Optimization

326 Appendix B

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% MATLAB code quadr.m

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

%

% x -> independent variable and symbolic variable (later)

% y -> exp(-x)

% syms -> symbolic object (MATLAB function)

% taylor -> MATLAB function

% subs -> symbolic substitution (MATLAB function)

% xlabel, ylabel, legend, plot, hold on -> MATLAB functions

%

x = -2:0.01:2;

y = exp(-x);

plot(x,y)

hold on

% Linear approximation

syms x

f = taylor(exp(-x),2);

x = -2:0.01:2;

z = subs(f);

plot(x,z,'r--')

% Quadratic approximation

syms x

f = taylor(exp(-x),3);

x = -2:0.01:2;

z = subs(f);

plot(x,z,'g--')

legend('exp(-x)','linear','quadratic')

xlabel('x')

ylabel('f(x)')

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

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

% MATLAB code quadr_examp12.m

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

%

% x1 -> radius of can

% x2 -> height of can

% area -> area of can

% pi -> MATLAB variable

% x,y,z -> array of design points

% vv -> user identified contour values

% xit, x2t -> identified point at which gradient required

% contour -> MATLAB function to generate contours

% syms -> symbolic object (MATLAB function)

% subs -> symbolic substitution (MATLAB function)

% gradient -> analytical value

% hessian -> analytical value

% xlabel, ylabel, legend, plot, hold on -> MATLAB functions

%

327Appendix B

clear all

clc

for x1 = 1:200

for x2 = 1:200

area = 2*pi*x1*x2 + 2*pi*x1*x1;

x(x1,x2) = x1;

y(x1,x2) = x2;

z(x1,x2) = area;

end

end

vv = [15000;50000;60000;70000;80000;90000;150000;200000];

[c, h] = contour(x,y,z,vv); clabel(c, h);

hold on

syms x1p x2p

gradient = [2*pi*x2p+4*pi*x1p;2*pi*x1p];

hessian = [4*pi 2*pi; 2*pi 0];

% Linear approximation

x1p = 60;

x2p = 72.629;

gf = subs(gradient);

for delx1 = 1:60

for delx2 = 1:60

x1 = x1p + delx1;

x2 = x2p + delx2;

area = 2*pi*x1p*x2p + 2*pi*x1p*x1p + gf’*[delx1;delx2]

+ 0.5*[delx1 delx2]*(hessian*[delx1;delx2]);

[x1 x2 area]

xx(delx1,delx2) = x1;

yy(delx1,delx2) = x2;

zz(delx1,delx2) = area;

end

end

vv1 = [50000;60000;70000;80000;90000];

[c, h] = contour(xx,yy,zz,vv1,'rd','LineWidth',3);

xlabel('x_1, mm')

ylabel('x_2, mm')

legend('Objective Function','Quadratic approx.')

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328 Appendix B

Chapter 2

Code Name Details

exhaustive.m Exhaustive search to locate the minimum of the test problem

bisection.m Bisection method

func.m Objective function to be coded here

newtonraphson.m Newton–Raphson method

secant.m Secant method

cubic.m Cubic polynomial t

golden.m Golden section method

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

% MATLAB code exhaustive.m

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

%

% delta -> step size for search

% T -> independent variable, temperature

% U -> cost function

% uvec -> vector of cost function evaluated at

% different temperatures

% minu -> minimum of cost function

% min -> MATLAB function

%

clear all

clc

uvec=[];

delta = 0.01;

for T = 40:delta:90

U = 204165.5/(330-2*T) + 10400/(T-20);

uvec = [uvec U];

plot(T,U)

hold on

end

xlabel('T');ylabel('U');

[minu,i]= min(uvec);

fprintf('Minimum Cost = %6.2f\n ',minu)

fprintf('occurs at T = %6.2f\n ',40+(i-1)*delta)

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

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

% MATLAB code bisection.m

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

%

% a -> lower bound of the design variable

% b -> upper bound of the design variable

329Appendix B

% 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

%

clear all

clc

a = 40;

b = 90;

epsilon = 0.01;

delx = 0.01;

fprintf(' a b \n')

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

for i= 1:100

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

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;

else

a = alpha;

end

if abs(a-b) < epsilon

break;

end

end

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

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

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

%

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

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

% MATLAB code func.m

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

%

% x -> input variable to the function

% fx -> output from the function

%

function fx = func(x)

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

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

330 Appendix B

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

% MATLAB code newtonraphson.m

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

%

% x -> initial guess of design variable

% delx -> ?x for central difference method

% derivative -> derivative using central difference method

% sec_derivative -> second derivative

% epsilon -> constant used to terminate the program

% xprev -> value of x stored from previous iteration

%

clear all

clc

x = 45;

delx = 0.01;

epsilon = 0.01;

fprintf(' x f(x) Deriv. Second deriv.\n')

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

for i = 1:100

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

sec_derivative =(func(x+delx)+func(x-delx)-2*func(x))/

(delx*delx);

fprintf('%8.3f %8.3f %8.3f %8.3f\n',x,func(x),derivative,

sec_derivative)

xprev = x;

x = x- derivative/sec_derivative;

if abs(x-xprev) < epsilon

break;

end

end

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

fprintf('Number of function calls = %3d\n',5*i)

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

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

% MATLAB code secant.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

%

clear all

clc