An optimization method:hummingbirds optimization algorithm

1.Introduction

Optimization problems are common in science,economics,Engineering,chemistry and other fields.However,in the face of increasingly complex optimization problems,especially high-dimensional ones with many local optimum points, gradient-based algorithms that use derivations of objective functions to direct the search are no longer applicable.

Stochastic algorithms optimize optimization problems randomly without substantial gradient information.Therefore, they are more suitable for complex optimization problems than gradient-based algorithms.In recent decades,stochastic algorithms have made great progress and have gradually replaced gradient-based algorithms to become the mainstream algorithms used for optimal calculations.

Stochastic algorithms include two main types:individual-based and population-based algorithms.Individual based algorithms generate only a single random solution and update during the whole optimization process.Although the algorithms have fewer function evaluations,the algorithms lack information sharing between individuals because of the small number of solutions,which makes it easy for the algorithm to fall into local optima.In contrast,population-based algorithms create multiple solutions randomly and ameliorate them during the course of optimization.Multiple solutions can enable population-based algorithms to search in different areas of space,and individuals can exchange information with each other.Therefore,compared with individual-based algorithms,population-based algorithms have the ability to jump out of local optima,at the expense of increasing the computational cost.

Most population-based algorithms have four basic characteristics:(i)they are nature-inspired;(ii)they use random variables;(iii)they do not require substantial gradient information;and(iv)they have several parameters that need to be tuned.According to the source of inspiration,the population-based algorithms can be divided into four categories:evolutionary,swarm intelligence-based,physics-based,and human behavior-based.Evolutionary algorithms are inspired by the evolutionary process of creatures in nature.The most representative algorithm is the genetic algorithm(GA)[1];it is based on Darwin’s theory of biological evolution,the main principle of which is survival of the fittest.Other popular algorithms are differential evolution(DE)[2],asexual reproduction optimization(ARO)[3],species co-evolutionary algorithm(SCEA)[4]and monkey king evolution(MKE)[5].

Swarm intelligence-based algorithms are inspired by the behavior of groups of creatures in nature.The most representative algorithm is particle swarm optimization(PSO)[6],which mimics the flocking behavior of birds.Other popular algorithms are cuckoo search algorithm(CS)[7],firefly algorithm(FA)[8],bat algorithm(BA)[9],grey wolf optimizer(GWO)[10],moth- flame optimization algorithm(MFO)[11],crow search algorithm(CSA)[12],cognitive behavior optimization algorithm(COA)[13],sperm whale algorithm(SWA)[14],grasshopper optimization algorithm(GOA)[15],and satin bower bird optimizer(SBO)[16].

Physics-based algorithms are inspired by physical rules in the universe.The most representative algorithm is gravitational search algorithm(GSA)[17],which works on the laws of gravity and mass interaction.Other popular algorithms are galaxy-based search algorithm(GbSA)[18],kinetic gas molecules optimization algorithm(KGMO)[19],water evaporation optimization(WEO)[20]and electrosearch algorithm(ES)[21].

whereis the dth dimension of the ith solution at the tth iteration.To sum up,the pseudo code of the HOA can be given as in Algorithm 1.

For the vast majority of population-based algorithms,how to more effectively balance exploration and exploitation is the key to improving the performance of the algorithms.Exploration represents the global search ability of population-based algorithms,and it is employed to enhance the ability to jump out of local optima by searching in new regions with some randomization method.Meanwhile,exploitation can improve the convergence speed of population-based algorithms,and it is applied to find a better solution in the vicinity of the current optimal solution.However,these two capabilities are conflicting,and no one knows the exact proportion of the two in the algorithm for any optimization problem.The existing viable balancing strategies can be grouped into three categories:(i)The initial phase of the algorithm is searched globally and locally searched later.(ii)Better individuals perform local search,and poor individuals perform global search.(iii)Global search and local search are randomly switched at a certain probability in the algorithm.

Moreover,almost all population-based algorithms contain several parameters that need to be tuned before runs.For example,DE requires scaling and crossover factors to be pre-set.HS should consider the value of the harmony memory considering rate,the pitch adjusting rate and the bandwidth of generation.PSO needs to determine the inertia weight,the maximum value of velocity,the cognition factors and the social learning factors.For different optimization problems,the algorithm often needs to set differentparameter values. This is time-consuming work.Therefore,compared to algorithms with multiple control parameters,an algorithm with fewer parameters is easier to implement and more adaptive to a wider variety of optimization problems.

满月酒那天，王树林一直纠结的块垒化解了许多。亲友朋友加同事来了不少，大家异口同声都说孩子的鼻子嘴巴特别像王树林，尤其是鼻梁，简直就是王树林的翻版，高高挺挺。王树林听后很高兴，兴致一下子提高了不少，还喝了很多酒。辛娜也很高兴，她旧话重提说，我说孩子会长变的吧？像你像你，以后孩子都归你管。

The rest of the paper is organized as follows:The mathematical model of HOA is described in detail in Section 2.Section 3 describes the experimental setting and demonstrates the experimental results.Eventually,the conclusion and suggestions for further work are presented in Section 4.

在大一新生入学时，重视宣传学科竞赛的重要性，组织大一新生赴现场观摩赛事，通过课堂表现和课外接触发现可造之材，鼓励有理想有潜力的新生参与到赛事之中，感受赛事的紧张程度，体验成就感和失落感，为日后成为参赛骨干打下基础。

根据地质情况及结合以往施工经验，采用CZ-8B型冲击钻机钻劈成槽，采用先Ⅰ序槽段的施工方法。在同一槽段内，先施工单号主孔，后施工双号副孔，主孔、副孔采用圆套方式劈打成槽。槽段造孔工作结束后，对造孔质量进行全面检查验收（包括孔位、孔深、孔径、孔斜等），检查合格后进行清槽换浆工序。

2.HOA

2.1 Inspiration

Hummingbirds are the smallest birds in the world,and most of them are 7 cm to 13 cm long.This unique bird is only distributed in the Americas,and the habitat of the vast majority of species is in tropical and subtropical Central America.Hummingbirds can fly at a speed of up to 45 km per hour.They can hover in mid-air at rapid wingflapping rates,which vary from approximately12beats per second in the largest species to in excess of 80 in some of the smallest[28].Hummingbirds need to rely on foraging to maintain the body’s high metabolism,and their main food comes from nectar.Fig.1 shows a hummingbird in nature.

Fig.1 A real hummingbird who is gathering nectar

There are two search phases during the foraging process of hummingbirds.The first is the self-searching phase.In this phase,the hummingbird can search according to its cognitive behavior without interacting with other indivi-duals in the population[29,30].Essentially,cognitive behavior is a selective search process that is based on the search experience accumulated by an individual.For example,hummingbirds often perform a targeted search using their experience and can thus anticipate the amount of nectar in flowers according to their shape or color.If the amount of nectar is determined to be plentiful,the hummingbird will further exploit this flower.However,they will quickly leave a flower and randomly search for the next target if it is not plentiful.It is undoubtedly a simple and effective search model when hummingbirds lack explicit information about food location.

The second is a guide-search phase.In addition to searching through experience,hummingbirds can also search by using various dominant individuals as guidance information.For example,in the territorial behavior of hummingbirds,after an individual occupies a rich food source as its territory,other hummingbirds will move rapidly to the territory [31,32].When these new arrivals are driven away by the first individual,they will follow other outstanding individuals.The advantages of this mode are to help hummingbirds have a clear searching direction and avoid blind searching.

Inspired by these two search phases,we propose a new optimization algorithm,the HOA,whose details are described in the next section.

Despite a variety of new algorithms emerging in an endless stream,the no-free-lunch theorem[27]has proven that no single algorithm can solve all optimization problems.In other words,if an algorithm performs well for some issues,it will inevitably perform poorly for other problems.Therefore,there are still many specific optimization problems that need to be addressed by new algorithms instead of the current optimization techniques.For this reason,a population-based algorithm named the hummingbirds optimization algorithm(HOA)is proposed to solve optimization problems.The algorithm is inspired by the foraging process of hummingbirds.

2.2 Proposed method description

As described above,our HOA includes two phases:a self searching phase and a guide-searching phase.The hummingbirds in HOA represent searchers,and their positions are corresponding to feasible solutions of the optimization problem.The quality of food sources are the values of the fitness functions,and the best food source is the optimal solution.The initialization of HOA is done by the following formula:

where Piis the position of the ith hummingbird in the population(i∈{1,2,3,...,N},N is the population size),and ub and lb represent the upper and lower bounds of the variables in the search space,respectively.rand is a random number between 0 and 1,and it has the same meaning in the following paragraphs.

2.2.1 Self-searching phase

To convert the cognitive behavior into a mathematical model,we treat the last preserved solution of the algorithm as the experience of the current search.When hummingbirdi continuously finds better sources of foodit means that the current search area is promising.

Thus,the hummingbird will exploit the area further.The new position of hummingbirdi can be obtainedvia the following formula:

where are the positions of the ith hummingbird at iteration t and t-1 respectively.is accepted if it gives a better function value,otherwise it keeps unchanged.

When hummingbirdi searches continuously,but fails to find better resultsit means that the hummingbird will know from experience that the current area is not worth continued exploitation.In this case,the hummingbird will randomly change the search direction.This procedure is implemented based on Levy flight. Levy flight is an important non-Gaussian random walk whose random step is subject to a heavy-tailed probability distribution[33].Due to the in finite and rapid growth of variance,the most important feature of this flight mode is that it can search the space to the greatest extent possible in an uncertain environment.Levy flight is more efficient for searching than regular random walk or Brownian motion.Fig.2 exhibits the movement tracks of 1 000 steps of Levy flight and Brownian motion in two dimensions.

Fig.2 Comparison of the movement tracks of Levy flight and Brownian motion in two dimensions

As shown in Fig.2,we can observe that Levy flight generates larger jumps than the Brownian motion and thereby explores the search space more broadly.Therefore,it is more suitable for large-scale search.

The new positions of hummingbirds are generated by performing Levy flight,as described as follows:

where α is a scale factor that should be related to the scales of the problem of interest.The⊕is the entry-wise multiplications.Accept if it gives a better fitness value.

According to[7,34],α and Levy(β)can be formulated as

whereindicates the best solution at iteration t.α0is a constant. μ and υ are selected from the normal distributionwith σμ==1.Here,Γ(z)is the Gamma function.α0is equal to 0.01 and β is equal to 1.5 in this work as the CS set in[7].

In the self-searching phase,hummingbirds generally learn according to the original gradient information,which can accelerate the convergence speed of the algorithm.However,when the algorithm may fall into a local optimum,the hummingbirds search extensively through Levy fl ight in the search space,which can enhance the global search capability of the algorithm.

2.2.2 Guide-searching phase

Combining the two states,the movement pattern of the following birds can be described as follows:

wherePT,tis the position of the territory birdat the tth iteration.rdis a random value between–1 and 1,which can adjust the search direction of the individual.λ is a scale factor that make the territory bird move slightly around its current position.Here,λ is set to 0.1(ub-lb).PT,t+1will replace PT,tif its fitness value is better than PT,t.Equation(5)is aimed to adding a slight perturbation to the best individual,allowing it to perform a fine search within its neighborhood.This method can help the best individual jump out of the local optimum,thus avoiding the stagnation of the evolution of the algorithm.

凸轮式基质填料装置的配套动力为90W的电子调速齿轮减速电机，传动方式为链传动；生产率为180～600盘/h，可调；料箱容量为40L；填料深度为36～40mm，可调；配套皮带输送机构使用电机为YCT-112型电磁调速电机，有效转速为125～1 250r/min。

Different from the territory bird,the movement of the following birds is divided into two states.

State 1 The territory bird does not find the approaching following bird j.In this case,the following bird j will move to the territory bird.Its new position is updated as follows:

《液化烃球罐紧急切断阀选型设计规定》中规定: 当罐区有可靠的仪表空气系统时，应选用气动紧急切断阀；当罐区无仪表空气系统、但有负荷分级为一级负荷的电力电源系统时，应选用电液执行机构或电动执行机构驱动的紧急切断阀。在实际应用中，电动控制阀响应动作慢，且不能实现故障安全状态，因此不建议作为紧急切断阀使用，在无仪表气源的情况下优先选用电液控制阀。当切断阀的执行机构为气动型时，首选故障安全型单作用气缸执行机构；若采用气动双作用气缸执行机构时应配储气罐；若必须采用电动型执行机构时，其电源应采用负荷分级为一级负荷中特别重要的电力电源供电。

whereis the position of the jth following bird at the tth iteration.MF is a mutation factor that can decide if the position of the following bird is to be changed.The value of this factor is 1 or 2,which is again a heuristic step selected randomly by equal probability as MF=round[1+rand{2-1}].As the population gradually converges in the search space,will approachIn the later stage of convergence,it is assumed that,then(6)can be transformed into the following forms:

From(7),it can be found that,will always be when MF=1.Meanwhile,(8)shows that the population remains exploratory and allows for exploration away from the spatially converged solutions.Based on the above analysis,MF can effectively balance the local search ability and global search ability of the algorithm.

State 2 The territory bird finds the following bird j close to itself and issues warnings.The following bird j is frightened and flies to the surrounding area. In this process,an individual is randomly selected from the other following birds.If the selected individual has a better position,the following bird j flies towards it.However,if the selected individual has a worse position,it is better to move away from that individual.This process is described by the following formula:

where j,k∈{1,2,3,...,N-1},j/=k.

随车一起来接迟羽的还有老太太。七哥扶着迟羽走在前面，老太太凑到我的耳边，喜滋滋地说：“总算踏实过日子了，我死都瞑目啦。”她脸上的皱纹写满了由衷安心的笑容，那是一个母亲最为知足的表情。

In this phase,the humming birds search in the environment by territorial behavior.The current best individual in HOA that occupies the territory is called the territory bird.The other individuals are called following birds. The position of the territory is the same as the position of the current optimal individual.After occupying the territory,the territory bird patrols around its territory ceaselessly.This process can be described as follows:

The comparative results obtained by all algorithms for the multimodal functions(f06–f10)are summarized in Table 3.Unlike the unimodal functions,multimodal functions include many local minima,the number of which increases exponentially as the problemsize increases.Therefore,these functions can undermine the exploration capability of the optimizers.From Table 3,we can find that HOA outperforms other algorithms on functions(f07–f10)and only loses in f06.For f06,ABC exhibits the best performance and HOA shows the second best performance.From the last row of Table 3,HOA ranks first among all the algorithms.As the convergence curves of all algorithms shown in Fig.4,HOA presents the promising performance in the convergence speed.

whereis the fitness value of and rankis the rank of following bird j among the other following birds in the population.PFtemphasizes that the better the individual is,the higher this probability is.It enables good individuals to follow the best individual with a greater probability,and the poor individuals follow randomly selected individuals with a greater probability.From the perspective of optimization,the proposed algorithm uses the current best individual to guide the population,which can rapidly reduce the search space and improve the ability of exploitation of the population.The randomly selected individuals are used as guidance information to enhance the global searching ability of the algorithm.Overall,the guide-searching phase can absorb information from different individuals and can thus better balance the exploitation and exploration ability of the algorithm,which makes the algorithm more suitable for finding the solution of the optimization problem.

第二，作业假做。作业假做分两种情况。一部分学生直接照抄别的同学的作业，从来不自己思考。没有知识在自己头脑的加工过程，这样自己只是做了一次没有含金量的搬运工。另一部分学生很认真，作业自己完成，却不懂学习方法。在做作业的过程中，参考教材，查阅课堂笔记。这样完成的作业，没有脱离笔记教材的帮助，其实不算自己完成，最后导致的结果是，平时做题都会，考试脱离了笔记的帮忙，什么都不会。长此以往，对孩子积极性打击极大。

To avoid being trapped into local optima,a role-switch mechanism during the search process is introduced:if a following bird finds an area that is better than that currently occupied by both other following birds and the territory bird,it will be converted into the new territory bird,and the original territory bird will be converted into a following bird in the next iteration.

In addition,individuals in HOA may search beyond the borders.Therefore,we add a border control strategy,whose specific process is as follows:

Human behavior-based algorithms are inspired by the social behavior of human beings.The most representative algorithm is teaching-learning-based optimization(TLBO)[22],which simulates the teaching behavior of teachers and the learning behavior of students in a classroom.Other popular algorithms are harmony search(HS)[23],interior search algorithm(ISA)[24],human behavior-based optimization(HBBO)[25]and human mental search(HMS)[26].

多媒体技术引进课堂对老师的阅读教学起到了非常大的帮助，然而过度使用不仅使老师在教学中依赖多媒体工具，学生也对它产生了依赖心理，多媒体功能的极大发挥减弱了学生和老师在教学中的作用。阅读是非常个人化的学习行为，它需要学生沉下心来进入文本构造的世界，这个过程是连续性的，学生的注意力也处在高度集中的状态。而多媒体的使用会分散学生的阅读注意力，打破阅读的连续性，严重影响了学生的阅读行为。老师对阅读内容的讲解也是十分深刻的，而老师在教学中过度使用多媒体，用不严谨的图片和视频进行阅读教学使很多教学内容无法进行层次的拔高，缺乏教学的严肃性。

Algorithm 1 HOA pseudo code

3.Experimental study

In total,ten classical benchmark functions and two well known engineering design optimization problems are employed to integrally evaluate the performance of HOA.All experiments are performed in the MATLAB 2016a software,and the simulations are run on a Core(TM)i7-6700HQ 2.60 GHz with 16 GB RAM.

3.1 Experiment I:unconstrained benchmarks

In this section,ten different benchmark functions,as given in[35],are used to evaluate the performance of HOA,and their results are compared with the results of 11 algorithms in two groups.In the first group,three classical and mainstream algorithms that have been applied in many fields,i.e.,PSO[36],artificial fish swarm algorithm(AFSA)[37],artificial bee colony(ABC)algorithm[38]are used.In the second group,eight state-of-the art algorithms that have been proposed in the recent literature:CS[7],GSA[18],FA[8],GWO[10],MFO[11],CSA[12],sine cosine algorithm(SCA)[39]and SBO[16]are employed.The benchmark functions include two categories:unimodal(f01–f05)and multimodal(f06–f10).The detailed information for the ten benchmarks is summarized in Table 1. The maximum number of function evaluations(MaxFEs)is set to 100 000 for f01–f10.Each algorithm is performed independently for the test functions 30 times.

Table 1 Description of the benchmark functions(U:unimodal;M:multimodal;Dim:dimension)

First,we compare HOA with mainstream algorithms in the first group.For a fair comparison,to compare all algorithms under the same number of maximum iteration for each test function,the population size of the HOA is set to 50 because it has two phases,and the population size of the other algorithms is set to 100 because of the inclusion of one phase.The control parameter settings for the compared algorithms in this test are detailed as follows:

(i)PSO:ωmax=0.9,ωmax=0.4 and c1=c2=2 as in[36];

(ii)AFSA:Visual=1,Step=10,try number=100 and δ=0.618 as in[37];

(iii)ABC:limit= (N·D)/2;size of employed bee=onlooker bee=(colony size)/2 as in[38].

Table 2 shows the optimization results obtained using the HOA and three compared algorithms over unimodal functions(f01 –f05),where“Best”,“Worst”,“Mean”,and“SD”,respectively,denote the best,worst,and mean fitness values with the standard deviation.Moreover,the algorithms are sorted on the mean solution from small to large,and the best results are highlighted for each function with boldface font.Finally,to intuitively exhibit the performance of the three algorithms,we add the average ranks and the over all rank in Table 2.According to Table2,it can be observed that our HOA has the best results among the compared algorithms,and the convergence curve shown in Fig.3 indicate that HOA has the fastest convergence speed.

Table 2 Statistical results of five unimodal functions for four algorithms

Function Metric PSO AFSA ABC HOA Best 4.4661E-09 5.6944E+00 6.0932E-11 0 Worst 5.3809E-07 8.0334E+00 1.5289E-09 0 f01 Mean 1.0592E-07 6.9494E+00 4.3182E-10 0 SD 1.3464E-07 7.0391E-01 4.0839E-10 0 Rank 3 4 2 1 Best 5.1065E-06 1.0288E+01 6.2532E-07 1.3252E-229 Worst 7.4840E-05 1.6562E+01 3.1236E-06 2.4570E-199 f02 Mean 1.9743E-05 1.4761E+01 2.0404E-06 8.3443E-201 SD 1.4384E-05 1.4411E+00 7.0388E-07 0 Rank 3 4 2 1 Best 3.6639E+01 2.6562E+01 5.4211E+03 0 Worst 1.3955E+02 5.0529E+01 1.4716E+04 0 f03 Mean 7.9331E+01 3.8830E+01 1.0391E+04 0 SD 2.6585E+01 5.9073E+00 2.0738E+03 0 Rank 3 2 4 1 Best 1.5490E+01 6.8621E+02 7.8929E-03 1.2122E-03 Worst 2.4689E+02 1.7763E+03 7.0644E+00 1.7633E+00 f04 Mean 6.1495E+01 1.1777E+03 2.0521E+00 2.1932E-01 SD 6.3004E+01 2.5344E+02 2.1603E+00 4.0761E-01 Rank 3 4 2 1 Best 2.0420E-02 1.2235E+01 8.0045E-02 1.0886E-05 Worst 1.3401E-01 2.7191E+01 2.3204E-01 2.5065E-03 f05 Mean 5.5557E-02 2.0114E+01 1.5399E-01 7.0678E-04 SD 2.3476E-02 3.4812E+00 3.2530E-02 6.2312E-04 Rank 2 4 3 1 Average rank 2.8 3.6 2.6 1 Overall rank 3 4 2 1

Fig.3 Convergence graphs of four unimodal functions for PSO,AFSA,ABC,and HOA

where PFtdenotes the probability that the following birds are found by the territory bird.In this work,PFtis determined as follows:

Table 3 Statistical results of five multimodal functions for four algorithms

Function Metric PSO AFSA ABC HOA Best –8.1474E+03 –9.0509E+03 –1.2451E+04 –9.1108E+03 Worst –5.1071E+03 –7.1022E+03 –1.2080E+04 –6.3858E+03 f06 Mean –6.6989E+03 –7.8385E+03 –1.2258E+04 –8.2889E+03 SD 7.4309E+02 4.9643E+02 1.0206E+02 6.4126E+02 Rank 4 3 1 2 Best 2.3879E+01 2.0079E+02 2.5841E-10 0 Worst 6.9647E+01 2.4204E+02 9.9497E-01 0 f07 Mean 4.0728E+01 2.2493E+02 6.6331E-02 0 SD 1.1002E+01 1.0023E+01 2.5243E-01 0 Rank 3 4 2 1 Best 2.1826E-06 3.3679E+00 2.8043E-06 0 Worst 2.5074E-05 3.9042E+00 1.8588E-05 0 f08 Mean 1.0056E-05 3.6668E+00 1.1395E-05 0 SD 5.5968E-06 1.4558E-01 4.4298E-06 0 Rank 2 4 3 1 Best 2.9294E-07 1.3534E+01 1.7291E-10 0 Worst 5.4083E-02 5.2048E+01 2.8638E-06 0 f09 Mean 1.4776E-02 3.5190E+01 2.5593E-07 0 SD 1.4790E-02 1.1422E+01 6.9536E-07 0 Rank 3 4 2 1 Best 5.6210E-01 7.5819E-01 1.1747E-12 7.2501E-05 Worst 4.5167E+00 1.2770E+00 8.2437E-11 1.2187E-02 f10 Mean 2.3753E+00 1.0301E+00 1.5396E-11 2.7549E-03 SD 1.0560E+00 1.5043E-01 1.7705E-11 2.9718E-03 Rank 4 3 1 2 Average rank 3.2 3.6 1.8 1.4 Overall rank 3 4 2 1

Fig.4 Convergence graphs of four multimodal functions for PSO,AFSA,ABC,and HOA

In order to further test the performance of the algorithm,the HOA is compared with the second group of advanced algorithms.To ensure the fairness of comparison, the population size of HOA and CS is set to 50 because they have two phases.For the remaining algorithms,the population size is equal to 100because they have only one phase.Specific control parameters are not required for the HOA,and the control parameters settings for all compared algorithms are described in Table 4.

Table 4 Control parameters of eight compared algorithms

Algorithm Specification CS β=1.5 and pa=0.25 as in[7]GSA GO=100 and α=20 as in[18]FA α =0.2,β0=1,γ =1 as in[8]GWO a=2-1×(2/MaxCycle)as in[10]MFO b=1 and flame no=round[N-l·(N-1)/T]as in[11]CSA AP=0.1 and fl=2 as in[12]SCA α =2,r3=2·rand as in[39]SBO α=0.94,Z=0.02 and mutation probability is 0.05 as in[16]

Table 5 provides the statistic results obtained by the nine algorithms for unimodal functions(f01–f05).From Table 5,we can observe that HOA can achieve better results than the other companion algorithms for f01–f04and is worse only for f05.The last row of Table 5 shows that HOA ranks first among all algorithms.GWO has promising performance for f05and ranks second compared with the other algorithms.The graphical results of the convergence rate and analysis of variance(ANOVA)test given in Fig.5 indicate that HOA outperforms its competitors in terms of convergence rate and solution quality.

Table 5 Statistical results of five unimodal functions for nine algorithms

Function Metric CS GSA FA GWO MFO Best 3.1873E-03 2.2356E-18 3.2175E-04 3.3152E-87 1.0556E-35 Worst 1.8546E-02 5.1571E-18 9.9751E-04 5.0603E-84 1.5940E-31 f01 Mean 8.8861E-03 3.8401E-18 5.5797E-04 3.3557E-85 1.1267E-32 SD 3.6808E-03 7.6262E-19 1.7113E-04 9.5395E-85 2.9352E-32 Rank 9 5 6 2 4 Best 4.6496E-01 7.7224E-09 1.1523E+01 2.5968E-50 3.4803E-21 Worst 3.0992E+00 1.3183E-08 1.2012E+02 7.2046E-49 4.2380E-19 f02 Mean 1.2720E+00 1.0131E-08 5.7526E+01 2.6832E-49 1.0203E-19 SD 5.6264E-01 1.2349E-09 3.3051E+01 2.0623E-49 9.7286E-20 Rank 8 5 9 2 4 Best 2.2965E+02 5.9359E+01 2.8976E+03 2.1684E-31 4.8183E-15 Worst 7.7555E+02 2.1078E+02 9.6717E+03 1.0079E-25 6.6667E+03 f03 Mean 4.8197E+02 1.0621E+02 5.5386E+03 6.9282E-27 2.2222E+02 SD 1.3122E+02 3.7795E+01 1.7061E+03 2.0630E-26 1.2172E+03 Rank 8 6 9 2 7

Continued

Function Metric CS GSA FA GWO MFO Best 2.9232E+01 2.5830E+01 2.2550E+01 2.4873E+01 8.1081E-01 Worst 7.0603E+01 9.3011E+01 2.8434E+04 2.7920E+01 3.0199E+03 f04 Mean 3.8122E+01 2.8263E+01 2.7620E+03 2.6117E+01 3.1340E+02 SD 1.0049E+01 1.2230E+01 6.0837E+03 7.3272E-01 9.1759E+02 Rank 5 4 9 3 8 Best 1.8749E-02 2.5682E-03 5.7134E+01 7.0008E-05 3.5000E-04 Worst 7.2482E-02 1.0297E-02 8.5993E+01 5.2054E-04 2.8998E-03 f05 Mean 3.5559E-02 5.3473E-03 8.5031E+01 2.2807E-04 1.5665E-03 SD 1.1799E-02 1.8592E-03 5.2689E+00 1.1727E-04 6.9356E-04 Rank 8 5 9 1 4 Average rank 7.6 5 8.4 2 5.4 Overall rank 8 4 9 2 5 Function Metric CSA SCA SBO HOA Best 4.6416E-04 3.4963E-44 7.2106E-04 0 Worst 2.2371E-02 6.8963E-35 4.8623E-03 0 f01 Mean 5.1425E-03 3.9021E-36 1.8193E-03 0 SD 5.5721E-03 1.3694E-35 9.5993E-04 0 Rank 8 3 7 1 Best 1.7892E-01 4.9622E-28 9.2507E-03 1.0777E-222 Worst 1.6354E+00 3.6748E-23 1.8586E-02 5.7755E-197 f02 Mean 7.8482E-01 4.9188E-24 1.1883E-02 1.9367E-198 SD 4.9658E-01 9.5048E-24 2.2940E-03 0 Rank 7 3 6 1 Best 3.1720E+00 5.6985E-25 2.7381E+01 0 Worst 2.1382E+01 7.7601E-14 1.2536E+02 0 f03 Mean 8.8877E+00 2.5954E-15 5.7587E+01 0 SD 5.0462E+00 1.4166E-14 2.1035E+01 0 Rank 4 3 5 1 Best 2.5915E+01 6.2365E+00 3.7705E+00 1.2702E-03 Worst 1.2433E+02 7.3753E+00 3.6187E+02 1.0433E+00 f04 Mean 4.3232E+01 6.8795E+00 7.1427E+01 1.7314E-01 SD 2.7732E+01 4.0504E-01 8.5066E+01 2.8961E-01 Rank 6 2 7 1 Best 5.0821E-03 2.3530E-05 1.4931E-02 1.7315E-05 Worst 1.9171E-02 1.1278E-03 3.6658E-02 2.9103E-03 f05 Mean 1.0681E-02 2.9819E-04 2.6566E-02 7.1999E-04 SD 3.7847E-03 2.8251E-04 5.1333E-03 6.5413E-04 Rank 6 2 7 3 Average rank 6.2 2.6 6.4 1.4 Overall rank 6 3 7 1

Fig.5 Convergence curve and ANOVA test of four unimodal functions

The optimization results obtained by all considered algorithms for the multimodal functions(f06–f10)are listed in Table 6.As shown in Table 6,HOA outperforms the other eight algorithms on f07–f09,but its performance is not superior for f06and f10.In other details,for f10,GWO can achieve the same excellent results as HOA,and MFO shows the best performance for f06and f10.According to the overall rank,HOA ranks first for the multimodal functions,which means that HOA has outstanding exploration ability.GWO and MFO rank second and third,respectively,and CS shows the worst performance among the nine algorithms.Fig.6 describes the graphical results for HOA and the other algorithms for four selected functions and shows that our HOA has better convergencespeed and accuracy.

入院第1天，2组大腿、小腿周径间差异无统计学意义。入院第7天，肌电生物反馈组大腿周径减少值明显小于常规康复组(P<0.05),2组小腿周径减少值差异无统计学意义。见表2。

Table 6 Statistical results of five multimodal functions for nine algorithms

Function Algorithm CS GSA FA GWO MFO Best –9.2290E+03 –4.2079E+03 –9.0932E+03 –7.5366E+03 –1.0904E+04 Worst –8.2272E+03 –2.4576E+03 –6.4878E+03 –3.9478E+03 –7.5818E+03 f06 Mean –8.6647E+03 –3.1353E+03 –7.7504E+03 –6.2036E+03 –9.1032E+03 SD 2.3800E+02 3.9793E+02 5.2527E+02 8.2229E+02 7.6905E+02 Rank 2 8 4 6 1 Best 6.0216E+01 2.9849E+00 2.8854E+01 0 1.9899E+00 Worst 1.1102E+02 1.1940E+01 1.0447E+02 0 3.6884E+01 f07 Mean 8.3875E+01 7.5285E+00 6.5800E+01 0 1.4472E+01 SD 1.1548E+01 2.1496E+00 1.7857E+01 0 1.0085E+01 Rank 9 4 8 1 5 Best 1.8952E+00 1.2627E-09 2.1203E+00 7.9936E-15 4.4409E-15 Worst 9.3048E+00 1.8461E-09 1.1949E+01 1.5099E-14 4.4409E-15 f08 Mean 4.5589E+00 1.5474E-09 4.0139E+00 1.0007E-14 4.4409E-15 SD 2.0045E+00 1.4669E-10 1.9598E+00 2.5861E-15 0 Rank 9 5 8 4 3 Best 4.0521E-02 1.1490E+00 2.9338E-03 0 3.4448E-02 Worst 2.4749E-01 2.5729E+00 6.8436E-02 1.7903E-02 2.6566E-01 f09 Mean 9.7669E-02 1.6150E+00 2.2698E-02 5.9675E-04 1.4386E-01 SD 4.8698E-02 3.6852E-01 1.3938E-02 3.2686E-03 7.4666E-02 Rank 6 9 4 2 8 Best 4.5344E-01 1.7571E-20 2.6879E+00 3.0032E-07 4.7116E-32 Worst 1.5468E+00 1.0367E-01 2.6805E+01 3.2467E-02 1.6909E-31 f10 Mean 9.6076E-01 1.7104E-02 1.0422E+01 1.2261E-02 5.5095E-32 SD 2.5509E-01 3.8910E-02 5.1881E+00 8.5137E-03 2.2900E-32 Rank 8 5 9 4 1 Average rank 6.8 6.2 6.6 3.4 3.6 Overall rank 9 7 8 2 3

Continued

Function Algorithm CSA SCA SBO HOA Best –9.1925E+03 –2.7944E+03 –8.1080E+03 –9.4367E+03 Worst –5.3633E+03 –2.0971E+03 –4.6926E+03 –7.0501E+03 f06 Mean –7.2524E+03 –2.4232E+03 –6.0269E+03 –8.1011E+03 SD 8.7286E+02 1.6963E+02 8.9981E+02 5.7228E+02 Rank 5 9 7 3 Best 2.9980E+00 0 2.6865E+01 0 Worst 3.4827E+01 2.3922E+01 5.5719E+01 0 f07 Mean 1.7269E+01 7.9740E-01 4.3258E+01 0 SD 7.5528E+00 4.3676E+00 7.5163E+00 0 Rank 6 3 7 1 Best 1.5019E+00 8.8818E-16 7.1657E-03 8.8818E-16 Worst 3.8363E+00 4.4409E-15 1.6477E-02 8.8818E-16 f08 Mean 2.6001E+00 4.0856E-15 1.0522E-02 8.8818E-16 SD 6.0866E-01 1.0840E-15 2.2961E-03 0 Rank 7 2 6 1 Best 9.2387E-03 0 1.9840E-03 0 Worst 9.8607E-02 2.7968E-01 7.8375E-01 0 f09 Mean 5.0072E-02 1.1920E-02 1.3485E-01 0 SD 2.5063E-02 5.2127E-02 1.8202E-01 0 Rank 5 3 7 1 Best 5.7406E-03 1.3029E-02 3.1810E-06 6.3640E-05 Worst 2.8469E+00 9.7563E-02 4.2273E-03 1.4864E-02 f10 Mean 6.7057E-01 4.7943E-02 1.5976E-04 2.7611E-03 SD 6.3986E-01 1.9819E-02 7.6901E-04 3.5082E-03 Rank 7 6 2 3 Average rank 6 4.6 5.8 1.8 Overall rank 6 4 5 1

Fig.6 Convergence curve and ANOVA test of four multimodal functions for different algorithms

Table 7 presents the statistical results of the Wilcoxon signed ranks test[40]at the 95%significance level for ten functions.As a nonparametric test,the Wilcoxon signed ranks can detect whether there is a significant difference between HOA and each compared algorithm.In Table 6 and Table 7,“+”means that HOA is significantly better than the other algorithm,and“–”means the opposite.“≈”indicates that HOA is not significantly different from the competitor.“R+”indicates the sum of ranks for which HOA exceeds the corresponding competitor,and“R–”denotes the sum of ranks for the opposite.From the results of“+/≈/–”of Table 5 and Table 6,we can observe clearly that HOA obtains a greater number of“+”than the other algorithms,which means that HOA shows a statistically excellent performance compared to its competitors in the Wilcoxon signed ranks test.

Table 7 Results of a Wilcoxon signed ranks test based on the best solution for each function with 30 independent runs(α=0.05)

CS vs HOA GSA vs HOA FA vs HOA GWO vs HOA Function p-Value R+ R- Win p-Value R+ R- Win p-Value R+ R- Win p-Value R+ R- Win f01 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f02 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f03 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f04 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f05 1.9209E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f06 1.8910E-0451414– 1.7344E-064650 + 3.3269E-02336129+ 1.2506E-0446419–f07 1.7344E-06 465 0 + 1.6416E-06 465 0 + 1.7344E-06 465 0 + 1.0000E+00 0 0 ≈f08 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 9.2745E-07 465 0 +f09 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 3.1731E-01 1 0 ≈f101.7344E-064650+ 5.7096E-02140325– 1.7344E-064650+ 2.3704E-0543827++/≈/– 9/0/1 9/0/1 10/0/0 7/2/1 MFO vs HOA CSA vs HOA SCA vs HOA SBO vs HOA Function p-Value R+ R- Win p-Value R+ R- Win p-Value R+ R- Win p-Value R+ R- Win f01 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f02 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f03 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f04 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 + 1.7344E-06 465 0 +f05 2.2248E-04 412 53 + 1.7344E-06 465 0 + 6.8359E-03 101 364 – 1.7344E-06 465 0 +f065.7924E-0537428– 2.6134E-0641055+ 1.7344E-064650+ 3.1817E-064596+f071.7224E-064650+1.7344E-064650+1.7971E-0130≈ 1.7344E-064650+f08 4.3205E-08 465 0 + 1.7344E-06 465 0 + 2.0346E-07 378 0 + 1.7344E-06 465 0 +f091.7344E-064650+1.7344E-064650+2.7708E-02210≈ 1.7344E-064650+f10 1.7344E-060465– 2.3534E-064623 + 1.7344E-064650 + 1.7344E-060465–+/≈/– 8/0/2 10/0/0 7/2/1 9/0/1

In general,HOA is faster than other algorithms in terms of convergence speed,due to full use of its own gradient information and global optimal individual information,which effectively improves the local exploitation capability of the algorithm.In addition,HOA also gets its own superiority in search accuracy.The main reason of that is that HOA contains two random search strategies:Levy flight strategy and random escape strategy,which enhance the exploration ability of the algorithm.At the same time,we can note that CS,FA,AFSA perform poorly among the above-stated methods.This is because the mutation of the CS algorithm is single,and the control parameters of the FA and AFSA algorithms are more difficult to adjust.Therefore,the combination of multiple mutation methods and the minimal control parameters may be the focus of future research on optimization algorithms.

3.1.1 Search behavior analysis of HOA

3.1.2 Computational complexity of the HOA

Table 8 Statistic results of HOA-S,HOA-G and HOA

HOA-S HOA-G HOA Function Mean SD Mean SD Mean SD f01 5.3855E-05 1.4262E-04 8.2585E+02 8.1274E+01 0 0 f02 1.4329E-03 1.6914E-03 6.3969E+01 3.1304E+01 2.9230E-194 0 f05 2.0396E-03 1.4584E-03 1.0719E+00 1.3378E-01 7.1633E-04 0 f07 1.3832E-13 7.4694E-13 1.3077E+02 6.9087E+00 0 0 f08 5.6283E-04 8.1531E-04 1.9364E+01 1.4999E-01 8.8818E-16 0 f09 1.4114E-05 3.6145E-05 8.3534E+00 7.3072E-01 0 0

Fig.7 Convergence curves of HOA-S,HOA-G and HOA for different benchmarks

According to Table 8 and Fig.6,we can easily draw several significant conclusions:First,HOA outperforms HOA-S and HOA-G in terms of accuracy and convergence rate,which means that the coexistence of the self-searching phase and the guide-searching phase is very important.Second,the performance of HOA-G is the worst among all algorithms,which suggests that the guide-searching phase has the greatest impact on the performance of HOA.Finally,although the impact of the self-searching phase on the algorithm is far less than that of the guide-searching phase,the former also has important effects on the performance of HOA.In general,the two phases influence the final optimization results of the algorithm to different degrees.When the two phases coexist,the algorithm is better at solving the optimization problem.Therefore,each behavior is essential for HOA.

To study the influence of the self-searching phase and the guide-searching phase,in this section,the proposed algorithm is compared with two different algorithms:(i)HOA without the self-searching phase(denoted HOA-S)and(ii)HOA without the guide-searching phase(denoted HOA-G).In addition,the population of the two tested algorithms is set to 100 because each of them has only one phase.These two algorithms and the standard HOA are tested by six typical functions from the above classic benchmark functions.The statistical results of 30 runs for all algorithms are presented in Table 8,and the convergence rates for each algorithm for four functions are depicted in Fig.7.

The computational complexity of our HOA relies on the population size N,dimension of the problem D,maximum number of iterations T,and sorting mechanism of the following birds in each iteration.During one iteration,the time complexity of all individuals update and border control strategy in the self-searching phase of HOA is O(2ND).In addition,in the guide-searching phase of HOA,the time complexity of the territory bird update and border control strategy is O(2D);the time complexity of probability PFtis O((N-1)2);the time complexity of the following birds update and border control strategy is O(2(N-1)D).Based on the analysis of the above,the overall computational complexity of HOA is defined as follows:

你说，世界上还有这么强横的人，我对小白忿忿地说。小白说，你现在可不能生气，一生气病情会出现反复。我笑着说，我不生气，我这个人从来不生气。

Table9lists the comparisons of computational complexity of HOA and several representative algorithms in our experiments.From Table 9,we can observe that the computational complexity of CSA is the smallest,and it is followed by the PSO and CS.Although the HOA ranks fifth,it is better than the ABC and GSA.However,based on the promising performance for the ten benchmarks,the computational complexity of the HOA is acceptable.

Table 9 Comparison of the computational complexity of seven algorithms

3.2 Experiment II:engineering design problems

Two well-known engineering design optimization problems,the three-bar truss design problem and the welded beam design problem,are chosen as constrained functions to further examine the performance of HOA.The penalty functions are used as constraint handling mechanisms because they are the simplest and have the lowest computational cost.The population size of HOA is set to 50,and the statistical results of HOA for each engineering problem are obtained in 30 independent runs.

3.2.1 Three-bar truss design problem

关于体育赛事的概念定义，学者黄海燕在其博士学位论文中作了详尽的归纳和论述[3]。具体而言，体育赛事的定义有狭义和广义之分。根据姚颂平、陈锡尧等学者对国际性重大体育赛事的划分[4-5]，本文所涉的国际性重大体育赛事主要包含3大类，分别是大型综合性体育赛事(如奥运会、世界大学生运动会等)、国际单项体育组织主办的高等级的国际性赛事(如世锦赛、F1、网球大师赛等)，以及由跨国公司或知名企业操办的具有重要影响的系列国际性商业赛事。

The three-bar truss design problem is a famous structural design problem in practical engineering applications,and is often used to assess the performance of different algorithms.A pictorial illustration of some components of the design problem is given in Fig.8 There are two structural variables,which are cross-sectional areas(A1and A2).This problem aims to acquire the least weight,but it also needs to be subject to several constraints,such as stress,deflection,and buckling constraints.The problem is formulated as follows:

Fig.8 Schematic of the three-bar truss design problem

Table 10 presents the comparisons of the best optimization results obtained by HOA and several previous methods.To date,this problem has been dealt with by numerous optimization algorithms,including:society and civilization(SC)[41],hybrid evolutionary algorithm and an adaptive constraint-handling technique(HEA-ACT)[42],DE with dynamic stochastic selection(DEDS)[43],PSODE[44],CS[45],mine blast algorithm(MBA)[46],and CSA[12].The comparisons of statistical results for HOA and the above mentioned optimizers are shown in Table11.From Table 11,it can be found that the best result is 2.638958433764684E+02 with only 13 000 function evolutions(FEs),which is the lowest among all methods.Specially,for the mean or standard deviation,even the worst value of our HOA with 20 000 FEs is the smallest among all the algorithms,which indicates that HOA is more robust than the other reported optimizers.

Table 10 Comparison of the best solution found by various studies for the three-bar truss design problem

e HEA-ACT[42] DEDS[43] PSO-DE[44] CS[45] MBA[46] CSA[12] HOA 0.788 68 0.788 675 0.788 675 1 0.788 67 0.788 565 0 0.788 675 128 4 0.788 675 142 372 453 0.408 23 0.408 248 0.408 248 2 0.409 02 0.408 559 7 0.408 248 308 0 0.408 248 268 465 375–0.000 000 1.77E-08 –5.29E-11 –0.000 29 –5.29E-11 –1.687539e-14 0–1.464 118 –1.464 101 –1.463 747 5 –0.268 53 –1.463 747 5 –1.464 101 595 2 –1.464 101 640 145 903–0.535 898 –0.535 898 36 –0.536 252 4 –0.731 76 –0.536 252 4 –0.535 898 404 8 –0.535 898 359 854 097 263.895 843 263.895 843 4 263.895 843 263.971 6 263.895 852 2 263.895 843 376 5 2.638958433764684E+02

Table 11 Comparison of statistical results found by different approaches for the three-bar truss design problem

Method Worst Mean Best SD FEs SC[41] 263.969 756 263.903 356 263.895 846 1.3E-02 17 610 HEA-ACT[42] 263.896 099 263.895 865 263.895 843 4.9E-05 15 000 DEDS[43] 263.895 849 263.895 843 263.895 843 9.7E-07 15 000 PSO-DE[44] 263.895 843 263.895 843 263.895 843 4.5E-10 17 600 CS[45] NA 264.066 9 263.971 56 9.00E-05 15 000 MBA[46] 263.915 983 263.897 996 263.895 852 3.93E-03 13 280 CSA[12] 263.895 843 377 0 263.895 843 376 5 263.895 843 376 5 1.0122543402E-10 25 000 2.638958445046976E+02 2.638958434379900E+02 2.638958433764684E+02 2.191366322234361E-07 13 000 HOA 2.638958433767606E+02 2.638958433764794E+02 2.638958433764684E+02 5.319778017413415E-11 20 000

3.2.2 Welded beam design problem

As Fig.9 indicates,this problem includes four variables:the design thickness of the weld h,the length of the clamped bar l,the height of the bar t,and the thickness of the bar b.The main purpose of this optimization process is to obtain the minimum cost of a welded beam subject to constraints such as weld stress,buckling load,beam deflection and beam bending stress.This problem can be expounded as follows:

Fig.9 Schematic of welded beam design problem

Table 12 compares the best designresults given by HOA and other published work.There are plenty of optimization methods to solve this problem in the literature,such as coevolutionary PSO(CPSO)[47],hybrid PSO(HPSO)[48],coevolutionary DE(CDE)[49],FA[50],GA[51],cultural algorithms with evolutionary programming(CAEP)[52],water cycle algorithm(WCA)[53],ABC[54],MBA[46],ISA[24],and improved global-best-guided PSO(IGPSO)[55].The statistical results using the abovementioned optimizers and HOA are shown in Table 13.As we can observe from Table 13,in terms of the best solution and the number of function evaluations,HOA apparently outperforms the published methods.The best total cost is 1.724 852 308 597 365,with 100 000 FEs obtained by HOA.In addition,the mean value of HOA is the best,and its SD is better than those of the other algorithms,except for MBA,which means that our HOA is quite competitive with the other compared methods.

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Table 12 Comparison of the best solution found by previous works for the welded beam design problem

Variable CPSO[47] HPSO[48] CDE[49] MBA[46] GA3[51] CAEP[52] WCA[53] HOA x1 0.202 369 0.205 73 0.203 137 0.205 729 0.205 986 0.205 700 0.205 728 0.205 729 639 786 079 x2 3.544 214 3.470 489 3.542 998 3.470 493 3.471 328 3.470 500 3.470 522 3.470 488 665 628 005 x3 9.048 210 9.036 624 9.033 498 9.036 626 9.020 224 9.036 600 9.036 620 9.036 623 910 357 633 x4 0.205 723 0.205 73 0.206 179 0.205 729 0.206 480 0.205 700 0.205 729 0.205 729 639 786 080 g1 –13.655 547 –0.025 399 –44.578 568 –0.001 614 –0.103 049 1.988 676 –0.034128 –3.637978807091713E-12 g2 –78.814 077 –0.053 122 –44.663 534 –0.016 911 –0.231 747 4.481 548 –3.49E-05 0 g3 –3.35E-03 0 –0.003 042 –2.40E-07 –5E-04 0.000 000 –1.19E-06 –1.387778780781446E-16 g4 –3.424 572 –3.432 981 –3.423 726 –3.432 982 –3.430 044 –3.433 213 –3.432 980 –3.432 983 785 362 248 g5 –0.077 369 –0.080 73 –0.078 137 –0.080 729 –0.080 986 –0.080 700 –0.080 728 –0.080 729 639 786 079 g6 –0.235 595 –0.235 540 –0.235 557 –0.235 540 –0.235 514 –0.235 538 –0.235 540 –0.235 540 322 584 754 g7 –4.472 858 –0.031 555 –38.028 268 –0.001 464 –58.646 888 –2.603 347 –0.013 503 –2.728484105318785E-12 f(x) 1.728 024 1.724 852 1.733 462 1.724 853 1.728 226 1.724 852 1.724 856 1.724 852 308 597 365

Table 13 Comparison of statistical results found by different works for the welded beam design problem

Method Worst Mean Best SD FEs CPSO[47] 1.782 143 1.748 831 1.728 024 1.29E-02 240 000 HPSO[48] 1.814 295 1.749 040 1.724 852 4.01E-02 81 000 CDE[49] 1.824 105 1.768 158 1.733 461 0.022 194 204 800 FA[50] 2.345 579 3 1.878 656 0 1.731 206 5 0.267 798 9 50 000 GA3[51] 1.993 408 1.792 654 1.728 226 7.47E-02 80 000 CAEP[52] 3.179 709 1.971 809 1.724 852 4.43E-01 50 020 WCA[53] 1.744 697 1.726 427 1.724 856 4.29E-03 46 450 ABC[54] NA 1.741 913 1.724 852 0.03100 30 000 MBA[46] 1.724 853 1.724 853 1.724 853 6.94E-19 47 340 ISA[24] 2.670 0 2.497 3 2.381 2 1.02E-1 30 000 IGPSO[55] 1.724 852 1.724 852 1.724 852 4.76378E-09 120 000 1.785 904 800 543 632 1.727 531 203 494 599 1.724 852 308 597 814 0.011 185 363 062 822 29 900 HOA 1.724 852 309 234 164 1.724 852 308 619 237 1.724 852 308 597 365 1.161694738829006E-10 100 000

4.Conclusions

With the development of society,studying new optimizationalgorithms for solving complex optimization problems has become a research hots pot.Based on the search behavior of hummingbirds during the foraging process,this paper proposes an HOA.The proposed algorithm can be divided into two phases:the self-searching phase and the guide-searching phase.The self-searching phase mainly simulates the cognitive behavior of hummingbirds.On the one hand,HOA uses gradient information for targeted searching to improve search efficiency.On the other hand,it uses Levy flight for random searching to jump out of local optima.The guide-searching phase mimics the territorial behavior of hummingbirds.In this phase,HOA uses the optimal and random individuals as guiding information,which effectively balances the exploitation and exploration abilities.Through the collaborative operation of these two search phases,the final solution is obtained.

For experiment study,ten classic benchmark functions are selected to test the performance of HOA.The results prove that HOA outperforms other classic and state-of the-art algorithms.Moreover,two engineering design optimization problems,the three-bar truss design problem and the welded beam design problem,are used as constrained problems to evaluate the performance of HOA.The results demonstrate that HOA can achieve the best performance among many published algorithms.Therefore,our HOA will most likely become a new optimization tool and can replace the existing optimization methods.

Future work can be expanded in these areas:First,the binary and multi-objective versions of HOA are worth researching.Second,the HOA can be hybridized with other optimization algorithms to improve its performance.Finally,studying the application of HOA in fields such as image processing,inverse geophysical problems,eco-nomic statistical design,and petroleum production optimization is of practical significance.

高中政治教材中收录了很多经典文章，并涉及方方面面的知识。对此，教师在组织教学活动时，应该深入挖掘有效的教学资源，让课堂教学内容更加丰富，这样可以帮助学生增长见识，在潜移默化中提升学生的政治综合素养。教师尤其要注重在课堂上运用微课，为学生带来新奇的体验，让学生尽情徜徉在知识的海洋中，从而开阔学生的视野，提升学生的政治学习能力。

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