-Define the objective function and the constraints

-Set the optimization options

-Set the initial starting point for the optimization

-Set the bounds (lower and upper) for the optimization variables

-Set the maximum number of iterations

-Initialize the best solution and its objective value

-Iterate until the maximum number of iterations is reached

-Evaluate the objective function and the constraints at the random point

-Check if the random point is feasible or change bounds

The graphical approach to solving nonlinear optimization problems involves plotting the objective function and the constraints on a graph and then visually identifying the optimal solution by examining the intersection of the objective function and the constraints. This approach can be effective when the problem is relatively simple and the objective function and constraints can be easily plotted on a graph. This approach can also help identify where the local extrema may be, this is crucial when deciding bounds.

However, the graphical approach can be limited in its ability to handle more complex nonlinear optimization problems, where the objective function and constraints may have multiple local mins or maxs, or where the problem may have a large number of variables. In these cases, the graphical approach may not be able to accurately identify the global optimum, leading to a potential error between the graphical solution and the true optimal solution.

%% problem 2 

%parameters

d1 = 1;

d2 = 10;

t1 = .1;

t2 = 1;

nop= 1000; %number of points

d = linspace (d1,d1,nop);

t = linspace (t1,t2,nop);

E = 900000;

H = 275;

P = 2000;

%% handle functions and set up of variables for plot

[D,T] = meshgrid(d,t);

% set up meshgrid for values

I = @(d,t)(pi./8).*d.*t.*((d.^2) +(t.^2));

%for grid and for stress equation

x = I(D,T);

%define I as x for next equation

bstress = @(d,t)((pi.^2).* x .* E)./ ((H.^2).* d.* t);

%for grid

y = bstress(D,T);

% equation for stress (z equation)

w = @(d,t)pi.*d.*t.*H.*.0025;

%for grid

z = w(D,T);

%specific points for marker at d1,t1 and d2,t2

I1 = (pi/8)*d1*t1 * ((d1^2) + (t1^2));

I2 = (pi/8)*d2*t2 * ((d2^2) + (t2^2));

b1stress = ((pi^2) * E * I1)/ ((H^2) * d1 * t1);

b2stress = ((pi^2) * E * I2)/ ((H^2) * d2 * t2);

w1 = pi.*d1*t1*H.*.0025;

w2 = pi.*d2*t2*H.*.0025;

maxx = max(x);

maxy = max(y);

maxz = max(z);

%plot

figure;

hold all;

contour3(x,y,z,'ShowText','on')

plot3(I1,b1stress,w1,'k*','MarkerSize',5,'LineWidth',2,'Color','r');

plot3(I2,b2stress,w2,'k*','MarkerSize',5,'LineWidth',2,'Color','r');

plot3(maxx,maxy,maxz,'k*','MarkerSize',5,'LineWidth',2,'Color','g');

xlabel('x')

ylabel('y')

• The surf function did not represent the values properly so the conto

% parameters for equation and bounds

x1 = 1;

x2 = 10;

y1 = .1;

y2 = 1;

nop= 100; %number of points

x = linspace (x1,x1,nop);

y = linspace (y1,y2,nop);

E = 900000;

H = 275;

P = 2000;

fun = @(x,y)(pi./8).*x.*y.*((x.^2) +(y.^2)); % equation used 

%parameter for max's

maxiter = 1e5; %maximum number of iterations

maxf = 0; % store the maximum value of the function

maxx = 0; % store the value of x that corresponds to function max.

maxy = 0; % store the value of y that corresponds to function max.

% select upper and lower bounds for x and y

xl = 1;

xu = 10;

yl = .1;

yu = 1;

for i = 1:maxiter

   rx = rand(1,1); % ranges from 0 to 1

   ry = rand(1,1); % ranges from 0 to 1

    x = xl + (xu - xl)*rx;

   y = yl + (yu - yl)*ry;

   f = fun(x,y);

   if( f > maxf )

       maxf = f;

       maxx = x;

       maxy = y;

   end

end

% vectors of x and y for solving z - same as before- rewrote because the code stopped working

xg = 1:.01:10;

yg = .1:0.001:1;

% turn vectors into matrices that represent entire 2-D space

[X,Y] = meshgrid(xg,yg);

% evaluate Z = f(x,y)

Z = pi./8.*X*Y*((X.^2) +(Y.^2));