CN103218658A - Population competition dynamics optimization method with horizontal nutrition structures - Google Patents

Abstract

A population competition dynamics optimization method with horizontal nutrition structures is a PCDO-HNS algorithm. A solution space of an optimization problem is regarded as an ecosystem which has a plurality of different sub-systems, each sub-system has a specific horizontal nutrition structure type which is one of a normal distribution type, a discrete distribution type, a nearest type, an even type and a monotone decreasing type, populations living in each sub-system mutually compete, and mutual learning behaviors, mutual influence behaviors, mutation occurring behaviors and close keeping behaviors also exist in an accompanying mode. A population competition dynamics model with the horizontal nutrition structures are used for constructing five population evolution operators of the normal distribution type, the discrete distribution type, the nearest type, the even type and the monotone decreasing type, and the operators are used for constructing evolution strategies of the populations. The algorithm has the advantages of being strong in search capability and having global convergence, and a solution scheme is provided for the solution of a complex function optimization problem.

Description

The population competition Dynamics Optimization method of the horizontal trophic structure of tool

Technical field

The present invention relates to intelligent optimization algorithm, be specifically related to the population competition Dynamics Optimization method-PCDO-HNS algorithm of the horizontal trophic structure of a kind of tool.

Background technology

Consider the generic function optimization problem

minf(X)

$s.t.\left\{\begin{array}{c}{g}_i\left(X\right)\≥0,i=1,2,\·\·\·,I\\ h_i\left(X\right)=0,i=\mathrm{1,2},\·\·\·,E\\ X\∈S\⋐R^n,X\≥0\end{array}\right.---\left(1\right)$

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, traditional mathematical optimization method based on function continuity and the property led can't address this problem.

The method for solving of above-mentioned optimization problem (1) is a Swarm Intelligent Algorithm, and this class algorithm 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 was used the method proposition of biogeography in 2008 by Dan Simon, 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.

Optimized Algorithm related to the present invention is competition Cooperative Evolutionary Algorithm (Competitive Coevolution Algorithm, CompCEA), this algorithm is by Rosin C.D., Belew R.K. is at document " New methods for competitive coevolution, Evolutionary Computation, 1997,5 (1): 1-29 " propose in, but this algorithm is the simulation of predation-quilt in the ecogenesis being eaten phenomenon.Stanley K.O., Miikkulainen R. is at document " Competitive coevolution through evolutionary complexification, Journal of Artificial Intelligence Research, 2004,21:63-100 " in considered that the evolution complicacy further carried out expansion research to the CompCEA algorithm; Tan T.G., Teo J., Lau H.K. is at document " Competitive coevolution with K-Random Opponents for Pareto multiobjective optimization, ICNC2007:Third International Conference on Natural Computation, Vol4, Proceedings, 2007:63-67 " in the CompCEA algorithm application in finding the solution the Pareto multi-objective optimization question; Tan T.G., Lau H.K., Teo J. is at document " Cooperative versus competitive coevolution for Pareto multiobjective optimization, Bio-Inspired Computational Intelligence and Applications, Springer Berlin, 2007:63-72 " will compete and the cooperation Cooperative Evolutionary Algorithm links together, be used to find the solution the Pareto multi-objective optimization question; McIntyre A.R., Heywood M.I. is at document " Multi-objective competitive coevolution for efficient GP classifier problem decomposition, 2007IEEE International Conference on Systems, Man and Cybernetics, Vols1-8,2007:2582-2589 " in multiple goal competed Cooperative Evolutionary Algorithm be used to find the solution GP sorter resolution problem.

Guo Yuanping is at document " the Cooperative Evolutionary Algorithm model of dynamic population scale, theory and application, China Science ﹠ Technology University's PhD dissertation, 2008 " general character from Cooperative Evolutionary Algorithm in is started with, interacting with the individuality in the fitness evaluation process is starting point, designed a dynamic population scale and regulated strategy with versatility, and on the basis of this strategy, made up a unified Cooperative Evolutionary Algorithm model, the Cooperative Evolutionary Algorithm model that is called dynamic population scale, be called for short the CEAD model, this model is from the essence of Cooperative Evolutionary Algorithm, described the common evolutionary mechanism of Cooperative Evolutionary Algorithm, its algorithm system has been contained and has been comprised state of conflict, cooperation type, mixed type is in interior various Cooperative Evolutionary Algorithm.

Cao Xianbin, Gao Juan, Wang Xufa at document " based on the hereditary intensified learning of Ecological Competition Model, the software journal, 1999,10 (6): 658-662 " use for reference biological growth pattern in the environmental ecology system in, propose a kind of Ecological Competition Model, this model realized in the subgroup that the congenital heredity of individual level is evolved and the day after tomorrow competitive learning, realize further competing intensified learning at the population level; Lee is green to have proposed a kind of new function optimization method based on the competition Cooperative Evolutionary Algorithm in " research of Cooperative Evolutionary Algorithm and application, South China Science ﹠ Engineering University's PhD dissertation, 2010 ".This algorithm combines competition Cooperative Evolutionary Algorithm and genetic algorithm.Individuality in the algorithm has two kinds of fitness: absolute fitness and relative adaptation degree.Fitness by the chromosome decision is absolute fitness, and absolute fitness determines individual problem-solving ability.Fitness by the performance decision of individuality in competition is the relative adaptation degree, the individual viability of relative adaptation degree decision, viability is decided by the adversary's that individuality is defeated quantity and outstanding degree, individual evolves gradually in the effort of defeating more more outstanding adversarys.

Mainly there are the following problems for above-mentioned these algorithms:

(1) do not have to consider that each subsystem has a specific horizontal trophic structure type at the total a plurality of different subsystems of an ecosystem;

(2) population that does not have consideration to live in a horizontal trophic structure subsystem is competed, but the behavior that also is attended by mutual study, influences each other, undergos mutation and keeps to seal exists;

(3) competition process is single, does not embody competitive relation complicated between dissimilar a plurality of populations common in the ecosystem;

(4) competition process does not embody intrinsic rate of increase, the natural mortality rate of dissimilar populations, the parameters such as Carrying capacity of population;

(5) global convergence of algorithm does not provide theoretical proof.

Summary of the invention

In order to solve the problem that above-mentioned prior art exists, the object of the present invention is to provide the population competition Dynamics Optimization method-PCDO-HNS algorithm of the horizontal trophic structure of a kind of tool, 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:

The population competition Dynamics Optimization method-PCDO-HNS algorithm of the horizontal trophic structure of a kind of tool, it is characterized in that: establishing the function optimization problem that will solve is:

minf(X)

$s.t.\left\{\begin{array}{c}{g}_i\left(X\right)\≥0,i=1,2,\·\·\·,I\\ h_i\left(X\right)=0,i=\mathrm{1,2},\·\·\·,E\\ X\∈S\⋐R^n,X\≥0\end{array}\right.---\left(1\right)$

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;

Regard the solution space of optimization problem (1) as an ecosystem, should be at the total a plurality of different subsystems of an ecosystem, each subsystem has a specific horizontal trophic structure type, but different subsystems can have identical horizontal trophic structure type; For each subsystem, there are some populations to live therein, a population can only have a kind of horizontal trophic structure type, but can occur at a plurality of subsystems, does not have unnecessary population not belong to any subsystem; Population can not shift between each subsystem, the population that in a horizontal trophic structure subsystem, lives with mutual competition, self study, be affected, sudden change and the existence of self-enclosed phenomenon, strong population continued growth, weak population then stops growing;

Described self study phenomenon is meant: population i is at subsystem Y _kBetween interior active stage, population i is in order to promote the competitive power of self, initiatively to subsystem Y _kOther interior some populations strongr than population i are learnt, and promptly population i is with subsystem Y _kIts interior PSI index is higher than some characteristic absorption of some populations of population i comes, and makes own strong purpose to reach;

The described phenomenon that is affected is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kSome crawler behavior of other interior population has caused influence to population i, i.e. subsystem Y _kSome feature of other interior population and the mean value of state value thereof have been passed to the character pair of population i, and it is affected;

Described jumping phenomenon is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kSome behavior of some interior special good variety population causes very big influence to population i, i.e. subsystem Y _kIn some very the difference of some feature of good variety population and weighting state value thereof passed to the character pair of population i, it is produced greatly changes;

Described self-enclosed phenomenon is meant: population i is at subsystem Y _kBetween interior active stage, some feature of population i is not subjected to subsystem Y _kAny influence of interior other population;

The search volume of optimization problem is corresponding with the ecosystem, and a population in this ecosystem is corresponding to the trial solution of an optimization problem, and a feature in the population is corresponding to a variable in the optimization problem trial solution; So the characteristic number of population is identical with the variable number of trial solution; The fitness index of population is the target function value of PSI index corresponding to optimization problem, and good trial solution correspondence has the population of higher PSI index, promptly strong population; The trial solution correspondence of difference has the population of low PSI index, i.e. Xu Ruo population.

The PCDO-HNS algorithm is to utilize the population competition dynamic law of the horizontal trophic structure of tool to construct population evolution operator, be that normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, even type competition operator and monotone decreasing type competition operator produce after the population of new generation, adopt and select operator that population of new generation and corresponding parent population are compared one to one, be saved in the colony of future generation than the superior; In case after new population formed, the PCDO-HNS algorithm continued by above-mentioned operator population constantly to be developed up to finding 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 ecosystem is corresponding with the search volume of optimization problem, and a population in this ecosystem is corresponding to the trial solution of an optimization problem, and the exploration disaggregation of N the pairing optimization problem of population is S={X ₁, X ₂..., X _N, wherein, X _i=(x _I1, x _I2..., x _In), i=1,2 ..., N; A feature j among the population i is corresponding to optimization problem trial solution X _iIn a variable x _IjSo the characteristic number of population i and trial solution X _iVariable number identical, all be n.The fitness index of population is the target function value of PSI index corresponding to optimization problem.Good trial solution correspondence has the population of higher PSI index, promptly strong population; The trial solution correspondence of difference has the population of low PSI index, i.e. Xu Ruo population.For optimization problem (1), the PSI index calculation method of population i is

PSI(X _i)=F _max-F(X _i) (3)

Population competition kinetic model with horizontal structure

In the ecosystem, the population interaction in same trophic level inside mainly is competition, competes very general at nature.The horizontal trophic structure of the so-called ecosystem refers to that a plurality of different classes of population institute consumed resources is same type in the ecosystem.

Be provided with u ecological factor (comprising time factor and envirment factor) and constitute u dimension resource space.The n that one population has been formed this space to u dimension utilization of resources scope ties up hypervolume, is called the u dimension ecological niche of this population.Suppose that population institute consumed resources represents its characteristic with certain parameter vector z.The quantity that resource consumption allowed with feature z is determined by certain function k (z).The set of z value is called resource spectrum (or resource space) among the function k (z).

Any population all has definite ecological zone, and they are survived in this zone and multiply.In fact, each population all occupies the ecological niche of oneself, and inlaying of ecological niche is typical phenomenon in the real biocommunity.Population not only coexists in same living environment, and is to coexist under the identical resources supplIes of consumption.The limitation of resource makes the resource mutual restriction of utilizing same population.Like this, the nature relation of constituting competition of inlaying of ecological niche, ecological niche is determining status and the effect of population in the structure of community of competition.

Suppose that a population resources consumption describes its characteristic with certain probability distribution, its density function f (z) is called utilization of resources function, has average z ₀With finite variance σ ²At this moment, the ecological niche of population depends on the some z on the resource spectrum ₀With given z ₀Near the density function f (z) of the stochastic distribution point.When a group is made of some populations that are at war with for common food resource, think the z that is suitable for different population naturally ₀Point has certain distance each other, and experimental observation has also confirmed this point.In this case, the population ecological niche inlay the warfare that caused will inevitably be in resource space corresponding utilization of resources function f _i(z) produce the phenomenon that field of definition intersects.

For the sake of simplicity, the dimension of later on total hypothetical resource spectrum is 1 dimension, and promptly z is 1 dimension, is expressed as z so the black matrix of z can be removed.The kinetic model of K population competition group is

$\frac{d{x}_i\left(t\right)}{\mathrm{dt}}=r_i{x}_i\left(t\right)-\frac{r_i}{k_i}\underset{j=1}{\overset{K}{\mathrm{\Σ}}}{\mathrm{\α}}_{\mathrm{ij}}^s{x}_i\left(t\right)x_j\left(t\right),i=\mathrm{1,2},\·\·\·,K---\left(4\right)$

In the formula: t represents period; x _i(t) be the scale of t population i in period, x _i(t) 〉=0; r _iBe the intrinsic rate of increase of population i, 0＜r _i＜1; k _iBe the ecological niche capacity of population i,

Coefficient of competition between population i and the population j is calculated by following formula

${\mathrm{\α}}_{\mathrm{ij}}^s=\∫f_i\left(z\right)f_i\left(z\right)\mathrm{dz}---\left(5\right)$

For simplicity, order A is called the competition matrix; S represents competition situation, promptly horizontal trophic structure type.When A gets different forms, represent different competition situations.Common competition matrix A has following form:

(1) normal distribution type.For each population i, have

$f_i\left(z\right)={\left(2\mathrm{\π}{w}_i^2\right)}^{-\frac{1}{2}}\exp (-\frac{{(z-z_i)}^2}{{2w}_i^2}),i=\mathrm{1,2},\·\·\·,K$

In the formula: z _iBe the central point of ecological niche,

Variance for normal distribution.If the central point of population i and population j ecological niche is each other at a distance of d _Ij, can calculate by formula (5) so

${\mathrm{\α}}_{\mathrm{ij}}^1=\frac{1}{2\mathrm{\π}{w}_i{w}_j}\underset{-\∞}{\overset{\∞}{\∫}}\exp \{-\frac{z^2}{{2w}_i^2}-\frac{{(z-d_{\mathrm{ij}})}^2}{{2w}_j^2}\}\mathrm{dz}=\frac{1}{\sqrt{2\mathrm{\π}(w_i^2+w_j^2)}}\exp \{-\frac{d_{\mathrm{ij}}^2}{2(w_i^2+w_j^2)}\}$

If all f _i(z) identical variance is all arranged And sequence numbering is adjacent just to mean that ecological niche is adjacent, then d _Ij=| i-j|d, and

${\mathrm{\α}}_{\mathrm{ij}}^1=\frac{1}{2w\sqrt{\mathrm{\π}}}\exp \{-\frac{{(i-j)}^{2}d}{4w}\}$

When the degree of inlaying certainly of regulation coefficient of competition is 1,

For i, j ∈ 1,2 ..., K} just has

${\mathrm{\α}}_{\mathrm{ij}}^1=a^{{(i-j)}^2}---\left(6\right)$

In the formula: s is horizontal trophic structure type, s=1 herein; A is two tolerance that adjacent ecological niche is inlayed, 0＜a＜1; Get during calculating

With

The value lower limit and the upper limit of expression a,

(a b) is illustrated in uniform random number of [a, b] interval generation to Rand.

It competes matrix A ₁Be expressed as

$A_1={\left[{\mathrm{\α}}_{\mathrm{ij}}^1\right]}_{K\×K}=\left[\begin{array}{ccccc}1& a& a^4& \·\·\·& a^{{(K-1)}^2}\\ a& 1& a& \·\·\·& a^{{(K-2)}^2}\\ a^4& a& 1& \·\·\·& a^{{(K-3)}^2}\\ \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·\\ a^{{(K-1)}^2}& a^{{(K-2)}^2}& a^{{(K-3)}^2}& \·\·\·& 1\end{array}\right]$

(2) Discrete Distribution type.Supposing K population is numbered, make them can be with the absolute value of subscript difference in the size of resource space mutual distance | i-j| measures.Be not difficult to think that warfare strengthens along with population ecological niche spacing and weakens, so coefficient of competition can be expressed as

${\mathrm{\α}}_{\mathrm{ij}}^2=\mathrm{\α}\left(\right|i-j\left|\right)$

In the formula: α (u)〉the 0th, the decreasing function of integer argument u=|i-j|.When being isolated from each other far more, it is more little that its ecological niche is inlayed degree.

If get b _iBe arbitrarily positive constant, and 0＜b _i＜1, for i, j ∈ 1,2 ..., K} has

In the formula: s is horizontal trophic structure type, s=2 herein,

Be the competition matrix A _sIn element, b _iBe two tolerance that adjacent ecological niche is inlayed, 0＜b _i＜1; get during calculating

With

Expression b _iThe value lower limit and the upper limit,

It competes matrix A ₂Be expressed as

$A_2={\left[{\mathrm{\α}}_{\mathrm{ij}}^2\right]}_{K\×K}=\left[\begin{array}{cccccc}1& b_1& b_2& \·\·\·& b_{K-1}& b_K\\ b_1& 1& b_1& b_2& \·\·\·& b_{K-1}\\ b_2& b_1& 1& b_1& \·\·\·& b_{K-2}\\ \·& \·& \·& & \·& \·\\ \·& \·& \·& \·\·\·& \·& \·\\ \·& \·& \·& & \·& \·\\ b_{K-1}& \·\·\·& b_2& b_1& 1& b_1\\ b_K& b_{K-1}& \·\·\·& b_2& b_1& 1\end{array}\right]$

(3) the most contiguous type.If each population in K population only with oneself the most contiguous population competition, and other populations competitions of getting along well.If at this moment with two tolerance that adjacent ecological niche is inlayed of c (0＜c＜1) expression, so, for i, j ∈ 1,2 ..., K} has

In the formula: s is horizontal trophic structure type, s=3 herein;

Be the competition matrix A _sIn element,

C is two tolerance that adjacent ecological niche is inlayed, 0＜c＜1; Get during calculating

With

The value lower limit and the upper limit of expression c,

It competes matrix A ₃Be expressed as

$A_3={\left[{\mathrm{\α}}_{\mathrm{ij}}^3\right]}_{K\×K}=\left[\begin{array}{cccccc}1& c& 0& 0& \·\·\·& 0\\ c& 1& c& 0& \·\·\·& 0\\ 0& c& 1& c& \·\·\·& 0\\ \·& \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·& \·\\ 0& \·\·\·& 0& c& 1& c\\ 0& \·\·\·& 0& 0& c& 1\end{array}\right]$

(4) even type.If each population in K population is all with identical degree d (0＜d＜1) and other population competition, for i, j ∈ 1,2 ..., K} has

In the formula: s is horizontal trophic structure type, s=4 herein;

Be the competition matrix A _sIn element,

D represents that each population is all with identical degree and the competition of other population, 0＜d＜1; Get during calculating

With

The value lower limit and the upper limit of expression d,

It competes matrix A ₄For

$A_4={\left[{\mathrm{\α}}_{\mathrm{ij}}^4\right]}_{K\×K}=\left[\begin{array}{ccccc}1& d& d& \·\·\·& d\\ d& 1& d& \·\·\·& d\\ \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·\\ \·& \·& \·& \·& \·\\ d& d& d& \·\·\·& 1\end{array}\right]$

(5) monotone decreasing type.If each population in K population is all with the degree e (0＜e＜1) and other population competition of monotone decreasing, for i, j ∈ 1,2 ..., K} has

${\mathrm{\α}}_{\mathrm{ij}}^5=e^{|i-j|}---\left(10\right)$

In the formula: s is horizontal trophic structure type, s=5 herein; Be the competition matrix A _sIn element,

E is that each population all competes 0 with degree and other population of monotone decreasing＜e＜1; Get during calculating

With The value lower limit and the upper limit of expression e,

It competes matrix A ₅For

$A_5={\left[{\mathrm{\α}}_{\mathrm{ij}}^5\right]}_{K\×K}=\left[\begin{array}{ccccc}1& e& e^2& \·\·\·& e^{n-1}\\ e& 1& e& \·\·\·& e^{n-2}\\ e^2& e& 1& \·\·\·& e^{n-3}\\ \·& \·& \·& & \·\\ \·& \·& \·& \·\·\·& \·\\ \·& \·& \·& & \·\\ e^{n-1}& e^{n-2}& e^{n-3}& ...& 1\end{array}\right]$

Horizontal trophic structure population competition dynamics operator

Supposing to divide in the ecosystem has M subsystem, is designated as Y={Y ₁, Y ₂..., Y _M, this ecosystem has 5 different horizontal trophic structure types, is designated as C={1, and 2,3,4,5}, wherein the implication of each element is corresponding one by one with { normal distribution type, Discrete Distribution type, the most contiguous type, even type, monotone decreasing type }; Each subsystem has a specific horizontal trophic structure type; For subsystem Y _k∈ Y supposes that its horizontal trophic structure type is s, and s ∈ C, the population number of surviving in this subsystem are K, and the set of planting group number in this subsystem is designated as B={1,2 ..., K}; There is not unnecessary population not belong to any subsystem.Further population of supposition can belong to a plurality of horizontal trophic structure subsystems, therefore has: N=MK.

The PCDO-HNS algorithm is to utilize the population competition dynamic law of the horizontal trophic structure of tool to construct population evolution operator, and with message exchange between the realization population, and then realization is to the search of solution space.

Though the dynamic law of K the population competition group that formula (4) is described is applicable to this feature of scale of population, but we think, feature relevant with its scale in the population has several, the described dynamic law of formula (4) is equally applicable to these features, thereby also is applicable to those corresponding variablees in the trial solution.Which feature is relevant with population scale in the population on earth, and we need not to be concerned about.And, for randomness and the popularity that strengthens search, we in addition can select those features relevant at random with population scale.That is to say that we are applied to the described dynamic law of formula (4) from (x _I1, x _I2..., x _In) in some variable of selecting at random.The evolutionary operator of PCDO-HNS algorithm is as described below.

For horizontal trophic structure type s (s=1,2,3,4,5), subsystem Y _kAdopted this structure type, its population number that comprises is K, and Y _k∈ Y, k=1,2 ..., M, i=1,2 ..., K, then

In the formula:

With

Be respectively period t and period t-1 population i the state value of l feature, and all be nonnegative real number; E ₀, E ₁, E ₂, E ₃, E ₄Be illustrated respectively in feature number set 1,2 ..., the probability upper limit of selecting a feature number to compete, learn mutually, influence each other, undergo mutation and keep sealing among the n} at random, 0 as population i＜E ₀, E ₁, E ₂, E ₃, E ₄≤ 1;

With

The same r of implication _iAnd s _i, just for different horizontal trophic structure types,

With

The value difference; During calculating, get

With The intrinsic rate of increase value lower limit and the upper limit of expression population,

With

The value lower limit and the upper limit of the ecological niche capacity of expression population,

G _SExpression is from subsystem Y _kIn its PSI index be higher than the L that random choose comes out in some populations of population i _SThe set of formed kind of group number of individual population; G _CExpression is from subsystem Y _kThe L that interior random choose comes out _CFormed kind of group number of individual population;

G _MExpression is from subsystem Y _kThe L that interior its PSI index comes out far above random choose in the good variety population of population i _MIndividual population number, V ∈ G _M, u ≠ v ≠ i; α _u, β _uBe constant, 0＜α _u, β _u＜1, get α during calculating _k=Rand (0,1), β _k=Rand (0,1); L _M=m _I+ m _E, m _I〉=2, m _E〉=1, m _IM _E

That the 1st formula in the formula (11) is described is subsystem Y _kCompetitive behavior between an interior K population; That the 2nd formula is described is subsystem Y _kThe learning behavior of interior population i; That the 3rd formula is described is subsystem Y _kThe behavior that influences of interior population i; That the 4th formula is described is subsystem Y _kThe sudden change behavior of interior population i; That the 5th formula is described is subsystem Y _kThe behavior of the maintenance sealing of interior population i;

Described learning behavior is meant: population i is at subsystem Y _kBetween interior active stage, population i is in order to promote the competitive power of self, initiatively to subsystem Y _kOther interior some populations strongr than population i are learnt, and promptly population i is with subsystem Y _kSome characteristic absorption of selecting at random that interior some its PSI indexes of selecting at random are higher than the population of population i are come, and make own strong purpose to reach.

Described interactive behavior is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kThe crawler behavior of other interior population has caused influence to population i, i.e. subsystem Y _kSome features selected at random of interior some populations of selecting at random and the mean value of state value thereof have been passed to the character pair of population i, and it is affected.

Described sudden change behavior is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kThe behavior that some interior its PSI indexes of selecting at random are higher than the special good variety population of population i causes very big influence to population i, i.e. subsystem Y _kIn some select at random but its PSI index has been passed to the character pair of population i far above the difference of some features selected at random of the special good variety population of population i and weighting state value thereof, it is produced greatly changes.

Described self-enclosed behavior is meant: population i is at subsystem Y _kBetween interior active stage, some features of selecting at random of population i are not subjected to subsystem Y _kAny influence of interior other population.

For convenience, the operator that is generated by normal distribution type, Discrete Distribution type, the most contiguous type, even type and monotone decreasing type is called normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, even type competition operator and monotone decreasing type competition operator.Its computing formula is formula (11), but for different horizontal trophic structure types, the parameter in the formula (11) has nothing in common with each other, and promptly for the normal distribution type, the parameter in the formula (11) is got by formula (6); For the Discrete Distribution type, the parameter in the formula (11) is got by formula (7); For the most contiguous type type, the parameter in the formula (11) is got by formula (8); For even type, the parameter in the formula (11) is got by formula (9); For the monotone decreasing type, the parameter in the formula (11) is got by formula (10).

Population is selected operator

The PCDO-HNS algorithm by utilize normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, evenly type competition operator and monotone decreasing type competition operator produce after the population of new generation, adopt and select operator that population of new generation and corresponding parent population are compared one to one, be saved in the colony of future generation than the superior.For optimization problem (1), it selects operator to be described as

In the formula:

In case after the formation of new population, the PCDO-HNS algorithm continues by normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, even type competition operator and monotone decreasing type competition operator population constantly to be developed up to finding optimum solution.

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 K initial solution _K(K ⁿ) construction algorithm.

Algorithm INIT produces the orthogonal arrage L of K initial solution _K(K ⁿ) construction algorithm

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

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

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

x _ij=y _jk，i=1，2，…，K，j=1，2，…，n

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

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

Described PCDO-HNS algorithm comprises the steps:

(1) initialization: make t=0 in period, all parameters that relate to by this algorithm of table 1 initialization;

The obtaining value method of table 1 parameter

(2) divide mating group randomly for M subsystem, make subsystem Y ₁, Y ₂..., Y _MOn population number be K population; A population is assigned in a plurality of subsystems population total N=MK;

(3) specify a kind of horizontal trophic structure type randomly for M subsystem, the horizontal trophic structure type of appointment is the normal distribution type, Discrete Distribution type, the most contiguous type, one of five kinds on even type and monotone decreasing type;

(4) population on each subsystem is carried out initialization according to the orthogonal Latin square generating algorithm, generate initial solution I=1,2 ..., K, k=1,2 ..., M;

(5) carry out following operation:

(6) make period t carry out following step (7)～step (24) from 1 to G circulation; Wherein G is the evolutionary period number;

(7) make subsystem number k carry out following step (8)～step (23) from 1 to M circulation;

(8) make evolution number of times w carry out following step (9)～step (22) from 1 to L circulation; Wherein L is the evolution of the phase weekly number of times of population in each subsystem;

(9) make population i carry out following step (10)～step (21) from 1 to K circulation;

(10) make the feature l of population carry out following step (11)～step (16) from 1 to n circulation;

(11) if subsystem Y _kBe the horizontal trophic structure type of normal distribution type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (6), obtain

(12) if subsystem Y _kBe the horizontal trophic structure type of Discrete Distribution type, then carry out the evolution operator by above-mentioned formula (11), but the model parameter in the formula (11) Determine by formula (7), obtain

(13) if subsystem Y _kBe the horizontal trophic structure type of the most contiguous type, then carry out the evolution operator, but model parameter is definite by formula (8), obtains by formula (11)

(14) if subsystem Y _kBe the horizontal trophic structure type of even type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (9), obtain

(15) if subsystem Y _kBe the horizontal trophic structure type of branch monotone decreasing type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (10), obtain

(16) make l=l+1,, otherwise change step (17) if l≤n then changes above-mentioned steps (11);

(17) component of separating that will exceed feasible zone pushes back feasible zone;

(18) select the trial solution of operational form (12) by population to all new acquisitions

With former trial solution

Carry out selection operation, obtain trial solution of future generation

(19) if the error between globally optimal solution that newly obtains and the last current globally optimal solution of having preserved satisfies minimum requirements ε, then change following step (25);

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

(21) make i=i+1,, otherwise change step (22) if i≤K then changes above-mentioned steps (10)

(22) make w=w+1,, otherwise change step (23) if w≤L then changes above-mentioned steps (9);

(23) make k=k+1,, otherwise change step (24) if k≤M then changes above-mentioned steps (8);

(24) make t=t+1,, otherwise change step (25) if t≤G then changes above-mentioned steps (7);

(25) finish.

Beneficial effect

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

But 1, disclosed by the invention is the novel optimization method-PCDO-HNS algorithm of the global convergence that constructs of a kind of population competition kinetic theory based on horizontal trophic structure.In the PCDO-HNS algorithm, suppose that each subsystem has a specific horizontal trophic structure type at the total a plurality of different subsystems of an ecosystem, but different subsystems can have identical horizontal trophic structure type.For each subsystem, there are some populations to live therein.A population can only have a kind of horizontal trophic structure type, but can occur at a plurality of subsystems, does not have unnecessary population not belong to any subsystem.Population can not shift between each subsystem.The population that in a horizontal trophic structure subsystem, lives with mutual competition, self study, be affected, sudden change and self-enclosed mode survive.Strong population continued growth, weak population then stops growing.The PCDO-HNS algorithm has very strong search capability, and has global convergence, for finding the solution of complex function optimization problem provides a solution.

2, the search capability of PPDO algorithm is very strong.The PCDO-HNS algorithm includes normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, evenly type is competed operator and monotone decreasing type competition operator, and these operators have significantly increased its search capability.

3, the model parameter value is simple.Adopt random device to determine the correlation parameter in each operator in the algorithm, both significantly reduced parameter input number, make model more can express actual conditions again.

4, the characteristics of PPDO algorithm of the present invention are as follows:

1) time complexity.The time complexity computation process of PCDO-HNS algorithm is as shown in table 2, and its time complexity and develop cycle G, ecosystem population scale N, variable number n, horizontal trophic structure type be population number K among 5, subsystem number M, each subsystem and cycle evolution number of times L thereof, it is relevant respectively to compete time complexity and other non-productive operations of operator.

The time complexity reckoner of table 2PCDO-HNS algorithm

Operation	Time complexity	Maximum cycle periods
			Initialization	O(6nN+M(n 2+5n+9)+2n)	1
Generate M horizontal trophic structure subsystem	O(1+6K)	GN
			Normal distribution type competition operator	O(L(nK+n 2+9n+7))	GN
Discrete Distribution type competition operator	O(L(nK+n 2+9n+7))	GN
			The most contiguous type competition operator	O(L(nK+n 2+9n+7))	GN
Evenly type is competed operator	O(L(nK+n 2+9n+7))	GN
			Monotone decreasing type competition operator	O(L(nK+n 2+9n+7))	GN
Objective function calculates	O(n 2)	GN
			Select operator	O(3n)	GN
Separate component and push back feasible zone	O(3n)	GN
			Result's output	O(n)	1

2) the PCDO-HNS algorithm has global convergence.The theoretical analysis of its reason is as follows:

Before the global convergence of proof PCDO-HNS algorithm, introduce earlier by Iisufescu M and in document " Finite Markov Processes and Their Applications, Wiley:Chichester, 1980 ", propose following theorem:

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 $P^{\′}=\left[\begin{array}{ccc}C& \·\·\·& 0\\ R& \·\·\·& T\end{array}\right],$ Wherein C is M rank basis stochastic matrix and R ≠ 0, and T ≠ 0 then has

$P^{\′\∞}=\underset{k\→\∞}{\lim }{P}^{\′k}=\underset{k\→\∞}{\lim }\left[\begin{array}{ccc}{C}^k& \·\·\·& 0\\ \underset{i=1}{\overset{k-1}{\mathrm{\Σ}}}{T}^{i}R{C}^{k-i}& \·\·\·& T^k\end{array}\right]=\left[\begin{array}{ccc}{C}^{\∞}& \·\·\·& 0\\ R^{\∞}& \·\·\·\·& 0\end{array}\right]$

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 ".Utilize theorem 1, it be easy to show that the global convergence of PCDO-HNS algorithm.Its proof procedure is as follows:

Known that by algorithm PCDO-HNS algorithm the ecosystem is a discrete space, with M subsystem, each subsystem is made up of K population, and all populations of each subsystem are rearranged into N population, and N=MK forms new population sequence and is

Each population

Be in continuous real number space value, each population is a trial solution of optimization problem (1), and its target function value is

(calculating by formula (2)), then the formed set of the state of all populations is

$F=\left\{F\left(X_i^t\right)\right|X_i^t\&Element;S\}$

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

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

Order

Expression

In the state of j population.Population is in evolutionary process, and (i, (k l) can be expressed as r j) to transfer to another state from a state ^{I, j}→ x ^{K, l}, then not imitative 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 p _{Ij, k}, from

In arbitrary state arrive In the transition probability of arbitrary state be p _{I, k}, then have $p_{\mathrm{ij},k}=\underset{l=1}{\overset{\left|X_S^k\right|}{\mathrm{\&Sigma;}}}{p}_{\mathrm{ij},\mathrm{kl}};$ $\underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}{p}_{\mathrm{ij},k}=1;$ p _i，k≥p _ij，k $p_{i,k}\&GreaterEqual;p_{\mathrm{ij},k}\&RightArrow;\underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}{p}_{i,k}\&GreaterEqual;\underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}{p}_{\mathrm{ij},k}=1,$ And $0\&le;\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{P}_{i,k}\&le;1,$ So $\underset{k=1}{\overset{N}{\mathrm{\&Sigma;}}}{p}_{i,k}=1---\left(B\right)$

Lemma 1 in PCDO-HNS algorithm, $\&ForAll;X^{i,j}\&Element;X_S^i,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N,j=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,\left|X_S^i\right|,$ Satisfy

$\&ForAll;k>i,p_{i,k}=0---\left(C\right)$

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

(1) proof of formula (C).If state i is t population X in period ^tState, this state i is exactly this population oneself best condition of reaching so far certainly.In PDO-HNS 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 this population oneself best condition that reaches) 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＞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, certainly must be this population oneself best condition of reaching up to now, at t+1 in period, this population selects to compete the evolution behavior at random in the hope of transferring on the better state, has 5 kinds of competition evolution behaviors such as normal distribution type, Discrete Distribution type, the most contiguous type, even type and monotone decreasing type available:

Situation 1: current population is selected normal distribution type competition evolution behavior, and its probability of happening is p ₁If this population can be transferred to the state more excellent than current state i and get on, then this population will be with probability P ₁=p ₁p _SBe updated to new state, obviously P ₁0, assign a topic to such an extent that demonstrate,prove.Wherein, p _SBe the probability of selecting operator to run succeeded.

Situation 2: current population is selected Discrete Distribution type competition evolution behavior, and its probability of happening is p ₂If this population can be transferred to the state more excellent than current state i and get on, then this population will be with probability P ₂=p ₂p _SBe updated to new state, obviously P ₂0, assign a topic to such an extent that demonstrate,prove.

Situation 3: current population is selected the most contiguous type competition evolution behavior, and its probability of happening is p ₃If this population can be transferred to the state more excellent than current state i and get on, then this population will be with probability P ₃=p ₃p _SBe updated to new state, obviously P ₃0, assign a topic to such an extent that demonstrate,prove.

Situation 4: current population is selected even type competition evolution behavior, and its probability of happening is p ₄If this population can be transferred to than the more excellent state of current good state i, then this population will be with probability P ₄=p ₄p _SBe updated to new state, obviously P ₄0, assign a topic to such an extent that demonstrate,prove.

Situation 5: current population is selected monotone decreasing type competition evolution behavior, and its probability of happening is p ₅If this population can be transferred to the state more excellent than current state i and get on, then this population will be with probability P ₅=p ₅p _SBe updated to new state, obviously P ₅0, assign a topic to such an extent that demonstrate,prove.

In any one time period, one of essential above-mentioned 5 kinds of behaviors of arbitrary population, i.e. p ₁+ p ₂+ p ₃+ p ₄+ p ₅=1, comprehensive above-mentioned various situations can get general probability and are

$P_d=\underset{i=1}{\overset{5}{\mathrm{\&Sigma;}}}{P}_i=(p_1+p_2+p_3+p_4+p_5)p_S=p_S$

Because of 0≤p _S≤ 1, easily know P _d〉=0; If P _d0, assign a topic to such an extent that demonstrate,prove; If P always _d=0, p always is described _S=0, promptly in the formula (12)

Probability of occurrence is 0 forever, and this shows that the population evolution has arrived globally optimal solution state X ^*, because of having revised current state once more forever.

Comprehensive above-mentioned situation can get

Card is finished.

Theorem 2PCDO-HNS 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

Get according to lemma 1 Chinese style (D) conclusion

But transition matrix P ' is N rank reduction stochastic matrixes 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 know that by formula (B) the probability sum of every row among the transition matrix P ' is 1.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 PCDO-HNS 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 PCDO-HNS algorithm.

(5) carry out computation optimization, the search globally optimal solution.

(6) optimum solution y _iAfter the acquisition, obtain x in the following method in profit _iGet 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.

Provide instantiation explanation embodiment below.

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

$\max f\left(X\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(-x_i^2+10\cos \left(2\mathrm{\&pi;}{x}_i\right)-10)$

s.t.-10≤x _i≤10,i＝1,2,…,n

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

$\min f\left(X\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(-x_i^2+10\cos \left(2\mathrm{\&pi;}{x}_i\right)+10)$

s.t.-10≤x _i≤10,i＝1,2,…,n

(3) make x _i=20y _i-10, Y=(y ₁, y ₂..., y _n), then

$\min f\left(Y\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(100{({2y}_i-1)}^2-10\cos \left(20\mathrm{\&pi;}({2y}_i-1)\right)+10)$

s.t.0≤y _i≤1,i＝1,2,…,n

(4) determine the parameter of algorithm.Each parameter value method and foundation thereof are as described below in the table 3.

Each parameter value of table 3 solution procedure

(5) find the solution with the PCDO-HNS algorithm, the solving result of optimization problem f (Y) is as shown in table 4.

Table 4PCDO-HNS test of heuristics effect

The optimum solution of (6) trying to achieve is at y _iWithin [0.49999989,0.50000012], get x after the conversion _iIn [4.0E-6,2.0E-6], i=1,2 ..., n.

Claims (1)

1. the population of the horizontal trophic structure of tool competes Dynamics Optimization method-PCDO-HNS algorithm, and it is characterized in that: establishing the function optimization problem that will solve is:

minf(X)

$s.t.\left\{\begin{array}{c}{g}_i\left(X\right)\&GreaterEqual;0,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,I\\ h_i\left(X\right)0,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,E\\ X\&Element;S\&Subset;R^n,X\&GreaterEqual;0\end{array}\right.---\left(1\right)$

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;

Regard the solution space of optimization problem (1) as an ecosystem, this ecosystem has a plurality of different subsystems, each subsystem has a specific horizontal trophic structure type, this specific horizontal trophic structure type is normal distribution type, Discrete Distribution type, the most contiguous type, evenly one of five kinds on type and monotone decreasing type, but different subsystems can have identical horizontal trophic structure type; For each subsystem, there are some populations to live therein, a population can occur at a plurality of subsystems, does not have unnecessary population not belong to any subsystem; Population can not shift between each subsystem, the population that lives in a subsystem is competed each other, but the behavior that also is attended by mutual study, influences each other, undergos mutation and keeps to seal exists, strong population continued growth, and weak population then stops growing;

Described learning behavior is meant: population i is at subsystem Y _kBetween interior active stage, population i is in order to promote the competitive power of self, initiatively to subsystem Y _kOther interior some populations strongr than population i are learnt, and promptly population i is with subsystem Y _kSome characteristic absorption of selecting at random that interior some its PSI indexes of selecting at random are higher than the population of population i are come, and make own strong purpose to reach;

Described interactive behavior is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kThe crawler behavior of other interior population has caused influence to population i, i.e. subsystem Y _kSome features selected at random of interior some populations of selecting at random and the mean value of state value thereof have been passed to the character pair of population i, and it is affected;

Described sudden change behavior is meant: population i is at subsystem Y _kBetween interior active stage, subsystem Y _kThe behavior that some interior its PSI indexes of selecting at random are higher than the special good variety population of population i causes very big influence to population i, i.e. subsystem Y _kIn some select at random but its PSI index has been passed to the character pair of population i far above the difference of some features selected at random of the special good variety population of population i and weighting state value thereof, it is produced greatly changes;

Described self-enclosed behavior is meant: population i is at subsystem Y _kBetween interior active stage, some features of selecting at random of population i are not subjected to subsystem Y _kAny influence of interior other population;

The search volume of optimization problem is corresponding with the ecosystem, and a population in this ecosystem is corresponding to the trial solution of an optimization problem, and a feature in the population is corresponding to a variable in the optimization problem trial solution; So the characteristic number of population is identical with the variable number of trial solution; The fitness index of population is the target function value of PSI index corresponding to optimization problem, and good trial solution correspondence has the population of higher PSI index, promptly strong population; The trial solution correspondence of difference has the population of low PSI index, i.e. Xu Ruo population;

The PCDO-HNS algorithm is to utilize the population competition kinetic model of the horizontal trophic structure of tool to construct population evolution operator, these operators comprise normal distribution type competition operator, Discrete Distribution type competition operator, the most contiguous type competition operator, evenly type is competed operator and monotone decreasing type competition operator, these operators are used to produce after the population of new generation, adopt and select operator that population of new generation and corresponding parent population are compared one to one, be saved in the colony of future generation than the superior; In case after new population formed, the PCDO-HNS algorithm continued by above-mentioned operator population constantly to be developed up to finding optimum solution;

Described PCDO-HNS algorithm comprises the steps:

(1) initialization: make t=0 in period, all parameters that relate to by this algorithm of table 1 initialization;

The obtaining value method of table 1 parameter

(2) divide mating group randomly for M subsystem, make subsystem Y ₁, Y ₂..., Y _MOn population number be K population; A population is assigned in a plurality of subsystems population total N=MK;

(3) specify a kind of horizontal trophic structure type randomly for M subsystem, the horizontal trophic structure type of appointment is the normal distribution type, Discrete Distribution type, the most contiguous type, one of five kinds on even type and monotone decreasing type;

(4) population on each subsystem is carried out initialization according to the orthogonal Latin square generating algorithm, generate initial solution I=1,2 ..., K, k=1,2 ..., M;

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)/(K-1)，i=1，2，…，n；j=1，2，…，K。

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

x _ij=y _jk，i=1，2，…，K，j=1，2，…，n

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

The determined K of an above-mentioned algorithm initial solution is

I=1,2 ..., K;

(5) carry out following operation:

(6) make period t carry out following step (7)～step (24) from 1 to G circulation; Wherein G is the evolutionary period number;

(7) make subsystem number k carry out following step (8)～step (23) from 1 to M circulation;

(8) make evolution number of times w carry out following step (9)～step (22) from 1 to L circulation; Wherein L is the evolution of the phase weekly number of times of population in each subsystem;

(9) make population i carry out following step (10)～step (21) from 1 to K circulation;

(10) make the feature l of population carry out following step (11)～step (16) from 1 to n circulation;

(11) if subsystem Y _kBe the horizontal trophic structure type of normal distribution type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (6), obtain

Described formula (6) is:

${\mathrm{\&alpha;}}_{\mathrm{ij}}^s=a^{{(i-j)}^2}---\left(6\right)$

In the formula: s is horizontal trophic structure type, s=1 herein;

Coefficient of competition when being s between population i and the population j for horizontal trophic structure type; A is two tolerance that adjacent ecological niche is inlayed, 0＜a＜1; Get during calculating

With

The value lower limit and the upper limit of expression a,

(a b) is illustrated in uniform random number of [a, b] interval generation to Rand;

Described formula (11) is:

In the formula:

With

Be respectively period t and period t-1 population i the state value of l feature, and all be nonnegative real number; E ₀, E ₁, E ₂, E ₃, E ₄Be illustrated respectively in feature number set 1,2 ..., the probability upper limit of selecting a feature number to compete, learn mutually, influence each other, undergo mutation and keep sealing among the n} at random, 0 as population i＜E ₀, E ₁, E ₂, E ₃, E ₄≤ 1;

With

The same r of implication _iAnd s _i, just for different horizontal trophic structure types,

With The value difference; During calculating, get

With The intrinsic rate of increase value lower limit and the upper limit of expression population,

With The value lower limit and the upper limit of the ecological niche capacity of expression population,

G _SExpression is from subsystem Y _kIn its PSI index be higher than the L that random choose comes out in some populations of population i _SThe set of formed kind of group number of individual population; G _CExpression is from subsystem Y _kThe L that interior random choose comes out _CFormed kind of group number of individual population;

G _MExpression is from subsystem Y _kThe L that interior its PSI index comes out far above random choose in the good variety population of population i _MIndividual population number,

V ∈ GM, u ≠ v ≠ i; α _u, β _uBe constant, 0＜α _u, β _u＜1, get α during calculating _k=Rand (0,1), β _k=Rand (0,1); L _M=m _I+ m _E, m _I〉=2, m _E〉=1, m _IM _E

That the 1st formula in the formula (11) is described is subsystem Y _kCompetitive behavior between an interior K population; That the 2nd formula is described is subsystem Y _kThe learning behavior of interior population i; That the 3rd formula is described is subsystem Y _kThe behavior that influences of interior population i; That the 4th formula is described is subsystem Y _kThe sudden change behavior of interior population i; That the 5th formula is described is subsystem Y _kThe behavior of the maintenance sealing of interior population i;

The 1st formula in the described formula (11) is from formula (4):

$\frac{{\mathrm{dx}}_i\left(t\right)}{\mathrm{dt}}=r_i{x}_i\left(t\right)-\frac{r_i}{k_i}\underset{j=1}{\overset{K}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{\mathrm{ij}}^s{x}_i\left(t\right)x_j\left(t\right),i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,K---\left(4\right)$

In the formula: t represents period; x _i(t) be the scale of t population i in period, x _i(t) 〉=0; r _iBe the intrinsic rate of increase of population i, 0＜r _i＜1; k _iBe the ecological niche capacity of population i, k _i1;

Coefficient of competition when being s between population i and the population j for horizontal trophic structure type;

(12) if subsystem Y _kBe the horizontal trophic structure type of Discrete Distribution type, then carry out the evolution operator by above-mentioned formula (11), but the model parameter in the formula (11)

Determine by formula (7), obtain

In the formula: s is horizontal trophic structure type, s=2 herein,

Be the competition matrix A _sIn element,

b _iBe two tolerance that adjacent ecological niche is inlayed, 0＜b _i＜1; get during calculating

With

Expression b _iThe value lower limit and the upper limit,

(13) if subsystem Y _kBe the horizontal trophic structure type of the most contiguous type, then carry out the evolution operator, but model parameter is definite by formula (8), obtains by formula (11)

In the formula: s is horizontal trophic structure type, s=3 herein;

Be the competition matrix A _sIn element,

C is two tolerance that adjacent ecological niche is inlayed, 0＜c＜1; Get during calculating

With

The value lower limit and the upper limit of expression c,

(14) if subsystem Y _kBe the horizontal trophic structure type of even type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (9), obtain

In the formula: s is horizontal trophic structure type, s=4 herein;

Be the competition matrix A _sIn element,

D represents that each population is all with identical degree and the competition of other population, 0＜d＜1; Get during calculating

With The value lower limit and the upper limit of expression d,

(15) if subsystem Y _kBe the horizontal trophic structure type of branch monotone decreasing type, then carry out the evolution operator by formula (11), but the model parameter in the formula (11)

Determine by formula (10), obtain

${\mathrm{\&alpha;}}_{\mathrm{ij}}^s=e^{|i-j|}---\left(10\right)$

In the formula: s is horizontal trophic structure type, s=5 herein; Be the competition matrix A _sIn element,

E is that each population all competes 0 with degree and other population of monotone decreasing＜e＜1; Get during calculating

With

The value lower limit and the upper limit of expression e,

(16) make l=l+1,, otherwise change step (17) if l≤n then changes above-mentioned steps (11);

(17) component of separating that will exceed feasible zone pushes back feasible zone;

(18) select the trial solution of operational form (12) by population to all new acquisitions

With former trial solution Carry out selection operation, obtain trial solution of future generation

In the formula: $V_i^k\left(t\right)=(v_{i1}^k\left(t\right),v_{i2}^k\left(t\right),\&CenterDot;\&CenterDot;\&CenterDot;,v_{\mathrm{in}}^k\left(t\right));$ $X_i^k(t-1)=(x_{\mathrm{il}}^k(t-1),x_{i2}^k(t-1),\&CenterDot;\&CenterDot;\&CenterDot;,x_{\mathrm{in}}^k(t-1));$ Function With

Calculate by formula (3)

$\mathrm{PSI}\left(X_i\right)=F_{\max }-F\left(X_i\right)---\left(3\right)$

In the formula: F _MaxBe very large arithmetic number, be used for the trial solution that does not satisfy constraint condition is punished; And F (X _i) calculate by formula (2):

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

(19) if the error between globally optimal solution that newly obtains and the last current globally optimal solution of having preserved satisfies minimum requirements ε, then change following step (25);

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

(21) make i=i+1,, otherwise change step (22) if i≤K then changes above-mentioned steps (10)

(22) make w=w+1,, otherwise change step (23) if w≤L then changes above-mentioned steps (9);

(23) make k=k+1,, otherwise change step (24) if k≤M then changes above-mentioned steps (8);

(24) make t=t+1,, otherwise change step (25) if t≤G then changes above-mentioned steps (7);

(25) finish.