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‎Unconstrained_Optimization/scripts/Newton_Opt_2.m

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function [x_k] = Newton_Opt(max_iter,min_tol)
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clear all;
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close all;
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% Initialization
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max_iter=50; % maximum number of iterations
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min_tol=10^-6; % tolerance
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x_k1= zeros(2,1);
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p_k= zeros(2,1);
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alpha=1;
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% Initial guess
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x_k=[-1.2;1];
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k=0; tol=10;
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xvec1(k+1)=x_k(1,1);
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xvec2(k+1)=x_k(2,1);
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x=x_k(1,1); y=x_k(2,1);
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f_k=F(x,y);
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fvec(k+1)=f_k;
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% Newton's algorithm
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while (k<=max_iter) && (tol>=min_tol)
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x=x_k(1,1); y=x_k(2,1);
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p_k=-1*(HF(x,y)\GradF(x,y));
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x_k1=x_k+(alpha*p_k);
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f_k=F(x,y);
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df_k=norm(GradF(x,y),Inf);
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x=x_k1(1,1); y=x_k1(2,1);
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f_k1=F(x,y);
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df_k1=norm(GradF(x,y),Inf);
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tol=abs(df_k1);
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fprintf('%3.0f\t %7.4f\t %7.4f\t %7.4f\t %7.4f\n',k, x_k,f_k,df_k);
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x_k=x_k1;
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k=k+1;
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xvec1(k+1)=x_k(1,1);
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xvec2(k+1)=x_k(2,1);
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fvec(k+1)=f_k1;
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end
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figure;
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plot(xvec1,xvec2,'bo-','LineWidth', 1.5)
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set(gca, 'fontsize', 14, 'fontname', 'times');
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xlabel('x_1')
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ylabel('x_2')
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grid;
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figure;
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plot3(xvec1,xvec2,fvec,'ro-','LineWidth', 2)
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set(gca, 'fontsize', 14, 'fontname', 'times');
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xlabel('x_1')
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ylabel('x_2')
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zlabel('f')
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grid;
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%The declaration of the objective function
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function f= F(x,y)
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f= 100*(y - x.^2)^2 + (1 - x)^2;
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%The declaration of the Grad
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function g = GradF(x,y)
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g= [400*x^3 - 400*x*y + 2*x - 2;
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200*(y - x^2)];
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%The declaration of the Hessian Matrix
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function h = HF(x,y)
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h= [1200*x^2 - 400*y + 2, -400*x;
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-400*x, 200];

‎Unconstrained_Optimization/scripts/Newton_Opt_4.m

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function [x_k] = Newton_Opt(max_iter,min_tol)
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clear all;
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close all;
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% Initialization
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max_iter=100; % maximum number of iterations
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min_tol=10^-6; % tolerance
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x_k1= zeros(4,1);
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p_k= zeros(4,1);
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alpha=1;
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% Initial guess
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%x_k=[1.1;1.2;1.3;1.4]; %easy start for Wood's f
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%x_k=[-3;-1;-3;-1]; %hard start for Wood's f
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x_k=[-3; -1; 0; 1]; %for Powell's f
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k=0; tol=10;
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xvec1(k+1)=x_k(1,1);
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xvec2(k+1)=x_k(2,1);
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xvec3(k+1)=x_k(3,1);
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xvec4(k+1)=x_k(4,1);
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x=x_k(1,1); y=x_k(2,1); w=x_k(3,1); z=x_k(4,1);
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f_k=F(x,y,w,z);
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fvec(k+1)=f_k;
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% Newton's algorithm
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while (k<=max_iter) && (tol>=min_tol)
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%x=x_k(1,1); y=x_k(2,1); w=x_k(3,1); z=x_k(4,1);
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p_k=-1*(HF(x,y,w,z)\GF(x,y,w,z));
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x_k1=x_k+(alpha*p_k);
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f_k=F(x,y,w,z);
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df_k=norm(GF(x,y,w,z), Inf);
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x=x_k1(1,1); y=x_k1(2,1); w=x_k1(3,1); z=x_k1(4,1);
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f_k1=F(x,y,w,z);
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df_k1=norm(GF(x,y,w,z),Inf);
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tol=abs(df_k1);
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fprintf('%3.0f\t %7.4f\t %7.4f\t %7.4f\t %7.4f\n',k, x_k,f_k,df_k);
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x_k=x_k1;
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k=k+1;
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xvec1(k+1)=x_k(1,1);
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xvec2(k+1)=x_k(2,1);
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xvec3(k+1)=x_k(3,1);
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xvec4(k+1)=x_k(4,1);
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fvec(k+1)=f_k1;
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end
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function f= F(x,y,w,z)
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%Wood's function
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%f= 100*(y - x.^2)^2 + (1 - x)^2 + 90*(z - w.^2)^2 + (1 - w)^2 + 10.1*((y - 1)^2 + (z - 1)^2) + 19.8*(y-1)*(z-1);
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%Powell's function
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f= (x + 10*y)^2 + 5*(w - z)^2 + (y - 2*w)^4 + 10*(x - z)^4;
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function g = GF(x,y,w,z)
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%Wood's GradF
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%g=[2*x - 400*x*(- x^2 + y) - 2
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% - 200*x^2 + (1101*y)/5 + (99*z)/5 - 40
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% 2*w - 360*w*(- w^2 + z) - 2
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% - 180*w^2 + (99*y)/5 + (1001*z)/5 - 40];
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%Powell's GradF
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g=[2*x + 20*y + 40*(x - z)^3
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20*x + 200*y - 4*(2*w - y)^3
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10*w - 10*z + 8*(2*w - y)^3
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10*z - 10*w - 40*(x - z)^3];
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function h = HF(x,y,w,z)
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%Wood's Hessian
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%h=[1200*x^2 - 400*y + 2, -400*x, 0, 0
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% -400*x, 1101/5, 0, 99/5
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% 0, 0, 1080*w^2 - 360*z + 2, -360*w
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% 0, 99/5, -360*w, 1001/5];
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%Powell's Hessian
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h=[120*(x - z)^2 + 2, 20, 0, -120*(x - z)^2
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20, 12*(2*w - y)^2 + 200, -24*(2*w - y)^2, 0
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0, -24*(2*w - y)^2, 48*(2*w - y)^2 + 10, -10
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-120*(x - z)^2, 0, -10, 120*(x - z)^2 + 10];

‎Unconstrained_Optimization/scripts/det_opt2.m

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syms x1 x2;
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assume(x1, 'real');
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assume(x2, 'real');
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tic
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f = x1^2 - 2*x1*x2^2 + x2^4 - x2^5; %obj function (c)
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%f = 8*x1^2 + 3*x1*x2 + 7*x2^2 - 25*x1 +31*x2 - 29; %obj function (a)
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%f = x1^2 + x1*x2 + 1.5*x2^2 - 2*log(x1) - log(x2); %obj function (b)
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%Calculate the Gradient of the objective function
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GradF = sym(zeros(2,1));
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GradF = [diff(f, x1); diff(f, x2)];
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%Calculate the Hessian Matrix of the objective function
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HF = sym(zeros(2,2));
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HF = hessian(f, [x1,x2]);
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%Solve GradF = 0 to find possible local minimizers/maximizers
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X = solve([diff(f, x1) == 0, diff(f, x2) == 0], [x1, x2]);
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sizeX = size(X.x1);
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%Store the solutions of GradF = 0
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solutions = sym(zeros(sizeX(1),2));
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for i = 1:sizeX(1)
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solutions(i,1) = X.x1(i);
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solutions(i,2) = X.x2(i);
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end
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%For every solution do the following..
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for i = 1:sizeX(1)
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flag_min = false;
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flag_max = false;
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%Calculate HF_star on the solution of GradF=0 that is being
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%investigated
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HF_star = subs(HF,{x1,x2}, {solutions(i,1), solutions(i,2)});
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%Calculate the det numbers of HF_star
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D1_star = HF_star(1,1);
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D2_star = det(HF_star);
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%Check the sign of the determinants calculated to make an assumption as
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%whether the point investigated is local min/max or nothing at all
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if D1_star > 0 && D2_star > 0
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fprintf('[%d, %d] is a local minimizer. \n', solutions(i,1), solutions(i,2));
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flag_min = true;
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elseif D1_star < 0 && D2_star >0
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fprintf('[%d, %d] is a local maximizer. \n', solutions(i,1), solutions(i,2));
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flag_max = true;
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elseif D1_star == 0 || D2_star == 0
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fprintf('Det value is found to be 0. This method cannot be used. \n')
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else
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fprintf('[%d, %d] is not a local minimizer/maximizer. \n', solutions(i,1), solutions(i,2));
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end
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%if a local min was found, check if it is also a global min
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if flag_min == true
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f_min = subs(f, {x1,x2}, {solutions(i,1), solutions(i,2)});
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if isAlways( f >= f_min, 'Unknown', 'false' ) == 0
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fprintf('[%d, %d] is not a global minimizer. \n', solutions(i,1), solutions(i,2));
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else
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fprintf('[%d, %d] is a global minimizer. \n', solutions(i,1), solutions(i,2));
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end
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%if a local max was found, check if it is also a global max
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elseif flag_max == true
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f_max = subs(f, {x1,x2}, {solutions(i,1), solutions(i,2)});
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if isAlways( f <= f_max, 'Unknown', 'false' ) == 0
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fprintf('[%d, %d] is not a global maximizer. \n', solutions(i,1), solutions(i,2));
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else
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fprintf('[%d, %d] is a global maximizer. \n', solutions(i,1), solutions(i,2));
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end
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end
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end
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timeElapsed = toc;
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fprintf('The elapsed time was: %f \n', timeElapsed);

‎Unconstrained_Optimization/scripts/det_opt3.m

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syms x1 x2 x3;
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assume(x1, 'real');
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assume(x2, 'real');
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assume(x3, 'real');
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tic
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f= x1^2 + 2*x2^2 + 5*x3^2 - 2*x1*x2 - 4*x2*x3 -2*x3; %obj function (d)
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%Calculate the Gradient of the objective function
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GradF = sym(zeros(3,1));
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GradF = [diff(f, x1); diff(f, x2); diff(f, x3)];
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%Calculate the Hessian Matrix of the objective function
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HF = sym(zeros(3,3));
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HF = hessian(f,[x1,x2,x3]);
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%Solve GradF = 0 to find possible local minimizers/maximizers
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X = solve([diff(f, x1) == 0, diff(f, x2) == 0, diff(f, x3) == 0], [x1, x2, x3]);
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sizeX = size(X.x1);
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%Store the solutions of GradF = 0
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solutions = sym(zeros(sizeX(1),3));
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for i = 1:sizeX(1)
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solutions(i,1) = X.x1(i);
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solutions(i,2) = X.x2(i);
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solutions(i,3) = X.x3(i);
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end
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%For every solution do the following..
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for i = 1:sizeX(1)
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flag_min = false;
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flag_max = false;
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%Calculate HF_star on the solution of GradF=0 that is being
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%investigated
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HF_star = subs(HF,{x1,x2,x3}, {solutions(i,1), solutions(i,2), solutions(i,3)});
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%Calculate the det numbers of HF_star
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D1_star = HF_star(1,1);
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D2_star = det(HF_star(1:2,1:2));
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D3_star = det(HF_star);
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%Check the sign of the determinants calculated to make an assumption as
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%whether the point investigated is local min/max or nothing at all
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if D1_star > 0 && D2_star > 0 && D3_star > 0
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fprintf('[%d, %d, %d] is a local minimizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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flag_min = true;
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elseif D1_star < 0 && D2_star > 0 && D3_star < 0
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fprintf('[%d, %d, %d] is a local maximizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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flag_max = true;
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elseif D1_star == 0 || D2_star == 0 || D3_star == 0
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fprintf('Det value is found to be 0. This method cannot be used. \n')
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else
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fprintf('[%d, %d, %d] is not a local minimizer/maximizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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end
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%if a local min was found, check if it is also a global min
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if flag_min == true
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f_min = subs(f, {x1,x2,x3}, {solutions(i,1), solutions(i,2), solutions(i,3)});
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if isAlways( f >= f_min, 'Unknown', 'false' ) == 0
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fprintf('[%d, %d, %d] is not a global minimizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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else
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fprintf('[%d, %d, %d] is a global minimizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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end
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%if a local max was found, check if it is also a global max
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elseif flag_max == true
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f_max = subs(f, {x1,x2,x3}, {solutions(i,1), solutions(i,2),solutions(i,3)});
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if isAlways( f <= f_max, 'Unknown', 'false' ) == 0
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fprintf('[%d, %d, %d] is not a global maximizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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else
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fprintf('[%d, %d, %d] is a global maximizer. \n', solutions(i,1), solutions(i,2), solutions(i,3));
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end
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end
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end
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timeElapsed = toc;
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fprintf('The elapsed time was: %f \n', timeElapsed);

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