A combinatorial particle swarm optimisation

Locating multiple optima using particle swarm optimizationR.Brits a ,A.P.Engelbrechta,*,F.van den Bergh b a Department of Computer Science,University of Pretoria,South Africab Meraka Institute,CSIR,Pretoria,South AfricaAbstractMany scientific and engineering applications require optimization methods to find more than one solution to multi-modal optimization problems.This paper presents a new particle swarm optimization (PSO)technique to locate and refine multiple solutions to such problems.The technique,NichePSO,extends the inherent unimodal nature of the standard PSO approach by growing multiple swarms from an initial particle population.Each subswarm represents a different solution or niche;optimized individually.The outcome of the NichePSO algorithm is a set of particle swarms,each representing a unique solution.Experimental results are provided to show that NichePSO can successfully locate all optima on a small set of test functions.These results are compared with another PSO niching algorithm,lbest PSO,and two genetic algorithm niching approaches.The influence of control parameters is investigated,including the relationship between the swarm size and the number of solutions (niches).An initial scalability study is also done.Ó2007Elsevier Inc.All rights reserved.Keywords:Particle swarm optimization;Niching;Speciation1.IntroductionFunction optimization is the process of finding an optimal solution to an objective function describing a problem.Optimization can be either a minimization or maximization task.Optimization problems can be broadly categorized into unimodal and multi-modal problems.Unimodal problems have a single global opti-mum,x *,subject to (assuming minimization)f ðx ÃÞ6f ðx Þ8x 2R n ;where f ðx Þ:R n !R is the objective function and n is the dimension of the search space.Multi-modal prob-lems,on the other hand,have more than one optimum.These optima may all be global optima,or a mixture of global and local optima.A local optimum,x ÃL ,is subject to (assuming minimization)f ðx ÃL Þ6f ðx Þ8x 2L ;where L &R n .0096-3003/$-see front matter Ó2007Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2006.12.066*Corresponding author.E-mail addresses:rbrits@cs.up.ac.za (R.Brits),engel@cs.up.ac.za (A.P.Engelbrecht),fvdbergh@csir.co.za (F.van denBergh).Applied Mathematics and Computation 189(2007)1859–18831860R.Brits et al./Applied Mathematics and Computation189(2007)1859–1883Many scientific and engineering optimization problems have convoluted search spaces with large numbers of optima.In the case of more than one global optimum,algorithms are needed to obtain all these solutions.It may also be beneficial to locate all,or as many as possible,local optima.Function optimization has received extensive research attention,and several machine learning techniques such as neural networks[3,16],evolutionary algorithms[1],and swarm intelligence-based algorithms [23,12,13],have been developed and applied successfully to solve a wide range of complex optimization prob-lems.By far the largest part of this research concentrated on developing algorithms that can locate only a single solution.However,evolutionary algorithms research has produced a number of approaches tofind multiple solutions[19,27,18].These evolutionary algorithms are generally referred to as niching or speciation algorithms. Each possible solution,known as a niche,is represented by a grouping of homogeneous GA individuals.Eberhart and Kennedy recently introduced the particle swarm optimization(PSO)approach[22].It is sim-ilar to evolutionary algorithms in that it evolves a group of candidate solutions.PSO,however,allows each individual to maintain a memory of the best solution that it has found and the best solution found in the indi-vidual’s neighborhood.Each individual’s traversal of the search space is then influenced by its own memory of best positions,with the individual moving towards a stochastically weighted average of these best positions [38,41].The PSO algorithm has been shown to successfully solve a variety of unimodal optimization problems [35].Several techniques have been proposed to improve the PSO algorithm’s traversal of the search space [21,25,30,36,38,39].Attempts have been made to solve multi-modal optimization problems with PSO,in the form of niching techniques.Parsopoulos and Vrahatis[31]proposed an extension to a PSO convergence rate improvement technique as a sequential niching approach.The technique locates multiple optima by adapting the objective function’sfit-ness landscape each time a new solution is located.The nbest PSO is a parallel niching technique used to locate multiple solutions to simple systems of equations[4,5].This paper presents and empirically analyses the niching particle swarm optimization algorithm,NichePSO. NichePSO is aimed at locating multiple,optimal solutions to multi-modal optimization problems.Niches are identified by monitoring thefitness of individual particles,and growing subswarms from the initial particle swarm ing a global optimization strategy,subswarms then refine the solution represented by the niche.While the focus of this paper is on niching techniques for locating multiple solutions to multi-modal opti-mization problems,niching techniques have also been used to solve multi-objective optimization problems.In the case of multiple objectives,the objective function consists of K(possibly conflicting)sub-objectives,where the task is to optimize these sub-objectives concurrently.The objective function,f,is expressed as fðxÞ¼f f1ðxÞ;f2ðxÞ;...;f KðxÞg.Multi-objective optimization algorithmsfind good trade-offs between conflict-ing objectives rather than a single solution.A set of such solutions is referred to as the Pareto-optimal set. Several evolutionary algorithm approaches to multi-objective optimization have been developed.For an extensive treatment refer to[7,9].Recently,PSO techniques have been developed for solving multi-objective problems[7,8,15,20,32].It is significant to make a clear distinction between multi-objective and multi-modal optimization problems: Where solutions to multi-objective problems represent a combination of(possibly conflicting)solutions to each of the subgoals f iðxÞin f,solutions to multi-modal optimization problems represent possible values for x under a single objective function f.This is the main focus of our study:niching algorithms for multi-modal,single-objective problems.The standard PSO algorithm and extensions toit as used within the NichePSO are discussed in Section2. Section3presents an introduction to niching and existing niching techniques,introduced both in thefields of genetic algorithms and particle swarm optimizers.NichePSO is presented in Section4,with experimentalfind-ings reported in Section5.Section5also includes an empirical comparison between niching algorithms.2.Particle swarm optimizersThis section presents and discusses the original PSO algorithm.Modifications and improvements to the PSO algorithm have been suggested by several authors.This section discusses PSO improvements used bythe NichePSO algorithm,as well as improvements which have influenced it.The section ends with a short dis-cussion on the ability of the standard PSO to locate multiple solutions.2.1.The standard PSOParticle swarm optimizers are optimization algorithms modeled after the social behavior of birds in aflock [22].PSO is a population based search process where individuals,referred to as particles,are grouped into a swarm.Each particle in a swarm represents a candidate solution to the optimization problem.In a PSO sys-tem,each particle is‘‘flown’’through the multidimensional search space,adjusting its position in search space according to its own experience and that of neighboring particles.A particle therefore makes use of the best position encountered by itself and that of its neighbors to position itself toward an optimal solution.The effect is that particles‘‘fly’’toward a minimum,while still searching a wide area around the best solution.The per-formance of each particle(i.e.the‘‘closeness’’of a particle to the global optimum)is measured using a prede-finedfitness function which encapsulates the characteristics of the optimization problem.Each particle i maintain a current position,x i,current velocity,v i,and personal best position,y i.For the purposes of this paper,x i represents a position in an unconstrained,continuous search space.The personal best position associated with a particle i is the best position that the particle has visited thus far,i.e.a position that yielded the highestfitness value for that particle.If f denotes the objective function to be minimized,then the personal best of a particle at a time step t is updated asy i ðtþ1Þ¼yiðtÞif fðx iðtþ1ÞÞP fðy iðtÞÞ;x iðtþ1Þif fðx iðtþ1ÞÞ<fðy iðtÞÞ:ð1ÞDifferent PSO models have been developed based on the neighborhood topology particles use to exchange information about the search space[24].In the gbest model,which is used in this paper,the best particle is determined from the entire swarm and all other particlesflock towards this particle.If the position of the best particle is denoted by the vector^y,then^yðtÞ2f y0;y1;...;ysgj fð^yðtÞÞ¼min f fðy0ðtÞÞ;fðy1ðtÞÞ;...;fðy sðtÞÞg;ð2Þwhere s is the total number of particles in the swarm.For each iteration of a gbest PSO,the j th-dimension of particle i’s velocity vector,v i,and its position vector,x i,is updated as follows:v i;jðtþ1Þ¼wv i;jðtÞþc1r1;jðtÞðy i;jðtÞÀx i;jðtÞÞþc2r2;jðtÞð^y i;jðtÞÀx i;jðtÞÞ;ð3Þx iðtþ1Þ¼x iðtÞþv iðtþ1Þ;ð4Þwhere w is the inertia weight,c1and c2are the acceleration constants and r1;jðtÞ;r2;jðtÞ$Uð0;1Þ.Upper and lower bounds are usually specified on v i to avoid too rapid movement of particles in the search space;that is, v i;j is clamped to the range½Àv max;j;v max;j .The inertia weight,w,was introduced by Shi and Eberhart[34]to control the influence of the velocity vector on a particle’s position.Decreasing w from a relatively large value to a small value over time,results in rapid initial exploration of the search space,and facilitates later explo-ration.Small w-values result in small adaptations to particle positions,effectively yielding a local search.The PSO algorithm performs repeated applications of the update equations until a specified number of iter-ations has been exceeded,or until velocity updates are close to zero.The reader is referred to[38,41]for a study of the relationship between the inertia weight and the acceler-ation constants in order to select values that will ensure convergent behavior.2.2.The guaranteed convergence particle swarm optimizerThe gbest algorithm exhibits an unwanted property:when x i¼y i¼^y(for any particle i),the velocity update in Eq.(3)depends only on the w v iðtÞterm.When a particle approaches the global best solution,its velocity approaches zero,implying that eventually all particles will stop moving.This behavior does not guar-antee convergence to a global best solution,or even a local best,only to a best position found thus far[38,41]. Van den Bergh et al.introduced a new algorithm,called the Guaranteed Convergence PSO(GCPSO)[38,40], R.Brits et al./Applied Mathematics and Computation189(2007)1859–188318611862R.Brits et al./Applied Mathematics and Computation189(2007)1859–1883to pro-actively counteract this behavior in a particle swarm.Let s be the index of the global best particle.The velocity and position updates for the global best particle are then redefined to bev s;jðtþ1Þ¼Àx s;jðtÞþ^y jðtÞþwv s;jðtÞþqðtÞð1À2r2;jÞ;ð5Þx s;jðtþ1Þ¼^y jðtÞþwv s;jðtÞþqðtÞð1À2r2;jÞ:ð6ÞThe termÀx s‘resets’the particle’s position to the global best position^y;w v s signifies a search direction,and qðtÞð1À2r2ðtÞÞadds a random search term to the equation.The parameter q(t)is dynamically adapted to con-trol the size of the bounding box around yˆwithin which a local search is conducted to force a change in the value of yˆ,thereby preventing the above problem[38,40].2.3.Niching ability of standard PSOThis section provides a short summary of[14],where it is shown that the standard gbest PSO cannot locate multiple solutions,and that the standard lbest PSO is inefficient in doing so.For the gbest PSO,Van den Bergh and Engelbrecht[38,41],and Clerc and Kennedy[6]formally proved that all particles converge on a single attractor,which is a weighted average of the global best position and the personal best position of the particle. These formal proofs clearly show that the gbest PSO cannot locate,in the same execution of the algorithm, more than one solution.If the gbest PSO is executed a number of times,each time from different initial con-ditions,it may be the case that more than one unique solution is found.However,there is no such guarantee. In fact,there is not even a guarantee that the solution found is a local optimum,as proven in[38].The proof of convergence to the weighted average of personal best and global best positions have been done with respect to gbest PSO,and cannot be directly applied to lbest PSO.Based on this proof,the only thing that can be said is that for each neighborhood,a particle converges to a weighted average of its personal best and neighborhood best solutions.Since neighborhoods are constructed based on particles indices with some over-lap between neighborhoods,it is intuitively expected that the neighborhoods(and therefore all particles),will eventually converge on the same point.However,no such proof exist.Engelbrecht et al.[14]empirically showed that,for the lbest PSO,particles do form subgroups,which can be perceived as niches.Even if these subgroups can be considered as niches,it was shown in[14]that only a small number of solutions are found, and only when large swarm sizes are used.The conclusion from[14]is that the standard lbest PSO is inefficient in locating multiple solutions.For sake of completeness,Section5summarizes the results from[14]in com-parison with the results of NichePSO.The inability of gbest PSO and the inefficiency of lbest PSO in locating multiple solutions motivates research in the modification of the standard PSO to promote niching within a single swarm.3.Niching techniquesA number of algorithms have been suggested tofind multiple solutions to multi-modal optimization prob-lems using genetic algorithms,and to a lesser extend,PSOs.In GA parlance,optimization techniques that locate multiple optima in multi-modal function optimization problems are known as niching techniques.Both GAs and PSOs use a population of‘agents’(individuals,particles)partitioned in some way to focus on and locate different possible solutions in a single search space.Each subgroup in the partitioned population is known as a species.The behavioral pattern of individuals competing for the use of a resource in a subgroup and between elements in a subgroup,is known as speciation.Section3.1defines niching,while the remainder of this section summarizes only those niching methods from which NichePSO borrowed concepts,and those methods used in the empirical analysis done for this paper.3.1.What is niching?In an environment where a large number of individuals compete for the use of available resources,behav-ioral patterns emerge where individuals are organized into subgroups based on their resource requirements. Horn defines niching as a‘‘form of cooperation aroundfinite,limited resources,resulting in the lack of compe-tition between such areas,and causing the formation of species for each niche’’[19].Niches are thus partitions of an environment,and species are partitions of a population competing within the environment.Localization of competition is introduced by simply sharing resources among individuals competing for it.The terms niche and species can be used interchangeably.As an example,a school offish that live in a certain part of the ocean compete with each other for access to a potentially limited food supply.Food may not be available everywhere in their environment.Certainfish may learn to live in a small area around a food source,while others may learn to roam their environment and only feed when they require nourishment.If there was to be a single food source,it is a reasonable expectation that allfish would eventually exhibit similar behavior.They would all be required tofind food in the same place,and encounter the same resistance from otherfish.The social interaction and adaptation of individuals in an environment around multiple resources form the basis for the study of niching techniques with evolutionary optimization algorithms.In the evolutionary con-text,Horn defines implicit niching as the sharing of resources,and explicit niching as the sharing offitness.Niching methods can be categorized as either being sequential or parallel:•Sequential niching(or temporal niching)techniques develop niches sequentially over time.As niches are dis-covered,the search space of a problem is adapted to repel other individuals from traversing the area around the recently located solution.The search is repetitively applied to the adapted search space in order to focus on unexplored areas[2].•Parallel niching forms and maintains several different niches simultaneously.The search space is not mod-ified.Parallel niching techniques therefore not only depend onfinding a good measure to locate possible solutions,but also need to organize individuals in a way that maintains their organization in the search space over time,to populate locations around solutions[27,17,19,29].Regardless of the way in which niches are found(i.e.in parallel or sequentially),the distribution of indi-viduals can be formalized in a number of ways,according to their speciation behavior[27]:•Sympatric speciation occurs when individuals form species that coexist in the same search space,but evolve to exploit different resources(or more formally,different ecological niches).•Allopatric speciation differentiates between individuals based on spatial isolation in a search space.No inter-species communication takes place,and subspecies can develop only through deviation from the available ‘genetic’information.•Parapatric speciation allows new species to form as a result of segregated species sharing a common border.Communication between the initial species may not have been encouraged or intended.The PSO niching approach presented in Section4may be classified as using an allopatric speciation approach.Allopatric speciation will therefore be a more prevalent issue of discussion,as it defines the goals of multi-modal function optimization.3.2.Fitness sharingFitness sharing is one of the earliest GA niching techniques,originally introduced as a population diversity maintenance technique[17].It is a parallel,explicit niching approach.The algorithm regards each niche as a finite resource,and shares this resource among all individuals in the niche.Individuals are encouraged to pop-ulate a particular area of the search space by adapting theirfitness based on the number of other individuals that populate the same area.Thefitness f i of individual i is adapted to its sharedfitness:f0 i ¼f iPjshðd i;jÞ:A common sharing function isshðdÞ¼1Àðd=r shareÞa if d<r share0otherwise:R.Brits et al./Applied Mathematics and Computation189(2007)1859–18831863The symbol d represents a distance calculated between individuals i and j.The distance measure may be geno-typic or phenotypic,depending on the optimization problem at hand.If the sharing functionfinds that d i;j is less than r share,it returns a value in the range[0,1],which increases as d i;j decreases.The more similar i and j, the lower their individualfitnesses will become.Sharing assumes that the number of niches can be estimated, i.e.it must be known prior to the application of the algorithm how many niches there are.It is also assumed that niches occur at least a minimum distance,2r share,from each other.3.3.Sequential nichingSequential niching(SN),introduced by Beasley et al.[2],identifies multiple solutions by adapting an opti-mization problem’s objective function’sfitness landscape through the application of a derating function at a position where a potential solution was found.A derating function is designed to lower thefitness appeal of previously located solutions.By repeatedly running the algorithm,all optima are removed from thefitness landscape.Sample derating functions,for a previous maximum x*,includeG1ðx;xÃÞ¼k xÀxÃkraif k xÀxÃk<r 1otherwise(andG2ðx;xÃÞ¼e log m rÀk xÀxÃkr if k xÀxÃk<r1otherwise;(ð7Þwhere r is the radius of the derating function’s effect.In G1,a determines whether the derating function is con-cave(a>1)or convex(a<1).For a=1,G1is a linear function.For G2,m determines‘concavity’.Noting that lim x!0logðxÞ¼À1,m must always be larger than0.Smaller values for m result in a more concave derating function.Thefitness function f(x)is then redefined to beM nþ1ðxÞ M nðxÞÂGðx;s nÞ;where M0ðxÞ fðxÞand s n is the best individual found during run n of the algorithm.G can be any derating function,such as G1and G2.3.4.CrowdingCrowding(or the crowding factor model),as introduced by de Jong[10],was originally devised as a diver-sity preservation technique.Crowding is inspired by a naturally occurring phenomenon in ecologies,namely competition amongst similar individuals for limited resources.Similar individuals compete to occupy the same ecological niche,while dissimilar individuals do not compete,as they do not occupy the same ecological niche. When a niche has reached its carrying capacity(i.e.being occupied by the maximum number of individuals that can exist within it)older individuals are replaced by newer(younger)individuals.The carrying capacity of the niche does not change,so the population size will remain constant.For a genetic algorithm,crowding is performed as follows:It is assumed that a population of GA individ-uals evolve over several generational steps.At each step,the crowding algorithm selects only a portion of the current generation to reproduce.The selection strategy isfitness proportionate,i.e.morefit individuals are more likely to be chosen.After the selected individuals have reproduced,individuals in the current population are replaced by their offspring.For each offspring,a random sample is taken from the current generation,and the most similar individual is replaced by the offspring individual.Deterministic crowding(DC)is based on de Jong’s crowding technique,but with the following improve-ments as suggested by Mahfoud[27]:–Phenotypic similarity measures are used instead of genotypic measures.Phenotypic metrics embody domain specific knowledge that is most useful in multi-modal optimization,as several different spatial positions can contain equally optimal solutions.1864R.Brits et al./Applied Mathematics and Computation189(2007)1859–1883–It was shown that there exists a high probability that the most similar individuals to an offspring are its parents.Therefore,DC compares an offspring only to its parents and not to a random sample of the population.–Random selection is used to select individuals for reproduction.Offspring replace parents only if the off-spring perform better than the parents.3.5.Objective function stretchingObjective function stretching[30]was applied as a sequential PSO niching technique[31],similar to that of Beasley et al.[2].A particle swarm is trained using the gbest algorithm.Once the PSO has identified a local minimum fðxÃÞ,through comparing particlefitnesses to a performance threshold value,the objective function is stretched such that for each point x,where fðxÞ<fðxÃÞ,x is unaffected.All other points,such that fðxÞP fðxÃÞholds,are stretched so that x*becomes a local maximum.All particles are then repositioned randomly.Thefitness function f(x)is redefined as H(x),whereHðxÞ¼GðxÞþc2signðfðxÞÀfðxÃÞÞþ1Þ2tanhðlðGðxÞÀGðxÃÞÞÞandGðxÞ¼fðxÞþc1k xÀxÃkðsignðfðxÞÀfðxÃÞÞþ1Þ2:For a minimization problem,the sign(Æ)function is defined assignðxÞ¼þ1;x>0;0;x¼0;À1;x<0; 8><>:where x is a scalar value.Stretching of f(x)to H(x)ensures that subsequent iterations of the PSO algorithm does not focus on previously located solutions.Although effective in global optimization and to a lesser extent in multi-modal optimization,the stretching technique introduces the following problems:•If multiple acceptable solutions are located close to each other,the effect of G(x)may cause these alternative solutions never to be detected.•The adaptation of f(x)close to x leads to the introduction of‘trenches’in f around remaining potential solutions withfitness similar to that of x[4].•For some combinations of c1,c2l,new local optima are introduced[38].3.6.The nbest particle swarm optimizerThe nbest PSO was developed to solve unconstrained systems of equations(SEs)[5].Solving SEs here spe-cifically refer to the process offinding points of intersection between the individual equations making up a SEs.Standard PSO techniques,such as gbest and lbest can quickly and accurately solve SEs that have a single solution.However,it is frequently the case that complex SEs have multiple solutions.Such problems can be classified as multi-modal problems.Solutions to the SEs all have optimalfitness,and thefitness of any other potential,non-optimal solution,depends on its proximity to one of the optimal solutions.For a potential solution x i,thefitness function for solving a SE can in general be written asfðx iÞ¼X Kk¼1j z kðx iÞj;R.Brits et al./Applied Mathematics and Computation189(2007)1859–18831865where z kðx iÞrepresents each one of the K equations in the SE;each equation is algebraically rewritten to be equal to zero.While thisfitness function works well when an unique solution exists,it fails for multiple solu-tions.For example,consider the following SE:y¼2xÀ3;y¼À3xÀ1;y¼Àxþ1:The SE has three solutions(i.e.there are three intersections between the solutions).Tofind them all,thefitness function should consider a particle’s relative distance to a possible solution.Thus,to evaluate thefitness of x for this SE,thefitness function is redefined asf ABCðxÞ¼min f f AB;ACðxÞ;f BA;BCðxÞ;f CB;CAðxÞg;wheref AB;ACðxÞis thefitness of particle x with respect to equations y¼2xÀ3and y¼À3xþ1,f BA;BCðxÞis thefitness of particle x with respect to equations y¼À3xþ1and y¼Àxþ1,f CB;CAðxÞis thefitness of particle x with respect to equations y¼Àxþ1and y¼2xÀ3.This formulation of thefitness function implicitly assumes that all the lines in the system of equations actually intersect,and rewards a particle for being close to one of the solutions,and does not penalize it if the particle is still far from the global best particle.If there are no intersections between lines in a system of equations,and therefore no solution to the sys-tem of equations,particles will eventually settle on locations where lines in the system are the closest to each other.The lbest model’s neighborhood definition is motivated by the fact that it tries to promote the spread of information regarding good solutions to all particles,without considering a particle’s current position.When searching for a single optimum solution,this model is appropriate as it allows for efficient evaluation of the search space while avoiding premature convergence.However,when searching for multiple possible solutions, lbest is biased towardsfinding a single optimum solution in the search space due to overlapping neighbor-hoods.It has been experimentally shown that it is possible for index-based particle neighborhoods to emerge for relatively high iterations of the lbest algorithm(see results presented in Section5.5).The propagation of information about a local best solution within an index-based neighborhood,where neighborhoods are unique to particles,also hampers convergence.Neighborhoods constantly change as particles do not pursue a com-mon goal.nbest PSO however attempts to compensate for this deficiency.For nbest PSO,a neighborhood best,^y i,is defined for each particle,x i,as the center of mass of the positions of all the particles in the topological neighborhood of x i.The topological neighborhood is defined as the n i closest particles to x i,where the closest particles are found by calculating the Euclidean distance between x i and all other particles in the swarm.Formally,for each particle define the set B i,where B i consists of the k closest particles to x i at any given time step t;^y i is then^y i¼1kX kj¼1B i j;where B ij is the current position of the j th particle in neighborhood B i of particle x i at time t;k is a user defined parameter.Considering the above formulation of^y i,k should not be too small,as it will force a particle to blindly trail its closest neighbor.Also,if k is too large,it would yield an algorithm similar to gbest,but where the goal position would be an average particle position conveying no information about a possible good result. Particle updates are done as for normal gbest PSO,but with the difference that the position of the global best particle,yˆ,is replaced with^y i,which represents the average of the best positions of particles in the topological neighborhood of particle x i.Experimental results showed that nbest successfully locates multiple solutions to SEs[5,4].1866R.Brits et al./Applied Mathematics and Computation189(2007)1859–1883。

Regrouping Particle Swarm Optimization: A New Global Optimization Algorithm with Improved Performance Consistency Across Bs and Mounir Ben Ghalia
I. INTRODUCTION While stochastic, population-based, heuristic optimization methods such as particle swarm optimization (PSO) [1, 2] are less susceptible to entrapment due to their stochastic nature and reliance directly upon function values rather than derivative information, they are nonetheless susceptible to premature convergence, which is especially the case for optimization problems of high dimensionality [3]. The more communication that occurs between the population agents, the more similar they tend to become until converging to the same region of the search space. In particle swarm, if the region converged to is a local well containing a local minimum, there may initially be hope for escape via a sort of momentum

the tyndall effect thus implies“The Tyndall Effect”is a phenomenon often observed in everyday life, in which the scattering of light by suspended particles in a medium leads to the appearance of a visible beam of light. In this article, we will explore the underlying principles behind the Tyndall Effect and delve into its implications in various fields.Firstly, let us understand the basic concept of the Tyndall Effect. Named after the 19th-century physicist John Tyndall, this effect occurs when light encounters particles within a medium, causing some of the light rays to scatter in different directions. The scattered light is then reflected or refracted, creating a visible beam or cone of light. This phenomenon is most noticeable when a beam of light passes through a cloudy liquid or a dusty room, where suspended particles are abundant.To comprehend why the Tyndall Effect occurs, we must delve into the behavior of light waves. Light is composed of electromagnetic waves, which consist of alternating electric and magnetic fields. When light interacts with particles in a medium, such as smoke particles or water droplets, the electric and magnetic fields can induce a dipole moment within the particles. As a result of thisinteraction, the light waves are scattered in various directions.The intensity and color of the scattered light depend on the size of the particles and the wavelength of light. If the particles are larger than the wavelength of incident light, the scattered light will contain various colors, resulting in white light. However, if the particles are smaller than the wavelength of light, the scattering will be more pronounced for shorter wavelengths, such as blue and violet light. This explains why the scattered light appears blue, while the transmitted light through the medium appears yellow or red, as blue light is scattered more strongly in the atmosphere.Now that we have grasped the fundamental principles of the Tyndall Effect, let us explore its implications in various fields. One significant area where the Tyndall Effect is commonly observed is in atmospheric science. This phenomenon plays a crucial role in the scattering of sunlight in the Earth's atmosphere, giving rise to the blue color of the sky. As sunlight encounters tiny molecules and particles in the atmosphere, the shorter blue and violet wavelengths of light are scattered more efficiently, creating the appearance of a blue sky.Additionally, the Tyndall Effect has significant applications in the field of medical diagnostics. This effect is often exploited in technologies such as turbidimetry and nephelometry, which measure the concentration of suspended particles in a liquid sample. By analyzing the scattered light, these techniques allow healthcare professionals to identify abnormalities or monitor the progress of certain diseases, such as kidney disorders or bacterial infections.Furthermore, the Tyndall Effect has numerous applications in industrial processes. For instance, in the field of cosmetics, manufacturers use this phenomenon to create shimmering or sparkling effects in products. By incorporating finely suspended particles that scatter light, such as mica or titanium dioxide, cosmetics can enhance the perceived appearance of skin or add an iridescent quality to lipsticks or nail polishes.In conclusion, the Tyndall Effect is a fascinating phenomenon that arises from the scattering of light by suspended particles in a medium. This effect has implications in various fields, ranging from atmospheric science to medical diagnostics and industrialapplications. By understanding the underlying principles behind the Tyndall Effect, we can appreciate the beauty of everyday occurrences and harness its potential in diverse areas of research and development.。

Section3 particle physics1．The nuclear atom(1)Mass number and atomic number(2)Alpha particle scattering experiment(3)Electrons are accelerated by E and B(4)Write equations using standard nuclear notating(5)Explain why high energies are required to break particles into their constituents and to investigate fine structure(6) de Broglie wavelength λ=h/p(201306R-7)The de Broglie wavelength associated with electrons moving at 2.5 × 106 m s–1 isA .2.9 × 10–4mB .2.4 × 10–8 mC .2.9 × 10–10mD .2.4 × 10–39m (201306-10) The de Broglie wavelength for neutrons used to study crystal structure is 1.2 nm. mass of a neutron = 1.67 × 10–27kgThe speed of these neutrons would beA .3.0 × 106 m s–1 B.3.3 × 102m s–1 C.3.0 × 10–3m s–1 D.3.3 × 10–7m s–1 (201306R-8 )Which of the following is not a valid conclusion from Rutherford’s alpha particle scattering experiment?A .The atom is mainly empty space.B .The mass of the atom is mostly concentrated in the nucleus.C. The nucleus must be positively charged.D .The nucleus must be very small compared to the atom.(201306R-9) Select the row in the table that correctly identifies the composition of a 235U+92(201306-1)The nucleus of one of the isotopes of nickel is represented by 6028Ni.（201201-10）The de Broglie wavelength of a moving tennis ball is calculated as 1 ×10-33 m. This means that the moving tennis ballA .diffracts through a narrow slit. B.does not behave as a particle.C.does not display wave properties.D.is travelling at the speed of light.(201306R-10).A transmissionelectron microscope passes abeam of electrons through atiny specimen to form an imageon a viewing screenDue to the wave nature ofelectrons, diffraction occurswhich can blur the image. Toreduce this effect whenviewing a smaller object thebeam must containA .more electrons per second.B .fewer electrons per second.C. faster moving electrons. D .slower moving electrons(201306-13).In an experiment to investigate the structureof the atom, α-particles are fired at a thin metal foil, which causes the α-particles to scatter.(a) (i) State the direction in which thenumber of α-particles detected will be amaximum(ii) State what this suggests about the structure of the atoms in the metal foil.(b) Some α-particles are scattered through 180°.State what this suggests about the structure of the atoms in the metal foil.(c) The diagram shows the path of an α-particle passing near to a single nucleus inthe metal foil.(i)Name the force that causesthe deflection of the α-particle.(ii) On the diagram, draw anarrow to show the direction ofthe force acting on the α-particle at the point where the force is a maximum. Label the force F.(iii) The foil is replaced by a metal of greater proton number.Draw the path of an α-particle that has the same initial starting point and velocity as the one drawn in the diagram.(201301-11) Early in the twentieth century physicists observed the scattering of alpha particles after they had passed through a thin gold foil. This scattering experiment provided evidence for the structure of the atom.(a) State why it is necessary to remove the air from the apparatus that is used for this experiment.(b) From the results of such an experiment give two conclusions that can be deduced about the nucleus of an atom.Conclusion 1Conclusion 2(c) The diagram shows three α-particles ,allwith the same kinetic energy .The path followedby one of the particles is shown . Add to thediagram to show the paths followed by the othertwo particles.(201302-12 )The electron in a hydrogen atom can be described by a stationary wave which is confined within the atom.This means that the de Broglie wavelength associated with it must be similar to the size of the atom which is of the order of 10–10 m.(a)(i)Calculate the speed of an electron whose de Broglie wavelength is1.00 × 10–10m. (ii) Calculate the kinetic energy of this electron in electronvolts(b)When βradiation was first discovered ,it was suggested that there were electrons in the atomic nucleus, but it was soon realised that this was impossible because the energy of such an electron would be too great.Suggest why an electron confined within a nucleus would have a much greater energy than the energy calculated in (a)(ii).2．high-energy collisions(1)The role of E in linear particle accelerators(2)r=p/BQ in B(3)⊿E=c2⊿m(4)Units:MeV,GeV,MeV/c2 ,GeV/c2,u(201301-10)The tubes of a linear accelerator (linac) get progressively longer down its length becauseA .the accelerating particles become relativistic.B .the frequency of the applied potential difference changes.C .the accelerating particles must spend the same time in each tube.D .the accelerating particles gain mass（201206-10）As a particle accelerates in a linac, it passes through drift tubes of increasing lengths. This is so thatA.the particle can be given more energy within each tube.B.the frequency of the accelerating voltage can be constant.C.the accelerating voltage can be as high as possible.D.the time spent in the tube by the particle is longer(201306R-6)Charged particles are travelling at a speed v, at right angles to a magnetic field of flux density B. Each particle has a mass m and a charge Q.Which of the following changes would cause a decrease in the radius of the circularpath of the particles?A .an increase inB B .an increase in mC .an increase in vD .a decrease in Q (201306-6)A muon has a mass of 106 MeV/c2. The mass of a muon, to two significant figures, isA .1.7 × 10–11 kgB .5.7 × 10–20 kgC .1.9 × 10–28 kg D.1.9 × 10–34 kg (201201-7)The rest mass of a proton is 1.67 ×10–27 kg. This mass, in MeV/c2 is approximatelyA .2.4 × 10–20 B.3.1 × 10–6 C.1.0 D .940(201106-10) The Large Hadron Collider is designed to accelerate protons to very high energies for particle physics experiments. Very high energies are required toA .annihilate hadrons. B.collide hadrons.C .create particles with large mass. D.produce individual quarks(201206-13) An electron and a positron annihilate with the emission of two photons of equal energy. Calculate the wavelength of the photons.(201201-17) (d) The technology suggested in the science fiction series, Star Trek, for powering the Starship Enterprise relied on antimatter. When an anti-hydrogen atom meets a hydrogen atom, they annihilate and produce energy.(i)How much energy, in joules, would be produced by the annihilation ofjust 1 milligram of anti-hydrogen atoms?(ii) Anti-protons are required to produce anti-hydrogen atoms. The total production of anti-protons on Earth over the past 25 years adds up to only a few nanograms.Suggest why so little anti-matter has been created3. Particle accelerators(1) E and B in cyclotron particle accelerators(2) Relativistic effects at speeds near c(201306-9)A cyclotron is a type of particle accelerator. It consists of two metal Dees which are connected to a high frequency voltage supply and are in a strong magnetic field. The particles change their speed becauseA.of the magnetic field they are in.B.the voltage supply is alternating.C.there is a potential difference between the two Dees.D.the magnetic field is at right angles to the Dees.(201306R-11)*11 The diagram shows the basic structure of a cyclotron.With reference to the magnetic field and the alternating potential difference explain how the cyclotron produces a beam of high speed particles.4. Particle theory(1) E and B in particle detectors(2) Charge energy and momentum are conserved in interactions between particles and hence interpret records of particle tracks(3) In the standard quark-lepton model each particle has a corresponding anti-particle meson:quark-anti-quark pairs baryon :quark triplets(4) Write and interpret equations using Standard symbols(201206-9)A pion can decay to produce two leptons. Which one of the following is possible?A.π+→e+ + v e B .π0→e– + v e C.π+→e++ e– D π0→π++ e–(201201-8)A positive kaon (K+) is a meson which includes a strange quark. Its structure could beA . u s B. us C. s d d D .usd(201201-9) The K+ is likely to decay toA.π++ μ– + vμB. π++π0Cπ++π– D π0 + μ–+vμ(201306-7 )The diagram shows the tracks from an event at a point P in a bubble chamber. A magnetic field is directed into the page. The tracks cannot show the production of a proton-antiproton pair with equal kinetic energies becauseA .the curvature is perpendicular to the magnetic field.B .the tracks curve in different directions.C. the tracks have different curvatures.D .there is no track before point P(201306-11)Scientists studying anti-matter recently observed the creation of a nucleus of anti-helium 4, which consists of two anti-protons and two anti-neutrons. The diagram represents the path of a proton through a magnetic field starting at pointX.Add to the diagram the path of an anti-helium 4 nucleus also startingat point X and initially travelling at the same velocity as theproton.Explain any differences between the paths.(201306-14)The photograph shows tracks in a particle detector.(a) Explain the role of a magnetic field in a particledetector(b) Explain how you can tell that track XY was produced by a particle moving fromX to Y rather than from Y to X.(c)The particle that produced track XY was a π+. Deduce the direction of the magnetic field in the photograph(d) At Y, the π+ decayed into a positively charged muon (μ+) and a muon neutrino. The μ+ has a very short range before decaying into various particles, including a positron which produced the final spiral.(i) Give two reasons why you can deduce that the muon neutrino is neutral.(ii) Explain the evidence from the photograph for the production of the muon neutrino at Y.(201306R-14) .Hadrons are a group of particles composed of quarks. Hadrons can be either baryons or mesons.(a) (i) State the quark structure of a baryon.(ii) State the quark structure of a meson.(b) State one similarity and one difference between a particle and its antiparticle. Similarity:Difference :+1.Write down the quark combination of the Κ+..(ii) The Κ+can decay in the following way Κ+→µ++ vµK–is the antiparticle of the K+ . Complete the equation below by changing each particle to its corresponding antiparticle in order to show an allowed decay for the K–meson.Κ－→(iii) The rest mass of the Κ+ is 494 MeV/c2Calculate, in joules, how much energy is released if a Κ+ meets and annihilates a K–(201306-12)The table gives some of the properties of the up, down and strange quarks.There are nine possible ways of combining u, d and s quarks and their antiquarks to make nine different mesons. These are listed belowu u u d u s d d d u d s s s s u s d (a) From the list select the four strange mesons and state the charge and strangeness ofSuggest what might distinguish these mesons from each other.*14 A bubble chamber is a particle detector which makes use of electric and magnetic fields. Explain the role of electric and magnetic fields in a particle detector.(201301-17)In 2011 physicists at the Relativistic Heavy Ion Collider (RHIC) announced the creation of nuclei of anti-helium-4 which consists of anti-protons and anti-neutrons instead of protons and neutrons(a) ‘Ordinary’ helium-4 is written as 42He.What do the numbers 4 and 2 represent?(b) In the RHIC experiment, nuclei of gold 19779Au travelling at speeds greater than2.99 × 108m s–1, in opposite directions, collided, releasing energies of up to 200 GeV. After billions of collisions, 18 anti-helium nuclei had been detected.(i) What is meant by ‘relativistic’ in the collider’s name?(ii) State why it is necessary to use very high energies in experiments such as these. (iii) Show that the mass of a stationary anti-helium nucleus is about 4 GeV/c2(iv) State why the small number of anti-helium nuclei produced only survive for a fraction of a second.(v) A slow moving anti-helium nucleus meets a slow moving helium nucleus. If they were to combine to produce 2 high energy gamma rays, calculate the frequency of each gamma ray.(c) There are two families of hadrons, called baryons and mesons. Baryons such as protons are made of three quarks.(i) Describe the structure of a meson(ii) Up quarks have a charge of +2/3e and down quarks a charge of –1/3e.Describe the quark composition of anti-protons and anti-neutrons and use this to deduce the charge on each of these particles.（201206-18）18 (a) Physicists were able to confidently predict the existence of a sixth quark. State why(b) The mass of the top quark was determined by an experiment. Collisions between protons and anti-protons occasionally produce two top quarks.(i) How do the properties of a proton and an anti-proton compare?(ii) After the collision the two top quarks move in opposite directions with the same speed. Explain why.(c) The two top quarks decay rapidly into two muons and four jets of particles. These can be detected and their momenta measuredThe diagram shows an end-on view of the directions of the four jets (J1 to J4) of particles. The two muons are shown as μ1and μ2. A muon neutrino is also produced but cannot be detected, so is not shown. Each momentum is measured inGeV/c.(i)Explain why GeV/c is a valid unit for momentum.(ii) The vector diagram shown below is not complete. Add to the diagram arrows to represent the momenta of J3 and J4.(iii) Complete the diagram to determine the magnitude of the momentum of the muon neutrino.(iv) Show that the total energy of all the products of this event is about 300 GeV. (v) Deduce the mass of a top quark in GeV/c2.(vi) Suggest why it took a long time to find experimental evidence for the top quark.。