GA对peaks函数进行极值寻优

peaks函数
$$f(x,y)=3(1-x{)}^2\exp (-x^2-(y+1{)}^2)-10(\frac{x}{5}-x^3-y^5)\exp (-x^2-y^2)-\frac{1}{3}\exp (-(x+1{)}^2-y^2)$$

使用GA Toolbox进行寻优计算，对比fmincon寻优结果

clc;
clear all;
close all;

peaks
problem = createOptimProblem('fmincon', 'objective', @(x) peaks(x(:, 1), x(:, 2)), 'nonlcon', @circularConstraint, ...
    'x0', [-1, 1], 'lb', [-3, 3], 'ub', [3 3]);

% fmincon for the optimal values
[x1, f1] = fmincon(problem);

% GA for global optimal values
problem.solver = 'ga';
problem.fitnessfcn = problem.objective;
problem.nvars = 2;
problem.options = gaoptimset('PopInitRange', [-3; 3]);

GA-PSO寻优

PSO在计算函数极值时，会出现早熟现象，导致求解函数极值存在较大的偏差，而遗传算法中对于函数采用了Select, Mutation, Cross等算子操作，直接以目标函数作为搜索信息，可以提高PSO算法的全局寻优能力，加快了算法的进化速度，且可以提高算法的收敛精度。

clc;
clear all;
close all;

%% initialize parameters
lenchrom = 3;
pc = 0.7; % probability of cross
pm = 0.3; % probability of mutation

% parameters of PSO
c1 = 1.49445;
c2 = 1.49445;

maxgen = 100;
popsize = 30;

% Velocity of update particles
Vmax = 1;
Vmin = -1;

% Population
popmax = 15;
popmin = 0;

% Range
bound = [popmin popmax; popmin popmax; popmin popmax];

% number of particles for optimizing
par_num = 3;

%% initial particles & velocity
for i=1:popsize
    % generate a population randomly
    pop(i, :) = 3*abs(rands(1, par_num));
    V(i, :) = rands(1, par_num);
    fitness(i) = func(pop(i, :));
end

% find the best chromosome
[bestfitness, bestindex] = min(fitness);
zbest = pop(bestindex, :);
gbest = pop;
fitnessgbest = fitness;
fitnesszbest = bestfitness;

% find the optimal value
for i=1:maxgen
    i
    for j = 1:popsize
        % Velocity: PSO
        V(j, :) = V(j, :)+c1*rand*(gbest(j, :)-pop(j, :))+c2*rand*(zbest- pop(j, :));
        V(j, find(V(j, :)>Vmax)) = Vmax;
        V(j, find(V(j, :)<Vmin)) = Vmin;
        
        % Population: PSO
        pop(j, :) = pop(j, :)+0.5*V(j, :);
        pop(j, find(pop(j, :)>popmax)) = popmax;
        pop(j, find(pop(j, :)<popmin)) = popmin;
        
        % Cross: GA
        GApop = Cross(pc, lenchrom, pop, popsize, bound);
        % Mutation: GA
        GApop = Mutation(pm, lenchrom, GApop, popsize, [i maxgen], bound);
        
        pop = GApop;
        
        % fitness --> constraint
        if pop(j, 1)-pop(j, 2)+pop(j, 3)<=20
            if 3*pop(j, 1)+2*pop(j, 2)+4*pop(j, 3)<=42
                if 3*pop(j, 1)+2*pop(j, 2)<=30
                    fitness(j) = func(pop(j, :));
                end
            end
        end
        
        % update the individual optimal value
        if fitness(j) < fitnessgbest(j)
            gbest(j, :) = pop(j, :);
            fitnessgbest(j) = fitness(j);
        end
        
        % update the population optimal value
        if fitness(j) < fitnesszbest
            zbest = pop(j, :);
            fitnesszbest = fitness(j);
        end
    end
    yy(i) = fitnesszbest;
end

%% results & plot
disp '*********************best particle number**********************'
zbest

plot(yy, 'linewidth', 1.5);
grid on;
title(['Fitness Curve ' 'Generations of Stopping: ' num2str(maxgen)]);
xlabel('Generation of evolution');
ylabel('Fitness');

Issue: Code

二进制编码的结构不能直接反映问题的固有结构，精度不高，个体长度大以及占用较多的内存。解决该问题的方法有：

1. 动态编码，在保持串长不变的情况下减少搜索区域，当算法收敛到某局部最优时增加搜索精度，使得算法可以在全局最优点附近进行更加精细化的搜索。
2. 对于问题变量时实向量的情况，可以直接采用实数编码
3. 复数编码是为了描述和求解二维问题