operation environment pollutes the optimization method that all living creatures that sows deposits the complicated optimum problem found the solution that kinetic theory constructs in the PSDO-EP method, and operation environment pollutes all living creatures that sows and deposits the dynamics basic model and proposed the pollution operator, disturbed operator and invade operators such as eliminating operator mutually in the method.The utilization of these operators has significantly promoted the performance of PSDO-EP algorithm, thereby the suitable complicated optimum problem of finding the solution.

8, the PSDO-EP method has global convergence.The theoretical analysis of its reason is as follows:

Know that by the PSDO-EP algorithm environmental system is a discrete space, but each population

Be in continuous real number space value.The total number of population is N, and each population is a trial solution of optimization problem (1), and its target function value is calculated as by formula (2)

Then the formed set of the state of all populations is

$F=\left\{F\left(X_i^t\right)\right|X_i^t\∈S\}$

Further order

F＝{F ₁,F ₂,…,F _N}，F ₁≤F ₂≤…≤F _N (A)

Be without loss of generality, make F ₁Be the globally optimal solution that we ask.The subscript of formula (A) is taken out set of formation, promptly

U={1，2，…，N}

Each population may residing state when the element among the set U was exactly random search.Suppose that our the best target function value that searches of phase at a time is F _i, its corresponding state is i.Obviously, (A) knows by formula, if shift to more excellent state k, then should satisfy k＜i when searching for next period; On the contrary,, then should satisfy k if shift to worse state k〉i, shown in Table A.

State transitions situation during the Table A random search

S is divided into nonvoid subset is:

$\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\left|X_S^i\right|=N;$

Order

Expression

In j state.Population during evolution, (i, (k l) can be expressed as X j) to transfer to other state from a state ^{I, j}→ X ^{K, l}, then supposition: from X ^{I, j}To X ^{K, l}Transition probability be p _{Ij, kl}, from X ^{I, j}Arrive

In the transition probability of arbitrary state be pi _{J, k}, from

In arbitrary state arrive

In the transition probability of arbitrary state be p _{I, k}, then have $p_{\mathrm{ij},k}=\underset{l=1}{\overset{\left|X_S^k\right|}{\mathrm{\Σ}}}{p}_{\mathrm{ij},\mathrm{kl}},$ $\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{p}_{\mathrm{ij},k}=1,$ p _i,k≥p _ij,k $p_{i,k}\≥p_{\mathrm{ij},k}\→\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{p}_{i,k}\≥\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{p}_{\mathrm{ij},k}=1,$ And $0\≤\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{p}_{i,k}\≤1,$ So have

$\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{p}_{i,k}=1---\left(B\right)$

Lemma 1 in the PSDO-EP algorithm,

Satisfy:

$\∀k>i,p_{i,k}=0---\left(C\right)$

$\&Exists;k<i,p_{i,k}>0---\left(D\right)$

(1) proof of formula (C).If state i is t population X in period ^tState, this state i is exactly the best condition that this population has reached so far certainly.In the PSDO-EP algorithm, carry out at every turn new evolution all always to this population current state i further to the renewal of better state, promptly have

The implication of following formula is: if i is the state (also must be the best condition that this population has reached) of t population in period, the evolution of this population of t+1 only can be upgraded to better state in period, so any other state that begins can not transfer to than the i difference from i gets on; (A) knows by formula, if want F _kF _i, then the state k than state i difference must satisfy k〉and i also is that best condition maintains the original state or can only upgrade (it is not poor step by step promptly to accomplish) to better state, shown in Table A.

(2) proof of formula (D).If the current state of certain population is i, must be the best condition that this population has reached up to now certainly, at t+1 in period, this population select at random to pollute operator, mutually disturb operator, invade operations such as eliminating operator and develop in the hope of transferring on the better state k.At this moment, have two kinds of situations to occur:

A) if i is global optimum's state, i.e. i=1, then next step shifts essential k=1 (getting on because can not transfer to the state also poorer than current state), promptly must be with Probability p _1,1=1 transfers to this global optimum's state gets on.Because of p _1,1=1〉0, assign a topic to such an extent that demonstrate,prove;

B) if i is not global optimum's state, then between global optimum's state 1 and current state i, must there be an intermediateness k (shown in Table A) at least, makes F ₁≤ F _k＜F _i, i.e. 1≤k＜i, this moment, current state i can transfer to state k get on (because new state k is more excellent than current state i), just p _{I, k}0, assign a topic to such an extent that demonstrate,prove.

Comprehensive above-mentioned situation can get

Card is finished.

Iisufescu M proposes following theorem in document " Finite Markov Processes and Their Applications, Wiley:Chichester, 1980 ":

But it is a n rank reduction stochastic matrix that theorem 1 is established P ', just by obtaining behind identical line translation and the rank transformation $P^{\&prime;}=\left[\begin{array}{ccc}C& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ R& \&CenterDot;\&CenterDot;\&CenterDot;& T\end{array}\right],$ Wherein C is m rank basis stochastic matrix and R ≠ 0, and T ≠ 0 then has

$P^{\&prime;\&infin;}=\underset{k\&RightArrow;\&infin;}{\lim }{P}^{\&prime;k}=\underset{k\&RightArrow;\&infin;}{\lim }\left[\begin{array}{ccc}{C}^k& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ \underset{i=1}{\overset{k-1}{\mathrm{\&Sigma;}}}{T}^i{\mathrm{RC}}^{k-i}& \&CenterDot;\&CenterDot;\&CenterDot;& T^k\end{array}\right]=\left[\begin{array}{ccc}{C}^{\&infin;}& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ R^{\&infin;}& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right]$

Above-mentioned matrix is a stable stochastic matrix and P ' ^∞=1 ' P ' ^∞, P ' ^∞=P ' ⁰P ' ^∞Unique definite and irrelevant with initial distribution, P ' ^∞Satisfy following condition:

$P^{\&prime;\&infin;}={\left[p_{\mathrm{ij}}\right]}_{n\&times;n},\left\{\begin{array}{c}{p}_{\mathrm{ij}}>\mathrm{0,1}\&le;i\&le;n,1\&le;j\&le;m\\ p_{\mathrm{ij}}=\mathrm{0,1}\&le;i\&le;n,m<j\&le;n\end{array}\right.$

The proof procedure of theorem 1 is very complicated, and concrete proof procedure can be referring to document " Finite Markov Processes and Their Applications, Wiley:Chichester, 1980 "

Theorem 2PSDO-EP algorithm has global convergence.

Proof: for each

It is a state on the limited Markov chain that N can be seen as, and can get according to the conclusion of lemma 1 Chinese style (C), and the transition matrix of this Markov chain is

(B) knows by formula, and every capable probability sum is 1 in P ' matrix.Get according to lemma 1 Chinese style (D) conclusion again

But P ' is N rank reduction stochastic matrix (a Markov probability matrix) as known from the above, satisfies the condition of theorem 1, so following formula is set up:

$P^{\&prime;\&infin;}=\underset{k\&RightArrow;\&infin;}{\lim }\left[\begin{array}{ccc}{C}^k& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ \underset{i=1}{\overset{k-1}{\mathrm{\&Sigma;}}}{T}^i{\mathrm{RC}}^{k-i}& \&CenterDot;\&CenterDot;\&CenterDot;& T^k\end{array}\right]=\left[\begin{array}{ccc}{C}^{\&infin;}& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ R^{\&infin;}& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right]$

Because of C ^∞=C=(1), T ^∞=0, so R must be arranged ^∞=(1,1 ..., 1) ^T, this is because the probability sum of every row is 1 among the Markov transition matrix P '.Therefore have

$P^{\&prime;\&infin;}=\left[\begin{array}{cccc}1& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ 1& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ 1& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right],$ And be stable stochastic matrix.

Following formula shows, when k → ∞, and Probability p _{I, 1}=1, i=1,2 ..., N, also promptly regardless of original state, at last can both convergence with probability 1 to global optimum's state 1.So

$\underset{t\&RightArrow;\&infin;}{\lim }p\{F\left(X_i^t\right)\&RightArrow;F\left(X^{*}\right)\}=1,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N$

Therefore, the PSDO-EP algorithm has global convergence, and card is finished.

Embodiment

The present invention is described in further detail below in conjunction with instantiation.

(1) the definite actual optimization problem that will find the solution transforms the described canonical form of an accepted way of doing sth (1) with this problem.Promptly

(2) if the actual optimization problem is to ask max f (X), then change min-f (X) into.

(3) all range of variables with the actual optimization problem compress and adjustment, promptly

If 0≤x _i≤ a _i, a _i〉=0, i=1,2 ..., n is then with x _i=a _iy _iSubstitution actual optimization problem, this is that the actual optimization problem is about variable y _iOptimization problem, 0≤y _i≤ 1.

If-a _i≤ x _i≤ 0, a _i〉=0, i=1,2 ..., n is then with x _i=-a _iy _iSubstitution actual optimization problem, this is that the actual optimization problem is about variable y _iOptimization problem, 0≤y _i≤ 1.

If-b _i≤ x _i≤ a _i, a _i〉=0, b _i〉=0, i=1,2 ..., n is then with x _i=(a _i+ b _i) y _i-b _iSubstitution actual optimization problem, this is that the actual optimization problem is about variable y _iOptimization problem, 0≤y _i≤ 1.

(4) determine the parameter of PSDO-EP algorithm by the described method of table 1.

(5) operation PSDO-EP algorithm is found the solution.

(6) optimum solution y _iAfter the acquisition, obtain xi in the following method in profit and get final product:

If by-b _i≤ x _i≤ a _iTransform, then x _i=(a _i+ b _i) y _i-b _i, i=1,2 ..., n;

If by 0≤x _i≤ a _iTransform, then x _i=a _iy _i, i=1,2 ..., n;

If by-a _i≤ x _i≤ 0 transforms, then x _i=-a _iy _i, i=1,2 ..., n.

(1) for the actual optimization problem, ask n=100,200,400,600,800,1000,1200 o'clock globally optimal solution.

$\max f\left(X\right)=-20-e+20\exp (-0.2\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i^2})+\exp \left(\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\cos \left(2\mathrm{\&pi;}{x}_i\right)\right)$

-100≤x _i≤100，i=1，2，…，n

(2) this optimization problem is transformed the described canonical form of an accepted way of doing sth (1), promptly

$\min f\left(X\right)=20+e-20\exp (-0.2\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i^2})-\exp \left(\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\cos \left(2\mathrm{\&pi;}{x}_i\right)\right)$

-100≤x _i≤100，i=1，2，…，n

(3) make x _i=200y _i-100, Y=(y ₁, y ₂..., y _n), then

$\min f\left(Y\right)=20+e-20\exp (-20\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{({2y}_i-1)}^2})-\exp \left(\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\cos \left(200\mathrm{\&pi;}({2y}_i-1)\right)\right)$

0≤y _i≤1，i=1，2，…，n

(4) determine the parameter of algorithm by the described method of table 1, as shown in table 3.

Each parameter value of table 3 solution procedure

(5) adopt PSDO-EP algorithm algorithm to find the solution, the gained result is as shown in table 4.

Table 4 result of calculation

The optimum solution of (6) trying to achieve is at y _iWithin [0.49999998,0.50000002], get x after the conversion _iIn [4.0E-6,2.0E-6], i=1,2 ..., n.

Claims (1)

1. the environmental pollution all living creatures that sows deposits Dynamics Optimization method-PSDO-EP algorithm, and it is characterized in that: establishing the function optimization problem that will solve is:

minf(X)

In the formula: R ⁿIt is n dimension Euclidean space; X=(x ₁, x ₂..., x _n) be a n dimension decision vector, variable x _i(i=1,2 ..., n) be nonnegative real number; S is non-negative search volume, claims solution space again; F (X) is an objective function; g _i(X) 〉=0 be i constraint condition, i=1,2 ..., I, I are inequality constrain condition number; h _i(X)=0 be i equality constraint, i=1,2 ..., E, E are the equality constraint number; Objective function f (X) and constraint condition g _i(X), h _i(X) do not need special restrictive condition;

The PSDO-EP algorithm adopts is that the environmental pollution all living creatures that sows deposits kinetic theory, environmental system is corresponding with the solution space of optimization problem (1), there is contamination phenomenon in this environmental system, several populations are wherein living, each population correspondence a trial solution of optimization problem, and the feature of population is corresponding to a variable in the trial solution; The population existence kinetic model that is applicable to some feature of population also be applicable in the trial solution to dependent variable, population constantly changes under the environmental pollution effect, the strong person that can resist environmental pollution obtains growth, valetudinarian, invalid stops growing; Because population is being shared the resource in the environmental system, so there is the phenomenon that influences each other between the population; These interactions between environment and population and population and the population are used to construct the growth strategy of population, utilize the environmental pollution all living creatures that sows to deposit kinetic model and come the structural evolution operator to realize message exchange between environment and population, population and the population; In the population evolution process, population is transferred to another kind of growth conditions from a kind of growth conditions and has realized the search of population to the optimization problem optimum solution;

Described PSDO-EP algorithm comprises the steps:

(1) initialization: a) make t=0 in period; By all parameters that this algorithm of table 1 initialization relates to, wherein t represents current the evolution to t in period; B) determine N initial population by orthogonal Latin square generating algorithm INIT;

The obtaining value method of table 1 parameter

Described orthogonal Latin square generating algorithm INIT is:

Step 1: the discrete point y that calculates each variable _Ij:

y _ij=l _i+(j-1)(u _i-l _i)/(N-1)，i=1，2，…，n；j=1，2，…，N。

Step 2: the generation method according to orthogonal Latin square is calculated initial solution x _Ij:

x _ij=y _jk，i=1，2，…，N，j=1，2，…，n

In the formula: k=(i+j-1) mod N; If k=0, then k=N;

The determined N of an above-mentioned algorithm initial solution is $X_i^0=(x_{i1},x_{i2},\&CenterDot;\&CenterDot;\&CenterDot;,x_{\mathrm{in}}),i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N;$

(2) make period t from 0 to G, following step (3)～(14) are carried out in circulation, and wherein G is the maximum epoch number that develops;

(3) make population i from 1 to N, following step (4)～(13) are carried out in circulation;

(4) the feature j that makes population is from 1 to n, and following step (5)～(9) are carried out in circulation;

(5) calculate p=Rand (0,1); Wherein p is for polluting operator, disturbing operator or invade and eliminate the actual probabilities that operator is performed mutually;

(6) if p≤WR ₀, then carry out and pollute operator by formula (5), obtain

In the formula:

With

Be respectively period t+1 and period population i the state value of feature j, and all be nonnegative real number;

Be population P _iRate of growth, c ^tBe the concentration of population absorption pollutant, 0＜c ^t＜1;

Be population P _iThe pollution mortality ratio, population P _iThe size of pollution mortality ratio and population absorb the degree c that pollutes ^tBe directly proportional; Be population P _iNatural mortality rate,

g ^tBe the efficient of population to environment drainage pollutant, 0＜g ^t＜1; h ^tThe part that in environment, eliminates for pollutant by other approach, 0＜h ^t＜1; m ^tBe the efficient of the pollutant that population absorbs, purification is eliminated, 0＜m ^t＜1; e ^tBe the concentration of environmental contaminants, 0＜e ^t＜1; K is the efficient of environmental pollution population, 0＜K ^t＜1; get during calculating

c ⁰=Rand (c _l, c _u), e ⁰=Rand (e _l, e _u),

g ^t=Rand (g _l, g _u), m ^t=Rand (m _l, m _u), h ^t=Rand (h _l, h _u); r _lAnd r _uExpression respectively

The lower limit and the upper limit of interval, and 1〉r _uR _l0; c _lAnd c _uRepresent c respectively ^tThe lower limit of interval and the upper limit, and 1〉c _uC _l0; e _lAnd e _uRepresent e respectively ⁰The lower limit of interval and the upper limit, and 1〉e _uE _l0; a _lAnd a _uExpression respectively

The lower limit of interval and the upper limit, and 1〉a _uA _l0; g _lAnd g _uRepresent g respectively ^tThe lower limit of interval and the upper limit, and 1〉g _uG _l0; m _lAnd m _uRepresent m respectively ^tThe lower limit of interval and the upper limit, and 1〉m _uM _l0; h _lAnd h _uRepresent h respectively ^tThe lower limit of interval and the upper limit, and 1〉h _uH _l0; K _lAnd K _uRepresent K respectively ^tThe lower limit of interval and the upper limit, and 1〉K _uK _l0; (a b) is illustrated in uniform random number of [a, b] interval generation to Rand; WR ₀Be high pollution probability, p is the selected probability of feature j; u ^tBe pollutant discharge amount, its computing method are: $u^t=\mathrm{Rand}(\min (x_{\mathrm{ij}}^t,x_j^{*t}),\max (x_{\mathrm{ij}}^t,x_j^{*t})),$ Wherein,

Be the current globally optimal solution X of t in period ^*J variable;

Described formula (5) is from formula (4):

$\left\{\begin{array}{c}\frac{{\mathrm{dx}}_i}{\mathrm{dt}}=x_i(r_i-r_{i}c-a_i{x}_i)\\ \frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{Ke}-N(g+m)c,\\ \frac{\mathrm{de}}{\mathrm{dt}}=-\mathrm{he}+u\end{array}\right.i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N---\left(4\right)$

In the formula: t represents period; U is the intensity of pollution source to environmental emissions; K is the efficient of environmental pollution population, 0＜K＜1; G is the efficient that population drains pollutant to environment, 0＜g＜1; H is the part that pollutant eliminates by other approach in environment, 0＜h＜1; M is the efficient that population absorbs, purifies the pollutant of being eliminated, 0＜m＜1; x _iBe population P _iQuantity, and x _i〉=0; r _iBe population P _iRate of growth, 0＜r _i＜1; r _iC is population P _iThe pollution mortality ratio, the size of polluting mortality ratio is directly proportional 0 with the contaminated degree c of population＜c＜1; a _iBe population P _iNatural mortality rate; E is the concentration of environmental contaminants, 0＜e＜1;

(7) if WR ₀＜p≤HR ₀, then carry out and disturb operator mutually by formula (6), obtain

In the formula: HR ₀Be the highest probability of disturbing mutually; s _kFor from 1,2 ..., the A that selects at random among the N} its contamination resistance is higher than population P _iPopulation, represent to be exactly { s with its numbering ₁, s ₂..., s _{| A|};

s _k≠ i; A is and population P _iProduce the good variety population number of disturbing mutually, be called the optimum population number of disturbing mutually, A 〉=1,

Be called the optimum rate of disturbing mutually, and

u _kFor from 1,2 ..., B the population P that selects at random among the N} _iPopulation, represent to be exactly { u with its numbering ₁, u ₂..., u _{| B|}; u _k≠ i; B is and population P _iProduce the ordinary population number of disturbing mutually, be called the ordinary population number of disturbing mutually, B 〉=1, Be called the ordinary rate of disturbing mutually, and

Get during calculating ${\mathrm{\&lambda;}}_{s_k}^0=\mathrm{Rand}({\mathrm{\&lambda;}}_l^0,{\mathrm{\&lambda;}}_u^0),$ ${\mathrm{\&lambda;}}_{u_k}^1=\mathrm{Rand}({\mathrm{\&lambda;}}_l^1,{\mathrm{\&lambda;}}_u^1),$

With Expression respectively

The lower limit and the upper limit of interval, and

With

Expression respectively

The lower limit and the upper limit of interval, and

(8) if HR ₀＜p≤JD ₀, then carry out to invade and eliminate operator by formula (7), obtain

In the formula: JD ₀For increased resistance invasion is eliminated probability, p ₀=Rand (0,1);

i _s, i _{2 (s-1)}, i _2s-1For from 1,2 ..., the numbering of the population that the N} random choose goes out; If s ≠ i, then i _s≠ i; α _s, β _sEliminate constant, 0 for invading＜α _s, β _s＜1, get α during calculating _s=Rand (α _l, α _u), β _s=Rand (β _l, β _u), α _lAnd α _uRepresent α respectively _sThe lower limit and the upper limit of interval, and 1〉α _uα _l0; β _lAnd β _uRepresent β respectively _sThe lower limit and the upper limit of interval, and 1〉β _uβ _l0; With

Eliminate constant for invading,

Get during calculating ${\mathrm{\&delta;}}_s^0=\mathrm{Rand}({\mathrm{\&delta;}}_l^0,{\mathrm{\&delta;}}_u^0),$ ${\mathrm{\&delta;}}_s^1=\mathrm{Rand}({\mathrm{\&delta;}}_l^1,{\mathrm{\&delta;}}_u^1),$

With

Expression respectively

The lower limit and the upper limit of interval, and

With

Expression respectively

The lower limit and the upper limit of interval, and

C is for implementing the population number that feature is invaded, C 〉=1; The population number that D is eliminated for its feature, D 〉=1; E is the implanted population number of feature, E 〉=1;

(9) make j=j+1, if j≤n then changes above-mentioned steps (5), otherwise changes following step (10);

(10) carry out accretive operatos by formula (8), obtain

In the formula: $X_i^{t+1}=(x_{\mathrm{il}}^{t+1},x_{i1}^{t+1},\&CenterDot;\&CenterDot;\&CenterDot;,x_{\mathrm{in}}^{t+1});$ $V_i^{t+1}=(v_{\mathrm{il}}^{t+1},v_{i1}^{t+1},\&CenterDot;\&CenterDot;\&CenterDot;,v_{\mathrm{in}}^{t+1});$ F (X _i) calculate by formula (2):

Symbol in the formula (2) is described in formula (1);

(11) if the error between globally optimal solution that newly obtains and the last globally optimal solution that obtains satisfies minimum requirements ε, then change following step (15);

(12) preserve the globally optimal solution that newly obtains;

(13) make i=i+1, if i≤N then changes above-mentioned steps (4), otherwise changes following step (14);

(14) make t=t+1, if t≤G then changes above-mentioned steps (3), otherwise changes following step (15);

(15) finish.