Add files via upload

NelderMead_main.m

@@ -0,0 +1,222 @@
+clc
+clear all
+close all
+
+% Optimization Problem
+% Simplex - Nelder Mead
+% Rosenbrok Function - 3 variables - initial point [1 -2 5]
+% 6/4/2023
+% By Amin Pirmohammadi
+
+% Define Function
+F = @(x) (1-x(1))^2 + (2-x(2))^2 + 65*(x(2)-x(1)^2)^2 + 65*(x(3)-x(2)^2)^2;
+
+% Initial Values
+alpha_r = .7;
+alpha_e = 3;
+alpha_q = -0.5;
+delta_j = 1;
+delta_sh = 0.7;
+epsilon = 1e-6;
+
+x_0 = [1; -2; 5];
+% x_0 = [1; 2; 3];
+% x_0 = [0; 0; 0];
+n = length(x_0);             % Number of Dimensions
+
+% Directions
+e_i = [1; 0; 0];
+e_j = [0; 1; 0];
+e_k = [0; 0; 1];
+
+x_1 = x_0 + delta_j*e_i;
+x_2 = x_0 + delta_j*e_j;
+x_3 = x_0 + delta_j*e_k;
+
+ConverganceTest = 10000;   % Just to define the value of Convergance
+
+Total_Iter = 0;
+reflection_Iter = 0;
+shrinkage_Iter = 0;
+expansion_Iter = 0;
+contraction_Iter = 0;
+
+while ConverganceTest > epsilon
+Total_Iter = Total_Iter+1;
+
+f0 = F(x_0);
+f1 = F(x_1);
+f2 = F(x_2);
+f3 = F(x_3);
+
+% NumberOfFuncEvaluation = n + 1;
+
+A = [f0 f1 f2 f3]
+A_sort = sort(A,'ascend')                            % Sort F(x) values without F(x_w)
+Max_f = A_sort(4);
+Min_f = A_sort(1);
+
+if Max_f == f0
+    x_w = x_0;
+    B = [x_1 x_2 x_3];                               % Sort x values without x_w  (B)
+    disp('x_w = x_0');
+elseif Max_f == f1
+    x_w = x_1;
+    B = [x_0 x_2 x_3];
+    disp('x_w = x_1');
+elseif Max_f == f2
+    x_w = x_2;
+    B = [x_1 x_0 x_3];
+    disp('x_w = x_2');
+elseif Max_f == f3
+    x_w = x_3;
+    B = [x_1 x_2 x_0];
+    disp('x_w = x_3');
+end
+
+if Min_f == f0
+    x_b = x_0;
+    C = [x_1 x_2 x_3];                               % Sort x values without x_b  (B)
+    disp('x_b = x_0');
+elseif Min_f == f1
+    x_b = x_1;
+    C = [x_0 x_2 x_3];
+    disp('x_b = x_1');
+elseif Min_f == f2
+    x_b = x_2;
+    C = [x_1 x_0 x_3];
+    disp('x_b = x_2');
+elseif Min_f == f3
+    x_b = x_3;
+    C = [x_1 x_2 x_0];
+    disp('x_b = x_3');
+end
+
+f1 = A_sort(1);
+f2 = A_sort(2);
+f3 = A_sort(3);                     % Sort f1 < f2 < f3 < f4(MAX_F)
+f4 = A_sort(4);
+
+x_c = (1/n)*([sum(B(1,:)); sum(B(2,:)); sum(B(3,:))])      % centroid point of n points
+
+% Convergance Test
+ConverganceTest = (sqrt((f1-F(x_c))^2 + (f2-F(x_c))^2 + (f3-F(x_c))^2 + (f4-F(x_c))^2))/(n+1);
+
+% Reflection
+x_r = (1 + alpha_r)*x_c - alpha_r*x_w;
+f_r = F(x_r);
+
+if f1 <= f_r && f_r < f3
+    % x_r is a good reflection
+    D = [B x_r];                                 % Accept and replacement
+    reflection_Iter = reflection_Iter + 1;
+elseif f_r >= f3
+    if f_r >= f3 && f_r < f4
+        % Calculate x_q by outside contraction
+        x_q = (1+alpha_q)*x_r - alpha_q*x_c;
+        f_q = F(x_q);
+        if f_q < f3
+            D = [B x_r];                         % Accept and replacement
+            contraction_Iter = contraction_Iter + 1;
+        else
+            % Shrinkage
+            C(:,1) = x_b + delta_sh*(C(:,1) - x_b);
+            C(:,2) = x_b + delta_sh*(C(:,2) - x_b);
+            C(:,3) = x_b + delta_sh*(C(:,3) - x_b);
+
+            D = [x_b C(:,1) C(:,2) C(:,3)];
+            shrinkage_Iter = shrinkage_Iter + 1;
+        end
+    elseif f_r >= f4
+        % Calculate x_q by inside contraction
+        x_q = (1+alpha_q)*x_c - alpha_q*x_w;
+        f_q = F(x_q);
+        if f_q < f3
+            D = [B x_r];                         % Accept and replacement
+            contraction_Iter = contraction_Iter + 1;
+        else
+            % Shrinkage
+            C(:,1) = x_b + delta_sh*(C(:,1) - x_b);
+            C(:,2) = x_b + delta_sh*(C(:,2) - x_b);
+            C(:,3) = x_b + delta_sh*(C(:,3) - x_b);
+
+            D = [x_b C(:,1) C(:,2) C(:,3)];
+            shrinkage_Iter = shrinkage_Iter + 1;
+        end
+    end
+elseif f_r < f1
+    %Expansion
+    x_e = (1+alpha_e)*x_c - alpha_e*x_w;
+    f_e = F(x_e);
+    if f_e < f_r
+        % x_e will be used as the new replacement point
+        D = [B x_e];
+        expansion_Iter = expansion_Iter + 1;
+    else
+        D = [B x_r];
+        reflection_Iter = reflection_Iter + 1;
+    end
+
+    each_nIter{Total_Iter} = Total_Iter;
+    f_eachIter{Total_Iter} = F(D(:,1));
+end
+
+x_0 = D(:,1);
+x_1 = D(:,2);
+x_2 = D(:,3);
+x_3 = D(:,4);
+
+x{Total_Iter} = [x_0, x_1, x_2, x_3];
+
+
+end
+
+f0 = F(x_0);
+f1 = F(x_1);
+f2 = F(x_2);
+f3 = F(x_3);
+
+x_optimum = D
+f_optimum = min([f0,f1,f2,f3])
+
+% Iterations
+Total_Iter
+reflection_Iter
+contraction_Iter
+shrinkage_Iter
+expansion_Iter
+
+%% Visualize the Process
+
+ShematicNelderMead;
+
+for i = 1:Total_Iter
+
+    NelderX = x{1,i}(1,:);
+
+    NelderY = x{1,i}(2,:);
+
+    NelderZ = x{1,i}(3,:);
+
+    figure(1)
+
+    grid on
+    Plot_Nelder_Mead
+
+    % Uncomment line below if you want to make plot axis constant.
+%     axis([NelderX(1)-1 NelderX(1)+1 NelderY(1)-1 NelderY(1)+1 NelderZ(1)-1 NelderZ(1)+1]) 
+
+ pause(.1)
+
+end
+
+% ---------------------------------Plot Convergance ----------------------------------------
+figure(2)
+AA = [each_nIter{:}];
+BB = [f_eachIter{:}];
+plot(AA,BB)
+axis([0 max(AA) -10 max(BB)])
+title('Convergance (function value in each iteration)')
+xlabel('Number of Iterations')
+ylabel('f value')
+print(gcf,'SimplexConverge.png','-dpng','-r600');

Plot_Nelder_Mead.m

@@ -0,0 +1,13 @@
+
+
+set(Line_1,'xdata',[NelderX(1) NelderX(2)],'ydata',[NelderY(1) NelderY(2)],'zdata',[NelderZ(1) NelderZ(2)])
+
+set(Line_2,'xdata',[NelderX(1) NelderX(3)],'ydata',[NelderY(1) NelderY(3)],'zdata',[NelderZ(1) NelderZ(3)])
+
+set(Line_3,'xdata',[NelderX(1) NelderX(4)],'ydata',[NelderY(1) NelderY(4)],'zdata',[NelderZ(1) NelderZ(4)])
+
+set(Line_4,'xdata',[NelderX(2) NelderX(3)],'ydata',[NelderY(2) NelderY(3)],'zdata',[NelderZ(2) NelderZ(3)])
+
+set(Line_5,'xdata',[NelderX(2) NelderX(4)],'ydata',[NelderY(2) NelderY(4)],'zdata',[NelderZ(2) NelderZ(4)])
+
+set(Line_6,'xdata',[NelderX(3) NelderX(4)],'ydata',[NelderY(3) NelderY(4)],'zdata',[NelderZ(3) NelderZ(4)])

ShematicNelderMead.m

@@ -0,0 +1,51 @@
+
+Line1_x=[
+    0 0];
+Line1_y=[
+    0 0];
+Line1_z=[
+    0 0];
+
+Line2_x=[
+    0 0];
+Line2_y=[
+    0 0];
+Line2_z=[
+    0 0];
+
+Line3_x=[
+    0 0];
+Line3_y=[
+    0 0];
+Line3_z=[
+    0 0];
+
+Line4_x=[
+    0 0];
+Line4_y=[
+    0 0];
+Line4_z=[
+    0 0];
+
+Line5_x=[
+    0 0];
+Line5_y=[
+    0 0];
+Line5_z=[
+    0 0];
+
+Line6_x=[
+    0 0];
+Line6_y=[
+    0 0];
+Line6_z=[
+    0 0];
+
+Line_1=patch('XData',Line1_x,'YData',Line1_z,'ZData',Line1_y,'facealpha',.9,'facecolor','b');
+Line_2=patch('XData',Line2_x,'YData',Line2_z,'ZData',Line2_y,'facealpha',.9,'facecolor','b');
+Line_3=patch('XData',Line3_x,'YData',Line3_z,'ZData',Line3_y,'facealpha',.9,'facecolor','b');
+Line_4=patch('XData',Line4_x,'YData',Line4_z,'ZData',Line4_y,'facealpha',.9,'facecolor','b');
+Line_5=patch('XData',Line5_x,'YData',Line5_z,'ZData',Line5_y,'facealpha',.9,'facecolor','b');
+Line_6=patch('XData',Line6_x,'YData',Line6_z,'ZData',Line6_y,'facealpha',.9,'facecolor','b');
+
+view(30,45)