Optimization

331Appendix B

a = 40;

b = 90;

epsilon = 0.001;

delx = 0.01;

flag = 0;

fprintf(' Alpha Deriv. \n')

fprintf('-------------------------\n')

for i = 1:100

alpha = (a+b)/2;

derivative = (func(a+delx) - func(a-delx))/(2*delx);

derivative_alpha = (func(alpha+delx)-func(alpha-delx))/

(2*delx);

if (derivative*derivative_alpha) < 0

b = alpha;

flag = 1;

else

a = alpha;

end

if flag == 1

break;

end

end

for j = 1:100

fprintf(' %7.3f %8.3f \n',alpha,derivative_alpha)

derivative_a = (func(a+delx) - func(a-delx))/(2*delx);

derivative_b = (func(b+delx) - func(b-delx))/(2*delx);

alpha = b - derivative_b*(b-a)/(derivative_b-derivative_a);

derivative_alpha = (func(alpha+delx) - func(alpha-delx))/

(2*delx);

if derivative_alpha > 0

b = alpha;

else

a = alpha;

end

if abs(derivative_alpha) < epsilon

break;

end

end

fprintf('-------------------------\n')

fprintf('x* = %7.3f Minimum = %8.3f\n',alpha,func(alpha))

fprintf('Number of function calls = %3d\n',4*i+6*j)

%

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

332 Appendix B

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

% MATLAB code cubic.m

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

%

% a -> lower bound of the design variable

% b -> upper bound of the design variable

% alpha -> midpoint of a and b

% delx -> ?x for central difference method

% derivative -> derivative using central difference method

% derivative_alpha -> derivative at x = alpha

% abs -> absolute of a number, MATLAB function

% flag -> set the flag when minimum is bracketed

% derivative_a -> derivative at point a

% derivative_b -> derivative at b

%

a = 40;

b = 90;

delx = 0.01;

flag = 0;

epsilon= 0.001;

fprintf(' a b \n')

fprintf('-------------------------\n')

for i = 1:100

alpha = (a+b)/2;

derivative = (func(a+delx) - func(a-delx))/(2*delx);

derivative_alpha = (func(alpha+delx)-func(alpha-delx))/

(2*delx);

if (derivative*derivative_alpha) < 0

b = alpha;

flag = 1;

else

a = alpha;

end

if flag == 1

break;

end

end

for j = 1:100

fprintf(' %7.3f %8.3f \n',a,b)

derivative_a = (func(a+delx) - func(a-delx))/(2*delx);

derivative_b = (func(b+delx) - func(b-delx))/(2*delx);

z = 3*(func(a)-func(b))/(b-a) + derivative_a + derivative_b;

w = ((b-a)/abs(b-a))*sqrt(z*z-derivative_a*derivative_b);

mew = (derivative_b+w-z)/(derivative_b-derivative_a+2*w);

if mew <= 1

x_opt = b - mew*(b-a);

else

x_opt = a;

end

333Appendix B

alpha1 = (func(x_opt+delx) - func(x_opt-delx) )/(2*delx);

if abs(alpha1) < epsilon

break;

else

if (derivative_a*alpha1) < 0

b = x_opt;

else

a = x_opt;

end

end

end

fprintf('-------------------------\n')

fprintf('x* = %7.3f Minimum = %8.3f\n',x_opt,func

(x_opt))

fprintf('Number of function calls = %3d\n',4*i+8*j)

%

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

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

% MATLAB code golden.m

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

%

% a -> lower bound of the design variable

% b -> upper bound of the design variable

% alpha -> midpoint of a and b

% falpha1 -> function value at x = alpha1

% falpha2 -> function value at x = alpha2

% epsilon -> constant used to terminate the algorithm

% abs -> absolute of a number, MATLAB function

% tau -> 2-golden number

%

clear all

clc

a = 40;

b = 90;

epsilon = 0.00001;

tau = 0.381967;

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

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

falpha1 = func(alpha1);

falpha2 = func(alpha2);

fprintf(' a b \n')

fprintf('-------------------------\n')

for i = 1:100

fprintf(' %7.3f %8.3f \n',a,b)

if falpha1 > falpha2

a = alpha1;

alpha1 = alpha2;

falpha1 = falpha2;

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

falpha2 = func(alpha2);

else

b = alpha2;

334 Appendix B

alpha2 = alpha1;

falpha2 = falpha1;

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

falpha1 = func(alpha1);

end

if abs(func(alpha1)-func(alpha2)) < epsilon

break;

end

end

fprintf('-------------------------\n')

fprintf('x* = %7.3f Minimum = %8.3f\n',alpha1,func(alpha1))

fprintf('Number of function calls = %3d\n',2+i)

%

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

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

% MATLAB code func.m

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

% objective function to be coded here

% different test functions

%

function fx = func(x)

% fx = 204165.5/(330-2*x) + 10400/(x-20);

% fx = 3*x^4+(x-1)^2;

% fx = -4*x*sin(x);

% fx = 2*(x-3)^2+exp(0.5*x*x);

fx = 3*(x)^2+12/(x^3)-5;

% fx = 2*x*x+16/x;

%

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

335Appendix B

Chapter 3

Code Name Details

golden_funct1.m Golden section method for a multivariable function

func_multivar.m Objective function to be coded here

rosenbrock.m Plot of Rosenbrock function

springsystem.m Finds minimum of the spring system problem

steep_des.m Steepest descent method

grad_vec.m Gradient vector computation

contour_testproblem.m Plots contour of the test problem function

newton.m Newton’s method

hessian.m Computes Hessian matrix

modied_newton.m Modied Newton’s method

levenbergmarquardt.m Levenberg–Marquardt’s method

conjugate.m Conjugate gradient method

DFP.m Davidon–Fletcher–Powell (DFP) method

BFGS.m Broyden–Fletcher–Goldfarb–Shanno (BFGS) method

powell.m Powell’s conjugate direction method

neldermead.m Nelder–Mead algorithm

\Robotics\ Directory containing codes for problems in robotics

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

% MATLAB code golden_funct1.m

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

%

% a -> lower bound of the design variable

% b -> upper bound of the design variable

% falpha1 -> function value at x = alpha1

% falpha2 -> function value at x = alpha2

% epsilon -> constant used to terminate the algorithm

% abs -> absolute of a number, MATLAB function

% tau -> 2-golden number

% func_multivar -> returns the value of a multivariable

% function

%

function [alpha1,falpha1] = golden_funct1(x,search)

a = -5;

b = 5;

tau = 0.381967;

epsilon = 1e-5;

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

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

falpha1 = func_multivar(x+alpha1*search);