Optimization

376 Appendix B

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% MATLAB code func_val1.m

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% computes function value

%

function y = func_val1(x)

y = (x(1)-1)^2 + (x(2)-2)^2;

%

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% MATLAB code eqconstr_val.m

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% computes value of equality constraint

%

function h = eqconstr_val(x)

h(1) = 2*x(1)-x(2);

%

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% MATLAB code ineqconstr_val.m

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% computes value of inequality constraint

%

function g = ineqconstr_val(x)

g(1) = x(1)-5;

%

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% MATLAB code grad_vec_f.m

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% computes gradient vector (obj. function)

%

function deriv = grad_vec_f(x,delx,n_of_var,scale_factor)

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_val1(xvec) - func_val1(xvec1))/(2*delx);

end

%

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377Appendix B

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% MATLAB code grad_vec_eqcon.m

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

% computes gradient vector (eq. constraint)

%

function deriv = grad_vec_eqcon(x,delx,n_of_eqcons)

xvec = x;

xvec1 = x;

for j = 1:n_of_eqcons

for i = 1:length(x)

xvec = x;

xvec1 = x;

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

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

h = eqconstr_val(xvec);

h1 = eqconstr_val(xvec1);

deriv(j,i) = (h(j) - h1(j))/(2*delx);

end

end

%

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% MATLAB code grad_vec_ineqcon.m

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

% computes gradient vector (ineq. constraint)

%

function deriv = grad_vec_ineqcon(x,delx,n_of_iqcons)

xvec = x;

xvec1 = x;

for j = 1:n_of_iqcons

for i = 1:length(x)

xvec = x;

xvec1 = x;

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

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

g = ineqconstr_val(xvec);

g1 = ineqconstr_val(xvec1);

deriv(j,i) = (g(j) - g1(j))/(2*delx);

end

end

%

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378 Appendix B

Chapter 7

Code Name Details

sqp.m (main program) Sequential quadratic programming (SQP) method modied for

weighted sum approach

func_val.m (function) Computes augmented Lagrangian function value

func_val1.m (function) Computes function value

sqp.m (main program) SQP method modied for solving multiobjective problems using

ε-constraint technique

func_val.m (function) Computes augmented Lagrangian function value

func_val1.m (function) Computes function value

ineqconstr_val.m Computes inequality constraint value

pso.m (main program) Particle swarm optimization (PSO) method with dynamic

weights

func1.m (function) Computes value of objective function

sqp.m (main program) Main program for solving reentry problem

dynamics.m (function) Computes area, volume, and X

cp

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% MATLAB code sqp.m

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

%

% n_of_var -> number of design variables

% x = [0.1 0.1] -> starting value of x

% epsilon1 -> constants used for terminating the algorithm

% delx -> required for gradient computation

% falpha_prev -> function value at first/previous iteration

% deriv -> gradient vector

% quadprog -> MATLAB function to solve quadratic programming

% LAMBDA -> Lagrange multipliers

clear all

clc

warning off

n_of_var = 2;

n_of_eqcons = 1;

n_of_iqcons = 1;

scale_factor = 1;

global LAMBDA RK BETA EQCONSTR ICONSTR FVALUE W1 W2

LAMBDA = zeros(1,n_of_eqcons);

BETA = zeros(1,n_of_iqcons);

X = [0.1 0.1];

RK = 1;

A = eye(length(X));

epsilon1 = 1e-6;

delx = 1e-3;

for kk = 1:101

X = [0.1 0.1];

W1 = (kk-1)/100;

W2 = 1 - W1;

379Appendix B

checkconstr = zeros(1,n_of_iqcons);

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

fprintf('___________________________________________________\n')

checkconstr = zeros(1,n_of_iqcons);

for i = 1:10

deriv_f = grad_vec_f(X,delx,n_of_var,scale_factor);

sec_deriv_f = hessian(X,delx);

deriv_eqcon = grad_vec_eqcon(X,delx,n_of_eqcons);

deriv_ineqcon = grad_vec_ineqcon(X,delx,n_of_iqcons);

options = optimset('Display','off');

x = quadprog(sec_deriv_f,deriv_f,deriv_ineqcon,-ineqconstr_val(X),

deriv_eqcon,-eqconstr_val(X),[],[],X,options);

fprev = func_val(X);

X = X+x';

yyy = func_val(X);

LAMBDA = LAMBDA + 2*RK*EQCONSTR;

BETA = BETA + 2*RK*(max([ICONSTR; -BETA./(2*RK)]));

fnew = func_val(X);

checkconstr1 = max([ineqconstr_val(X);checkconstr]);

disp([i X FVALUE norm([checkconstr1 eqconstr_val(X)])]);

if abs(fnew-fprev) < epsilon1

break

end

end

fprintf('___________________________________________________\n')

plot(0.5*(X(1)^2+X(2)^2) , 0.5*((X(1)-1)^2 + (X(2)-3)^2),'r*')

hold on

end

xlabel('f1');

ylabel('f2');

%

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

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

% MATLAB code func_val.m

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

% computes augmented Lagrangian function value

%

function y = func_val(x)

global LAMBDA RK BETA EQCONSTR ICONSTR FVALUE W1 W2

y = W1*0.5*(x(1)^2+x(2)^2) + W2*0.5*((x(1)-1)^2 + (x(2)-3)^2);

g = ineqconstr_val(x);

h = eqconstr_val(x);

EQCONSTR = h;

ICONSTR = g;

FVALUE = y;

y = y + LAMBDA*EQCONSTR' + RK*EQCONSTR*EQCONSTR' + sum(BETA.*

max([ICONSTR; -BETA./(2*RK)])) + sum(RK*(max([ICONSTR;

-BETA./(2*RK)])).^2);

%

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380 Appendix B

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% MATLAB code func_val1.m

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

% computes function value

%

function y = func_val1(x)

global W1 W2

y = W1*0.5*(x(1)^2+x(2)^2) + W2*0.5*((x(1)-1)^2 + (x(2)-3)^2);

%

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

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

% MATLAB code sqp.m modified for eps-constraints method

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

%

% n_of_var -> number of design variables

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

% epsilon1 -> constants used for terminating the algorithm

% delx -> required for gradient computation

% falpha_prev -> function value at first/previous iteration

% deriv -> gradient vector

% quadprog -> MATLAB function to solve quadratic programming

% LAMBDA -> Lagrange multipliers

clear all

clc

warning off

n_of_var = 1;

n_of_eqcons = 1;

n_of_iqcons = 5;

scale_factor = 1;

global LAMBDA RK BETA EQCONSTR ICONSTR FVALUE W1 W2

LAMBDA = zeros(1,n_of_eqcons);

BETA = zeros(1,n_of_iqcons);

X = [1 1];

RK = 1;

A = eye(length(X));

epsilon1 = 1e-6;

delx = 1e-3;

for kk = 1:100

W1 = (kk-1)/100;

W2 = 1 - W1;

checkconstr = zeros(1,n_of_iqcons);

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

fprintf('___________________________________________________\n')

checkconstr = zeros(1,n_of_iqcons);

for i = 1:10

deriv_f = grad_vec_f(X,delx,n_of_var,scale_factor);

sec_deriv_f = hessian(X,delx);

deriv_eqcon = grad_vec_eqcon(X,delx,n_of_eqcons);

deriv_ineqcon = grad_vec_ineqcon(X,delx,n_of_iqcons);

options = optimset('Display','off');

x = quad prog(sec_deriv_f,deriv_f,deriv_ineqcon,-ineqconstr_val(X),

deriv_eqcon,-eqconstr_val(X),[],[],X,options);

fprev = func_val(X);

X = X+x';