最近整理以前的代碼，將以前老師上課的作業代碼重新整理，分享出來，作業的內容是編寫單純形法，對測試函數進行尋優(極大值或者極小值)。
首先介紹一下單純形法：將上課的ppt轉化爲圖片。ppt藍色背景，眼睛快看瞎了
按照ppt的描述編寫算法如下：
clear all;
clc;
% mode可以選擇測試函數
% mode = 'exp_test';
% mode = 'Schaffer';
mode = 'Rosen_Brock';
if strcmp(mode,'Schaffer')
x=-4:0.1:4;
y=-4:0.1:4;
[X,Y]=meshgrid(x,y);
Z = 0.5-((sin(sqrt(X.^2+Y.^2)).^2)-0.5)./(1+0.001.*(X.^2+Y.^2)).^2;
figure(1);
surf(X,Y,Z);
title('Schaffer Function');
xlabel('X-軸');
ylabel('Y-軸');
zlabel('Z-軸');
end
if strcmp(mode,'exp_test')
x=-4:0.1:4;
y=-4:0.1:4;
[X,Y]=meshgrid(x,y);
% Z = 1400-(20*(X.^2-Y.^2).^2-(1-Y).^2-3*(1+Y).^2+0.3);
% Z = 5*sin(X.*Y)+X.^2+Y.^2;
Z = X.*exp(-X.^2-Y.^2)+1;
figure(1);
surf(X,Y,Z);
title('exp_test Function');
xlabel('X-軸');
ylabel('Y-軸');
zlabel('Z-軸');
end
if strcmp(mode,'Rosen_Brock')
x=-2.048:0.1:2.048;
y=-2.048:0.1:2.048;
[X,Y]=meshgrid(x,y);
% Z = 1400-(20*(X.^2-Y.^2).^2-(1-Y).^2-3*(1+Y).^2+0.3);
% Z = 5*sin(X.*Y)+X.^2+Y.^2;
% Z = X.*exp(-X.^2-Y.^2)+1;
Z = 100.*(Y-X.^2).^2+(1-X).^2;
figure(1);
surf(X,Y,Z);
title('Rosen Brock valley Function');
xlabel('X-軸');
ylabel('Y-軸');
zlabel('Z-軸');
end
if strcmp(mode,'Rosen_Brock')
variable_min_x = -2.048; % 變量的上下限
variable_max_x = 2.048;
variable_min_y = -2.048; % 變量的上下限
variable_max_y = 2.048;
else
variable_min_x = -4; % 變量的上下限
variable_max_x = 4;
variable_min_y = -4; % 變量的上下限
variable_max_y = 4;
end
% variable_min_x = -4; % 變量的上下限
% variable_max_x = 4;
% variable_min_y = -4; % 變量的上下限
% variable_max_y = 4;
yita = 0.5;
alpha = 1.3;
precision = 10^(-3);
% Schaffer function
% x=variable_min_x:0.1:variable_max_x;
% y=variable_min_y:0.1:variable_max_y;
% [X,Y]=meshgrid(x,y);
% Z = 0.5-((sin(sqrt(X.^2+Y.^2)).^2)-0.5)./(1+0.001.*(X.^2+Y.^2)).^2;
% surf(X,Y,Z);
% 複合型法尋找schafer functiong 最大值
%% 產生初始的頂點 sige
% M=4;
% N=3;
x=zeros(4,2);
for i=1:4
x_1 = variable_min_x+rand*(variable_max_x-variable_min_x);
x_2 = variable_min_y+rand*(variable_max_y-variable_min_y);
x(i,:)=[x_1,x_2];
end
max_iteraton = 250; % 最大的迭代次數
%%
% 判斷索取點是否滿足要求：
index_error = 1;
error = [];
optimization_path = [];
for gen=1:max_iteraton
% 判斷點是否在可行域內
alpha = 1.3;
for i=1:4
if ((x(i,1)<variable_min_x)||(x(i,1)>variable_max_x))||((x(i,2)<variable_min_y)||(x(i,2)>variable_max_y))
error(index_error) = i; % 不滿足的點
index_error = index_error+1;
end
end
% error中存放不滿足條件的點的索引值
if isempty(error) == 0 % error 非空
% 對不滿足的點進行處理
norm = []; % 滿足約束條件的點
abnorm = []; % 不滿足約束條件的點
%%%%%%%%%%%-----
len_error = length(error);
for i=1:len_error
abnorm(i,:) = x(error(i),:); % 壞點
end
kk = 1;
for m = 1:4
%kk = kk+1;
for n = 1:len_error
if x(m,:) == abnorm(n,:)
break;
end
if n >= len_error
norm(kk,:) = x(m,:);
kk = kk+1;
end
end
end
% 將不滿足條件的點進行調整
% 正常點的形星
norm_center = 0;
for m = 1:size(norm,1)
norm_center = norm_center + norm(m,:);
end
norm_center = norm_center/(size(norm,1));
for j = 1:size(abnorm,1)
abnorm_new(j,:) = norm_center + yita*(abnorm(j,:)-norm_center);
end
% 組成新的點
x = [norm;abnorm_new];
end
% 計算函數值
for k = 1:4
f(k) = test_function(x(k,:),mode);
end
[f_min,index_1] = min(f); % 尋找最好的點
[f_max,index_2] = max(f);
% 最差的點X_h 次差的點X_g 最好的點X_l
X_h = x(index_1,:);
X_l = x(index_2,:);
cnt = 1;
left_points = [];
left_two_points = [];
for i=1:4
if (i == index_1)||(i == index_2)
continue;
else
left_points(cnt) = f(i); % 除過最好的點和最差的點
left_two_points(cnt,:) = x(i,:);
cnt = cnt + 1;
end
end
[left_points_min,index_3] = min(left_points);
X_g = left_two_points(index_3,:);
% 求形星：
cnt = 1;
for i=1:4
if i==index_1
continue;
else
points_except_worst(cnt,:) = x(i,:);
cnt = cnt + 1;
end
end
% 計算形星
sum = 0;
for i = 1:length(points_except_worst)
sum = sum + points_except_worst(i,:);
end
X_c = sum/length(points_except_worst);
% 判斷形星是否在可行域內
if ((X_c(1)>variable_min_x)&&(X_c(1)<variable_max_x))&&((X_c(2)>variable_min_y)&&(X_c(2)<variable_max_y)) % 在可行域內
% 求反射點
X_r = X_c + alpha*(X_c - X_h);
while 1
if ((X_r(1)>variable_min_x)&&(X_r(1)<variable_max_x))&&((X_r(2)>variable_min_y)&&(X_r(2)<variable_max_y)) % 反射點在可行域內
break;
else
if alpha < precision
break;
end
alpha = alpha*0.5;
X_r = X_c + alpha*(X_c - X_h);
end
end
else % 形星不在可行域內
% 重新確定上下界
if X_l(1) <= X_c(1)
variable_min_x = X_l(1);
variable_max_x = X_c(1);
else
variable_min_x = X_c(1);
variable_max_x = X_1(1);
end
if X_l(2) <= X_c(2)
variable_min_y = X_l(2);
variable_max_y = X_c(2);
else
variable_min_y = X_c(2);
variable_max_y = X_l(2);
end
% 產生新的複合型
for i=1:4
x_1 = variable_min_x+rand*(variable_max_x-variable_min_x);
x_2 = variable_min_y+rand*(variable_max_y-variable_min_y);
x(i,:)=[x_1,x_2];
end
for k = 1:4
f(k) = Schaffer(x(k,:));
end
[f_min,index_1] = min(f); % 最差的點
[f_max,index_2] = max(f); % 尋找最好的點
% 最差的點X_h 次差的點X_g 最好的點X_l
X_h = x(index_1,:);
X_l = x(index_2,:);
cnt = 1;
left_two_points = [];
for i=1:4
if (i == index_1)||(i == index_2)
continue;
else
left_points(cnt) = f(i); % 除過最好的點和最差的點
left_two_points(cnt,:) = x(i,:);
cnt = cnt + 1;
end
end
[left_points_min,index_3] = min(left_points);
X_g = left_two_points(index_3,:);
% 求形星：
cnt = 1;
for i=1:4
if i==index_1
continue;
else
points_except_worst(cnt,:) = x(i,:);
cnt = cnt + 1;
end
end
% 由新的點產生的形星
sum = 0;
for i = 1:length(points_except_worst)
sum = sum + points_except_worst(i,:);
end
X_c = sum/length(points_except_worst);
% 求反射點
X_r = X_c + alpha*(X_c - X_h);
while 1
if ((X_r(1)>variable_min_x)&&(X_r(1)<variable_max_x))&&((X_r(2)>variable_min_y)&&(X_r(2)<variable_max_y)) % 反射點在可行域內
break;
else
if alpha < precision
break;
end
alpha = alpha*0.5;
X_r = X_c + alpha*(X_c - X_h);
end
end
% 反射點計算結束
end
% x X-r X_h X_l X_g
f_X_r = test_function(X_r,mode);
flag_1 = 0;
if f_X_r > f_min
x(index_1,:) = X_r; % 替換最差的點
else
% 重新計算反射點
while test_function(X_r,mode) < f_min
alpha = alpha*0.5;
X_r = X_c + alpha*(X_c - X_h);
if alpha < precision
flag_1 = 1;
break;
end
end
% 重新替換
if flag_1 == 0 % 新的反射點滿足約束條件
x(index_1,:) = X_r;
else % alpha 減小到足夠的範圍任然沒有找到合適的點
% 使複合型向最好的點收縮 再重新計算一次新的複合型
%Nop();
%disp('nono');
x(index_1,:) = X_g;
end
end
% x 更新完畢 如果想畫成凸多邊形，要處理x
figure(2);
for i=1:4-1
plot([x(i,1),x(i+1,1)], [x(i,2),x(i+1,2)]);
hold on;
end
plot([x(4,1),x(1,1)],[x(4,2),x(1,2)]);
hold off;
pause(0.01);
disp([gen, alpha]);
function_value(gen) = test_function(X_l,mode);
max_value = test_function(X_l,mode);
optimization_path(gen,:) = X_l;
end
figure(3);
plot(function_value);
title(['函數值變化曲線',' 最大值：',num2str(max_value)]);
xlabel('迭代次數');
ylabel('函數值');
figure(4);
plot(optimization_path(:,1),optimization_path(:,2));
title('尋優軌跡');
xlabel('最優點-X');
ylabel('最優點-Y');
% function f = Schaffer(buf)
% f = 0.5-((sin(sqrt(buf(1).^2+buf(2).^2)).^2)-0.5)./(1+0.001.*(buf(1).^2+buf(2).^2)).^2;
% end
% function f = test_function(buf,mode)
% if strcmp(mode,'Schaffer')
% f = 0.5-((sin(sqrt(buf(1).^2+buf(2).^2)).^2)-0.5)./(1+0.001.*(buf(1).^2+buf(2).^2)).^2;
% end
%
% % if mode=='Rastrigin'
% % f = (buf(1).^2-10*cos(2*pi*buf(1))+10) + (buf(2).^2-10*cos(2.*pi.*buf(2))+10);
% % end
%
% if strcmp(mode,'exp_test')
% f = buf(1)*exp(-buf(1)^2-buf(2)^2)+1;
% end
%
% if strcmp(mode,'Rosen_Brock')
% f = 100*(buf(2)-buf(1).^2).^2+(1-buf(1)).^2;
% end
%
% end
test_function.m文件
function f = test_function(buf,mode)
if strcmp(mode,'Schaffer')
f = 0.5-((sin(sqrt(buf(1).^2+buf(2).^2)).^2)-0.5)./(1+0.001.*(buf(1).^2+buf(2).^2)).^2;
end
% if mode=='Rastrigin'
% f = (buf(1).^2-10*cos(2*pi*buf(1))+10) + (buf(2).^2-10*cos(2.*pi.*buf(2))+10);
% end
if strcmp(mode,'exp_test')
f = buf(1)*exp(-buf(1)^2-buf(2)^2)+1;
end
if strcmp(mode,'Rosen_Brock')
f = 100*(buf(2)-buf(1).^2).^2+(1-buf(1)).^2;
end
end
代碼運行結果：
1.Schaffer函數（函數是不是很有特點呢）
尋優的值：
2. Rosen_Brock函數（是不是更有特點呢）
尋優的值：
尋優點在x-y平面上的尋優軌跡：
從尋優的結果發現，算法陷入了局部最優而沒有跳出來，這是一大問題。
2018-12-2 週日。重度霧霾，閒來無事
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期待後續有版本更新