Optimization

346 Appendix B

% delx -> required for gradient computation

% falpha_prev -> function value at first/previous iteration

% deriv -> gradient vector

% deltag -> difference in gradient vector (over previous

% iteration)

% A -> approximation of the hessian matrix

% search -> search direction

%

clear all

clc

n_of_var = 2;

x = [-3 2];

delx = 0.001;

A = eye(length(x));

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 = 1:50

if i==1

deriv_prev = grad_vec(x,delx,n_of_var);

search = -deriv_prev;

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

if abs(falpha-falpha_prev)<0.001

break;

end

falpha_prev = falpha;

x = x + alpha*search;

fpri ntf('%3d %8.3f %8.3f % 8.3f %8.3f \n',i,x,falpha_

prev,norm(deriv_prev))

else

deltax = (alpha*search);

if i>2

deltax = deltax';

search = search';

end

deriv = grad_vec(x,delx,n_of_var);

deltag = deriv-deriv_prev;

term1 = (deltag'*deltag)/(deltag*deltax');

term2 = (deriv_prev'*deriv_prev)/(deriv_prev*search');

A = A + term1 + term2;

search = -inv(A)*deriv';

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

fpri ntf('%3d %8.3f %8.3f % 8.3f %8.3f \n',i,x+alpha*search',

falpha,norm(deriv))

if ab s(falpha-falpha_prev)<epsilon1 ||

norm(deriv)<epsilon2

347Appendix B

break;

end

falpha_prev = falpha;

deriv_prev = deriv;

x = x+alpha*search';

end

end

fprintf('__________________________________________\n')

%

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

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

% MATLAB code powell.m

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

% n_of_var -> number of design variables

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

% epsilon -> constant used for terminating the algorithm

% term -> linearly independent search directions

% falpha_prev -> function value at first/previous iteration

% search -> search direction

%

clear all

clc

n_of_var = 2;

x = [-3 2];

epsilon = 1e-6;

falpha_prev = func_multivar(x);

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

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

fprintf('__________________________________________\n')

for i = 1:n_of_var

for j = 1:n_of_var+1

if (i==j)

term(i,j)=1;

else

term(i,j) = 0;

end

end

end

for i = 1: n_of_var

search{i} = (term(:,i))';

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

x = x + alpha*search{i};

end

search{i+1} = (term(:,i+1))';

for k = 1:200

xini = x;

i = 1;

while i<n_of_var+1

348 Appendix B

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

x = x + alpha*search{i};

i = i+1;

end

if abs(falpha-falpha_prev) < epsilon

break;

end

search{i} = (x-xini);

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

x = x + alpha*search{i};

temp = search;

for i = 1:n_of_var

search{i} = temp{i+1};

end

falpha_prev = falpha;

fprintf('%3d %8.3f %8.3f % 8.3f \n',k,x,falpha)

end

fprintf('__________________________________________\n')

%

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

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

% MATLAB code neldermead.m

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

% n_of_var -> number of design variables

% lb, ub -> lower/upper bound of variable

% (optional for generating initial feasible points randomly)

% ybest -> best value of the objective function in theiteration

% ysecondbest -> second best value of the objective function

% yworst -> worst value of the objective function in the

% iteration

% xworst -> corresponding value of the variable for yworst

% xc -> centroid of the polygon

% fcentroid -> function value at xc

% deviation -> sum square deviation of function values from

% centroid

% xr => reflected point

% freflec => function value at reflected point

% xe => expanded point

% fexp => function value at expanded point

% xcont => contracted point

% fcont => function value at contracted point

%

clear all

clc

n_of_var = 2;

epsilon = 1e-4;

alpha = 1;

gamma = 2;

349Appendix B

rho = -0.5;

lb = [-1 -1];

ub = [1 1];

fprintf(' Iteration Deviation f(x) \n')

fprintf('__________________________________________\n')

for JJ = 1:50

for i = 1:length(lb)

for j = 1:n_of_var+1

a(i,j) = lb(i) + (ub(i)-lb(i))*rand;

end

end

if JJ~=1

a = x';

end

for i = 1:n_of_var+1

for j = 1:n_of_var

x(i,j) = a(j,i);

end

fval(i) = func_multivar(x(i,:));

end

[yworst,I] = max(fval);

[ybest,II] = min(fval);

% compute centroid

for i = 1:length(lb)

sum(i) = 0;

for j = 1:n_of_var+1

if (j ~= I)

sum(i) = sum(i) + a(i,j);

else

xworst(:,i) = a(i,j);

end

end

end

xc = sum./n_of_var;

fcentroid = func_multivar(xc);

sum1 = 0;

for i = 1:n_of_var+1

sum1 = sum1 + (fcentroid-fval(i))^2;

end

deviation = sqrt(sum1/(n_of_var+1));

if deviation < epsilon

break;

end

fval(I) = [];

[ysecondworst,Isw] = max(fval);

xr = xc + alpha*(xc-xworst);

freflec = func_multivar(xr);

if freflec < ybest

%expansion

350 Appendix B

xe = xc + gamma*(xc-xworst);

fexp = func_multivar(xe);

if fexp < freflec

x(I,:) = xe;

else

x(I,:) = xr;

end

else

if freflec < ysecondworst

x(I,:) = xr;

else

xcont = xc + rho*(xc-xworst);

fcont = func_multivar(xcont);

if fcont < yworst

x(I,:) = xcont;

end

end

end

fprintf('%3d %15.4f %15.3f \n',JJ,deviation,ybest)

end

fprintf('__________________________________________\n')

xc

%

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

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

% MATLAB code robotics_nominal_traj.m

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

%

% Generates nominal trajectory for the robotics arm problem

clear all

clc

% generate 100 points in t

t = -pi:.063:pi

px = 30*cos(t);

py = 100*sin(t);

pz = 10*t + 66.04;

plot3(px,py,pz,'b-','LineWidth',3)

xlabel('px')

ylabel('py')

zlabel('pz')

%

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