Optimization

336 Appendix B

falpha2 = func_multivar(x+alpha2*search);

for i= 1:1000

if falpha1 > falpha2

a = alpha1;

alpha1 = alpha2;

falpha1 = falpha2;

alpha2 = tau*a + (1-tau)*b;

falpha2 = func_multivar(x+alpha2*search);

else

b = alpha2;

alpha2 = alpha1;

falpha2 = falpha1;

alpha1 = tau*b + (1-tau)*a;

falpha1 = func_multivar(x+alpha1*search);

end

if abs(func_multivar(x+alpha1*search)-

func_multivar(x+alpha2*search)) < epsilon

break;

end

end

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code func_multivar.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

function fx = func_multivar(x)

fx = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code rosenbrock.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% plots the Rosenbrock’s function

%

clear all

clc

[x1,x2] = meshgrid(-2:.03:2,-2:.03:2);

z = 100*(x2-x1.^2).^2+(1-x1).^2

surf(x1,x2,z)

shading interp

view (170,20)

xlabel('x1')

337Appendix B

ylabel('x2')

zlabel('f(x1,x2)')

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code springsystem.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

clear all

clc

zprev = inf;

i = 0;

j = 0;

for x = -1:0.01:1

i = i+1;

for y = -1:0.01:1

j = j+1;

z = 100*(sqrt(x^2+(y+1)^2)-1)^2 + 90*(sqrt(x^2+(y-

1)^2)-1)^2 -(20*x+40*y);

if z < zprev

zprev = z;

xbest = x;

ybest = y;

end

end

end

fprintf('Minimum Potential = %7.4f\n ',zprev)

fprintf('occurs at x1,x2 = %10.4f %10.4f\n',xbest,ybest)

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code steep_des.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% n_of_var -> number of design variables

% x = [-1.5 1.5] -> starting value of x

% epsilon1,epsilon2 -> constants used for terminating the

% algorithm

% delx -> required for gradient computation

% falpha_prev -> function value at first/previous iteration

% deriv -> gradient vector

% search -> search direction (set to negative of gradient)

%

clear all

clc

n_of_var = 2;

x = [-3 2];

338 Appendix B

epsilon1 = 1e-6;

epsilon2 = 1e-6;

delx = 1e-3;

falpha_prev = func_multivar(x);

fprintf('Initial function value = %7.4f\n ',falpha_prev)

fprintf(' No. x-vector f(x) Deriv \n')

fprintf('__________________________________________\n')

for i = 1:3000

deriv = grad_vec(x,delx,n_of_var);

search = -deriv;

[alpha,falpha] = golden_funct1(x,search);

if abs(falpha-falpha_prev)<epsilon1 || norm(deriv)<epsilon2

break;

end

falpha_prev = falpha;

x = x + alpha*search;

fprintf('%3d %8.3f %8.3f % 8.3f %8.3f

\n',i,x,falpha,norm(deriv))

end

fprintf('__________________________________________\n')

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code grad_vec.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% compute gradient vector using central difference method

% xvec, xvec1 -> vector of design variables

% deriv(i) -> derivative w.r.t. ith variable

%

function deriv = grad_vec(x,delx,n_of_var)

xvec = x;

xvec1 = x;

for i = 1:length(x)

xvec = x;

xvec1 = x;

xvec(i) = x(i) + delx;

xvec1(i) = x(i) - delx;

deriv(i) = (func_multivar(xvec) - func_multivar(xvec1))/

(2*delx);

end

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

339Appendix B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code contour_testproblem.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% plots contour of the test problem

% surfc -> Matlab function

clear all

clc

i = 0;

j = 0;

for x = -5:.02:5

i = i+1;

for y = -5:.02:5

j = j+1;

z(i,j) = 100*(sqrt(x^2+(y+1)^2)-1)^2 + 90*(sqrt(x^2+

(y - 1)^2)-1 )^2 -(20*x+40*y);

t1(i,j) = x;

t2(i,j) = y;

end

j = 0;

end

surfc(t1,t2,z)

shading interp

xlabel('x1')

ylabel('x2')

zlabel('f(x1,x2)')

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code newton.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% n_of_var -> number of design variables

% x = [-3 2] -> starting value of x

% epsilon1, epsilon2 -> constant used for terminating

% the algorithm

% delx -> required for gradient computation

% f_prev -> function value at first/previous iteration

% deriv -> gradient vector

% sec_deriv -> hessian matrix

%

clear all

clc

n_of_var = 2;

x = [-3 2];

epsilon1 = 1e-7;

epsilon2 = 1e-7;

delx = 1e-3;

340 Appendix B

f_prev = func_multivar(x);

fprintf('Initial function value = %7.4f\n',f_prev)

fprintf('No. x-vector f(x) Deriv \n')

fprintf('__________________________________________\n')

for i = 1:50

f_prev = func_multivar(x);

deriv = grad_vec(x,delx,n_of_var);

sec_deriv = hessian(x,delx,n_of_var);

x = (x' - inv(sec_deriv)*deriv')';

f = func_multivar(x);

if abs(f-f_prev)<epsilon1 || norm(deriv)<epsilon2

break;

end

fprintf('%3d %8.3f %8.3f % 8.3f %8.3f

\n',i,x,f,norm(deriv))

end

fprintf('%3d %8.3f %8.3f % 8.3f %8.3f \n',i,x,f,norm(deriv))

fprintf('__________________________________________\n')

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% MATLAB code hessian.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

%compute hessian matrix

% sec_deriv -> second derivative matrix

% func_multivar() -> multivariable function

% temp -> temporary variable

% Note that n_of_var = length(x)

%

function sec_deriv = hessian(x,delx,n_of_var)

for i = 1:length(x)

for j = 1:length(x)

if i == j

temp = x;

temp(i) = x(i) + delx;

term1 = func_multivar(temp);

temp(i) = x(i) - delx;

term2 = func_multivar(temp);

term3 = func_multivar(x);

sec_deriv(i,j) = (term1-2*term3+term2)/(delx^2);

else

temp = x;

temp(i) = x(i) + delx;

temp(j) = x(j) + delx;

term1 = func_multivar(temp);

temp = x;