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DungDaj 发表于 2022-9-28 07:55

智能优化算法的优化性能判断

文章背景

知乎给我推荐了优化算法 | Gaining Sharing Knowledge based Algorithm（附MATLAB代码），xxx优化算法实在太多了。这些优化算法的名字花里胡哨，配上一个听起来很合理的解释，包装一下，就是一个新的算法。那这些算法到底行不行呢，是骡子还是马，得拉出来溜溜。
Benchmark函数

选择4个比较经典的优化测试函数：Langermann，Damavandi，Ackley，Michalewicz 作为需要被优化的目标函数。



Two-dimensional Langermann function



Two-dimensional Damavandi function



Two-dimensional Ackley function



Two-dimensional Michalewicz function

Langermann函数定义域有一个局部极小值 f(7,9)=-3 ，同时还有很多正负切换的沟壑，这些也是局部极小值。Damavandi函数定义域内有一个局部极小值 f(7,7)=2 ，比较具有欺骗性，一旦陷入这个局部极小值，就很难跳出来了。Ackley函数图像上有不计其数的局部极小值，Michalewicz函数图像上有纵横交错的区域，这些都是陷阱。性能良好的智能优化算法，得跳出这些陷阱，找到全局最小值。
性能测试

一般的，智能优化算法用于离线优化居多，所以这里的性能测试仅针对优化性能：算法能不能找到全局最小值。实时性在离线优化时并不是很重要，而且还和算法的迭代次数有关，而如何定义不同的智能优化算法一次迭代也没有一个统一的标准，所以本文的测试不涉及到智能优化算法的实时性。
以同样的的参数，分别优化上面给到的目标函数多次（50次），统计多次优化之后的目标函数最小值，并进行比较，下面给出性能测试代码：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Gaining-Sharing Knowledge Based Algorithm for Solving Optimization
%% Problems: A Novel Nature-Inspired Algorithm
%% Authors: Ali Wagdy Mohamed, Anas A. Hadi , Ali Khater Mohamed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all;
clear;
clc;

format long;
Alg_Name='GSK';
n_problems=1;
ConvDisp=1;
Run_No=50;%重复求解多少次

problem_size = 2;
max_nfes = 300 * 100;
rand('seed', sum(100 * clock));
val_2_reach = 10^(-8);

%% Target function
flag = 1;
if flag == 1
    xmin = ;
    xmax = ;
    fun = @langermann;
elseif flag ==2
    xmin = ;
    xmax = ;
    fun = @damavandi;
elseif flag == 3
    xmin = [-2,-2];
    xmax = ;
    fun = @ackley;
else
    xmin = ;
    xmax = ;
    fun = @michal;
end
max_region = xmin(1);
min_region = xmax(2);
lu = ;
analysis= zeros(n_problems,6);

%% Core process
optimum = flag * 100.0;
%% Record the best results
outcome = [];
fprintf('\n-------------------------------------------------------\n')
fprintf('Function = %d, Dimension size = %d\n', flag, problem_size)
dim1=[];
dim2=[];
bsf_fit_var_record = cell(Run_No,1);
bsf_solution_record = cell(Run_No,1);
for run_id = 1 : Run_No
    bsf_error_val=[];
    run_funcvals = [];
    pop_size = 100;
    G_Max=fix(max_nfes/pop_size);
   
    %% Initialize the main population
    popold = repmat(lu(1, :), pop_size, 1) + rand(pop_size, problem_size) .* (repmat(lu(2, :) - lu(1, :), pop_size, 1));
    pop = popold; % the old population becomes the current population
   
    fitness = fun(mat2cell(pop, ones(size(pop, 1), 1), problem_size));
   
    nfes = 0;
    bsf_fit_var = 1e+300;
   
    %%%%%%%%%%%%%%%%%%%%%%%% for out
    for i = 1 : pop_size
      nfes = nfes + 1;
      %%      if nfes > max_nfes; exit(1); end
      if nfes > max_nfes; break; end
      if fitness(i) < bsf_fit_var
            bsf_fit_var = fitness(i);
            bsf_solution = pop(i, :);
      end
      run_funcvals = ;
    end
   
    %%%%%%%%%%%%%%%%%%%%%%%% Parameter settings%%%%%%%%%%
    KF=0.3;% Knowledge Factor
    KR=0.6;% Knowledge Ratio
    K=10*ones(pop_size,1);%Knowledge Rate
   
    g=0;
    %% main loop
    while nfes <= max_nfes
      g=g+1;
      D_Gained_Shared_Junior=ceil((problem_size)*(1-g/G_Max).^K);
      D_Gained_Shared_Senior=problem_size-D_Gained_Shared_Junior;
      pop = popold; % the old population becomes the current population
      
       = sort(fitness, 'ascend');
       = Gained_Shared_Junior_R1R2R3(indBest);
      
       = Gained_Shared_Senior_R1R2R3(indBest);
      R01=1:pop_size;
      Gained_Shared_Junior=zeros(pop_size, problem_size);
      ind1=fitness(R01)>fitness(Rg3);
      
      if(sum(ind1)>0)
            Gained_Shared_Junior (ind1,:)= pop(ind1,:) + KF*ones(sum(ind1), problem_size) .* (pop(Rg1(ind1),:) - pop(Rg2(ind1),:)+pop(Rg3(ind1), :)-pop(ind1,:)) ;
      end
      ind1=~ind1;
      if(sum(ind1)>0)
            Gained_Shared_Junior(ind1,:) = pop(ind1,:) + KF*ones(sum(ind1), problem_size) .* (pop(Rg1(ind1),:) - pop(Rg2(ind1),:)+pop(ind1,:)-pop(Rg3(ind1), :)) ;
      end
      R0=1:pop_size;
      Gained_Shared_Senior=zeros(pop_size, problem_size);
      ind=fitness(R0)>fitness(R2);
      if(sum(ind)>0)
            Gained_Shared_Senior(ind,:) = pop(ind,:) + KF*ones(sum(ind), problem_size) .* (pop(R1(ind),:) - pop(ind,:) + pop(R2(ind),:) - pop(R3(ind), :)) ;
      end
      ind=~ind;
      if(sum(ind)>0)
            Gained_Shared_Senior(ind,:) = pop(ind,:) + KF*ones(sum(ind), problem_size) .* (pop(R1(ind),:) - pop(R2(ind),:) + pop(ind,:) - pop(R3(ind), :)) ;
      end
      Gained_Shared_Junior = boundConstraint(Gained_Shared_Junior, pop, lu);
      Gained_Shared_Senior = boundConstraint(Gained_Shared_Senior, pop, lu);
      
      
      D_Gained_Shared_Junior_mask=rand(pop_size, problem_size)<=(D_Gained_Shared_Junior(:, ones(1, problem_size))./problem_size);
      D_Gained_Shared_Senior_mask=~D_Gained_Shared_Junior_mask;
      
      D_Gained_Shared_Junior_rand_mask=rand(pop_size, problem_size)<=KR*ones(pop_size, problem_size);
      D_Gained_Shared_Junior_mask=and(D_Gained_Shared_Junior_mask,D_Gained_Shared_Junior_rand_mask);
      
      D_Gained_Shared_Senior_rand_mask=rand(pop_size, problem_size)<=KR*ones(pop_size, problem_size);
      D_Gained_Shared_Senior_mask=and(D_Gained_Shared_Senior_mask,D_Gained_Shared_Senior_rand_mask);
      ui=pop;
      
      ui(D_Gained_Shared_Junior_mask) = Gained_Shared_Junior(D_Gained_Shared_Junior_mask);
      ui(D_Gained_Shared_Senior_mask) = Gained_Shared_Senior(D_Gained_Shared_Senior_mask);
      
      children_fitness = fun(mat2cell(ui, ones(size(ui, 1), 1), problem_size));
      
      for i = 1 : pop_size
            nfes = nfes + 1;
            if nfes > max_nfes; break; end
            if children_fitness(i) < bsf_fit_var
                bsf_fit_var = children_fitness(i);
                bsf_solution = ui(i, :);
            end
            run_funcvals = ;
            bsf_fit_var_record{run_id} = ;
            bsf_solution_record{run_id} = ;
      end
      
       = min(, [], 2);
      
      popold = pop;
      popold(Child_is_better_index == 2, :) = ui(Child_is_better_index == 2, :);
      
      fprintf('NFES: %d, bsf_fit: %1.6e, D_Gained_Shared_Junior: %2.2e, D_Gained_Shared_Senior: %2.2e\n', nfes/100,bsf_fit_var,problem_size*sum(sum(D_Gained_Shared_Junior))/(pop_size*problem_size),problem_size*sum(sum(D_Gained_Shared_Senior))/(pop_size*problem_size))
%         bsf_fit_var_record{run_id} = ;
%         bsf_solution_record{run_id} = ;
    end % end while loop
   
    bsf_error_val = bsf_fit_var - optimum;
    if bsf_error_val < val_2_reach
      bsf_error_val = 0;
    end
   
%   fprintf('%d th run, best-so-far error value = %1.8e\n', run_id , bsf_error_val)
    outcome = ;
   
    %% plot convergence figures
    if (ConvDisp)
      run_funcvals=run_funcvals-optimum;
      run_funcvals=run_funcvals';
      dim1(run_id,:)=1:length(run_funcvals);
      dim2(run_id,:)=log10(run_funcvals);
    end
    %%%%%%%%%%%%%%%%%%%%%%%%%%%
end %% end 1 run

%% save ststiatical output in analysis file%%%%
func = 1;
analysis(func,1)=min(outcome);
analysis(func,2)=median(outcome);
analysis(func,3)=max(outcome);
analysis(func,4)=mean(outcome);
analysis(func,5)=std(outcome);
median_figure=find(outcome== median(outcome));
analysis(func,6)=median_figure(1);

%% print statistical output and save convergence figures%%%
% fprintf('%e\n',min(outcome));
% fprintf('%e\n',median(outcome));
% fprintf('%e\n',mean(outcome));
% fprintf('%e\n',max(outcome));
% fprintf('%e\n',std(outcome));
% dim11=dim1(median_figure,:);
% dim22=dim2(median_figure,:);

%% plot bsf_fit_var
bsf_fit_var_best_record = cellfun(@min, bsf_fit_var_record);
plot(bsf_fit_var_best_record, 'r', 'Linewidth', 1.5);
xlabel('run number');
ylabel('object function');
grid on;
set(gca,'GridLineStyle','--');
fprintf('Mean and Std of solution\n');
display();
[~, run_id_best] = min(bsf_fit_var_best_record);

figure(2)
plot(bsf_fit_var_record{run_id_best}, 'r', 'Linewidth', 1.5);
xlabel('iteration');
ylabel('object function');
grid on;
set(gca,'GridLineStyle','--');
xlim();
title(['bsf\_solution: ', num2str(bsf_fit_var_record{run_id_best}(end))]);

figure(3); hold on;
scatter(bsf_solution_record{run_id_best}(:, 1), bsf_solution_record{run_id_best}(:, 2), 'MarkerFaceColor', 'y', 'MarkerEdgeColor', 'k');
scatter(bsf_solution_record{run_id_best}(end, 1), bsf_solution_record{run_id_best}(end, 2), 'MarkerFaceColor', 'b', 'MarkerEdgeColor', 'k');
xlim();
ylim();
grid on;
set(gca,'GridLineStyle','--');
xlabel('$x_1$', 'interpreter', 'latex');
ylabel('$x_2$', 'interpreter', 'latex');

% print bsf_fit_var and bsf_solution
if exist('bsf_fit_var', 'var') && exist('bsf_solution', 'var')
    fprintf('bsf_solution: %.4f\n', bsf_fit_var_record{run_id_best}(end));
    fprintf('bsf_solution: x1 = %.4f\n',bsf_solution_record{run_id_best}(end, 1));
    fprintf('bsf_solution: x2 = %.4f\n',bsf_solution_record{run_id_best}(end, 2));
end

function lang = langermann(x)
a = ;
b = ;
c = ;

lang =zeros(size(x,1),1);
for j =1:size(x,1)
    s = 0;
    xj = x{j};
    x1 = xj(1);
    x2 = xj(2);
    for i = 1:5
      zi = (x1-a(i)).^2 + (x2 - b(i)).^2;
      s = s - c(i)*cos(pi.*zi)./exp(zi./pi);
    end
    lang(j) = s;
end
end

function dama = damavandi(x)
dama = zeros(size(x,1),1);
for j = 1:size(x,1)
    xj = x{j};
    x1 = xj(1);
    x2 = xj(2);
    dama(j)=(1-(abs((sin(pi*(x1-2))*sin(pi*(x2-2)))/(pi*pi*(x1-2)*(x2...
      -2)))).^5)*(2+(x1-7).^2+2*(x2-7).^2);
end
end

function ackley = ackley(x)
ackley =zeros(size(x,1),1);
for j =1:size(x,1)
    xj = x{j};
    x1 = xj(1);
    x2 = xj(2);
    e = exp(1);
    s = 20 + e -20.*exp(-0.2*sqrt((x1^2+x2^2)/2)) - exp(1/2*(cos(2*pi*x1)+cos(2*pi*x2)));
    ackley(j) = s;
end
end

function mi = michal(x)
mi = zeros(size(x,1),1);
for j = 1:size(x,1)
    xj = x{j};
    x1 = xj(1);
    x2 = xj(2);
    s = sin(x1)*sin(x1^2/pi)^20 + sin(x2)*sin(2*x2^2/pi)^20;
    mi(j) = -s;
end
end结果分析

目标函数全局最小值局部极小值优化均值优化标准差Langerman5.1621-3-5.009190.06424Damavandi022.335860.48733Ackley0-0.872810.50766Michalewicz-1.8013--1.418390.25701


Langerman函数的50次优化结果



Damavandi函数的50次优化结果



Ackley函数的50次优化结果



Michalewicz的50次优化结果

测试的算法为Gaining Sharing Knowledge based Algorithm，测试的4个目标函数中，只有对Langerman函数的优化接近其全局最小值，其余的三个函数，50次中最多只有3次优化结果比较接近全局最小值，全局最优解命中率比较低。从50次求解的结果来看，每次求解的结果不一致，波动比较大。
总结

各种层出不穷的智能优化算法或多或少都有以下问题：

[*]全局最优解搜索能力有限，使用时一定要擦亮眼睛
[*]优化求解结果具有不一致性，重复运行多次得到的最优解不一定一致
[*]优化求解精度有限，即使解分布在真实的全局最优解附近，波动也不小

查看完整版本: 智能优化算法的优化性能判断