Particle swarm optimization
Particle swarm optimization (PSO) was proposed in 1995 by Dr Eberhart and Dr Kennedy, based on a study of the predatory behaviour of flocks of birds. Its basic core is to make use of the information sharing of individuals in the group so that the movement of the whole group can evolve from disorder to order in the problem solving space, so as to obtain the optimal solution of the problem. Consider the scene: a flock of birds are foraging for food, and there is a field of corn in the distance. None of the birds know exactly where the field is, but they know how far away they are from it. So the best strategy for finding a cornfield, the simplest and most effective strategy, is to search the area around the nearest flock.
In PSO, the solution of each optimization problem is a bird in the search space, called a “particle”, and the optimal solution of the problem corresponds to the “corn field” in the bird flock. All particles have a position vector (the position of the particle in the solution space) and a velocity vector (which determines the direction and speed of the next flight), and the fitness value of the current position can be calculated according to the objective function, which can be understood as the distance from the “corn field”. In each iteration, the examples in the population can learn not only from their own experience (historical location), but also from the “experience” of the optimal particles in the population, so as to determine how to adjust and change the direction and speed of flight in the next iteration. In this way, the whole population of examples will gradually approach the optimal solution.
The above explanation may seem abstract, but a simple example is used to illustrate it
There are two people in a lake who can communicate with each other and can detect the lowest point in their position. The initial position is shown in the picture above, and since the right side is deep, the person on the left will move the boat to the right.
Now it’s deeper on the left, so the person on the right will move the boat a little bit to the left
The process is repeated until the two boats meet
A locally optimal solution is obtained
Each individual is represented as a particle. The position of each individual at a given time is x(t), and the direction is v(t).
P (t) is the optimal solution of x individual at time t, g(t) is the optimal solution of all individuals at time t, v(t) is the direction of the individual at time t, and x(t) is the position of the individual at time T
The next position is shown above and is determined by x, P and g
The particles in the population can find the optimal solution of the problem by continuously learning from the historical information of themselves and the population.
However, in subsequent studies, the table shows that there is a problem in the original formula above: the update of V in the formula is too random, so that the global optimization ability of the whole PSO algorithm is strong, but the local search ability is poor. In fact, we need PSO to have strong global optimization ability at the early stage of algorithm iteration, while in the later stage of algorithm, the whole population should have stronger local search ability. Therefore, based on the above disadvantages, Shi and Eberhart modified the formula by introducing inertia weight, and thus proposed the inertia weight model of PSO:
The components of each vector are represented as follows
W as PSO inertia weight, it values between [0, 1] interval, general applications adopt adaptive accessor methods, namely the beginning w = 0.9, makes the PSO global optimization ability is stronger, with the deepening of the iteration, diminishing parameter w, so that the PSO with strong local optimization ability, at the end of an iteration, w = 0.1. The parameters C1 and c2 are called learning factors and are generally set to 1,4961. R1 and r2 are random probability values between [0,1].
The algorithm framework of the whole particle swarm optimization algorithm is as follows:
Step1 population initialization, random initialization or specific initialization method can be designed according to the optimized problem, and then the individual adaptive value is calculated to select the local optimal position vector of the individual and the global optimal position vector of the population.
Step2 set iteration: set the iteration number and set the current iteration number to 1
Step3 Speed update: Update the speed vector of each individual
Step4 Position update: Update the position vector of each individual
Step5 update local position and global position vector: update the local optimal solution of each individual and the global optimal solution of the population
Step 6 Judgment of termination conditions: when judging the number of iterations, the maximum number of iterations is reached. If so, output the global optimal solution; otherwise, continue the iteration and jump to Step 3.
The application of particle swarm optimization algorithm is mainly about the design of velocity and position vector iterative operator. The effectiveness of the iterator will determine the performance of the whole PSO algorithm, so how to design the iterator of PSO is the focus and difficulty of the application of PSO algorithm.
2. Improved Particle Swarm Optimization Algorithm based on Nonlinear Decreasing Inertia Weight (IMPSO)
Against in the premature convergence of particle swarm optimization algorithm and the convergence problem, put forward a kind of based on natural selection and nonlinear decreasing inertia weight of the improved particle swarm algorithm, in the process of iteration algorithm, the speed of particle boundary nonlinear decreasing maximum speed change strategy is used to limit, nonlinear decreasing inertia weight change is used to balance the early stage of the particle population late global search and local search Superior ability; In order to keep the diversity of the population in the process of evolution, the second-order oscillation strategy is used in the standard particle swarm optimization algorithm to keep the diversity of the population in the process of evolution. On this basis, the selection mechanism of genetic algorithm and particle swarm optimization algorithm are combined to improve the applicability of the algorithm. The proposed algorithm is verified by several benchmark functions and compared with other existing algorithms. Experimental results show that the proposed algorithm has more obvious advantages in search accuracy and optimization ability, especially in multi-dimensional, multi-peak and other complex nonlinear optimization problems, and has strong competitiveness.
Three, part of the code
function [gbest,gbestval,fitcount]= CLPSO_new_func(fhd,Max_Gen,Max_FES,Particle_Number,Dimension,VRmin,VRmax,varargin) % [gbest, gbestval fitcount] = CLPSO_new_func (" f8 ", 3500200, 000,30,30, 5.12, 5.12) rand (' state ', the sum (100 * clock)); me=Max_Gen; ps=Particle_Number; D=Dimension; cc=[1 1]; %acceleration constants t=0:1/(ps-1):1; t=5.*t; Pc = 0.0 + (0.5 0.0). * (exp (t) - exp (t (1))). / (exp (ps) (t) - exp (t (1))); Pc % = 0.5. * 'ones (1, ps); m=0.*ones(ps,1); Iwt = 0.9 - (1: me) * (0.7 / me); % iwt = 0.729 - (1: me) * (0.0 / me); Cc = [1.49445 1.49445]; if length(VRmin)==1 VRmin=repmat(VRmin,1,D); VRmax=repmat(VRmax,1,D); End the mv = 0.2 * (VRmax - VRmin); VRmin=repmat(VRmin,ps,1); VRmax=repmat(VRmax,ps,1); Vmin=repmat(-mv,ps,1); Vmax=-Vmin; pos=VRmin+(VRmax-VRmin).*rand(ps,D); for i=1:ps; e(i,1)=feval(fhd,pos(i,:),varargin{:}); end fitcount=ps; vel=Vmin+2.*Vmax.*rand(ps,D); %initialize the velocity of the particles pbest=pos; pbestval=e; %initialize the pbest and the pbest's fitness value [gbestval,gbestid]=min(pbestval); gbest=pbest(gbestid,:); %initialize the gbest and the gbest's fitness value gbestrep=repmat(gbest,ps,1); stay_num=zeros(ps,1); ai=zeros(ps,D); f_pbest=1:ps; f_pbest=repmat(f_pbest',1,D); for k=1:ps ar=randperm(D); ai(k,ar(1:m(k)))=1; fi1=ceil(ps*rand(1,D)); fi2=ceil(ps*rand(1,D)); fi=(pbestval(fi1)<pbestval(fi2))'.*fi1+(pbestval(fi1)>=pbestval(fi2))'.*fi2; bi=ceil(rand(1,D)-1+Pc(k)); if bi==zeros(1,D),rc=randperm(D); bi(rc(1))=1; end f_pbest(k,:)=bi.*fi+(1-bi).*f_pbest(k,:); end stop_num=0; i=1; while i<=me&fitcount<=Max_FES i=i+1; for k=1:ps if stay_num(k)>=5 % if round(i/10)==i/10%|stay_num(k)>=5 stay_num(k)=0; ai(k,:)=zeros(1,D); f_pbest(k,:)=k.*ones(1,D); ar=randperm(D); ai(k,ar(1:m(k)))=1; fi1=ceil(ps*rand(1,D)); fi2=ceil(ps*rand(1,D)); fi=(pbestval(fi1)<pbestval(fi2))'.*fi1+(pbestval(fi1)>=pbestval(fi2))'.*fi2; bi=ceil(rand(1,D)-1+Pc(k)); if bi==zeros(1,D),rc=randperm(D); bi(rc(1))=1; end f_pbest(k,:)=bi.*fi+(1-bi).*f_pbest(k,:); end for dimcnt=1:D pbest_f(k,dimcnt)=pbest(f_pbest(k,dimcnt),dimcnt); end aa(k,:)=cc(1).*(1-ai(k,:)).*rand(1,D).*(pbest_f(k,:)-pos(k,:))+cc(2).*ai(k,:).*rand(1,D).*(gbestrep(k,:)-pos(k,:)); %~~~~~~~~~~~~~~~~~~~~~~ vel(k,:)=iwt(i).*vel(k,:)+aa(k,:); vel(k,:)=(vel(k,:)>mv).*mv+(vel(k,:)<=mv).*vel(k,:); vel(k,:)=(vel(k,:)<(-mv)).*(-mv)+(vel(k,:)>=(-mv)).*vel(k,:); pos(k,:)=pos(k,:)+vel(k,:); if (sum(pos(k,:)>VRmax(k,:))+sum(pos(k,:)<VRmin(k,:)))==0; e(k,1)=feval(fhd,pos(k,:),varargin{:}); fitcount=fitcount+1; tmp=(pbestval(k)<=e(k)); if tmp==1 stay_num(k)=stay_num(k)+1; end temp=repmat(tmp,1,D); pbest(k,:)=temp.*pbest(k,:)+(1-temp).*pos(k,:); pbestval(k)=tmp.*pbestval(k)+(1-tmp).*e(k); %update the pbest if pbestval(k)<gbestval gbest=pbest(k,:); gbestval=pbestval(k); gbestrep=repmat(gbest,ps,1); %update the gbest end end end % if round(i/100)==i/100 % plot(pos(:,D-1),pos(:,D),'b*'); hold on; % for k=1:floor(D/2) % plot(gbest(:,2*k-1),gbest(:,2*k),'r*'); % end % hold off % title(['PSO: ',num2str(i),' generations, Gbestval=',num2str(gbestval)]); % axis([VRmin(1,D-1),VRmax(1,D-1),VRmin(1,D),VRmax(1,D)]) % drawnow % end if fitcount>=Max_FES break; end if (i==me)&(fitcount<Max_FES) i=i-1; end end gbestvalCopy the code4. Simulation results
Five, reference and code private message blogger
[1] Hua Yong, WANG Shuangyuan, Bai Guozhen, Li Bingchu. Improved particle swarm optimization algorithm based on nonlinear decreasing inertia weights, Journal of Chongqing Technology and Business University (Natural Science Edition), 2021 (02).