Learn Optimization Techniques with MATLAB

I'm facing a significant challenge with an assignment that has been ongoing for some time. After considerable effort, I believe that incorporating MATLAB could provide the most effective solution. However, the problem involves non-linear, constrained, and multi-objective optimization, which requires a graphical approach. Unfortunately, despite extensive online research and video tutorials, I haven't found resources specifically addressing this scenario. Most materials seem to focus on linear programming problems or utilize MATLAB's optimization toolbox, which doesn't align with the unique demands of my assignment.

I would be immensely grateful for any guidance on relevant resources that could assist me in solving and learning about this approach. Furthermore, I would be incredibly appreciative of mentorship from someone willing to teach me advanced engineering problem-solving techniques using MATLAB. While not directly related to the problem itself, I wanted to share the complexity in the image below I'm facing to provide context.

Thank you for your time and consideration.

[Donne]

So this is the actual problem I am facing. I have been able to partly solve my problem. The issue is that, I am not getting the graph I want.

`% this is to solve the optimization of a two two-bar truss system

close all; clear; clc

%% DEFINED SCALAR PARAMETERS

Aref = 1; %in^2

E = 30 * 10^6;%psi

rho = 0.283;%lb/in^3

P = 10000;%lb

sigma_0= 20000;%psi

h = 100;%in

%% DEFINED PHYSICAL CONSTRAINTS

x1 = linspace(0.1,2.0,1000);

x2 = linspace(0.1,2.5,10000);

[X, Y] = meshgrid(x1, x2);

%% DEFINED OBJECTIVE functions

% Objective function for the weight

obj_fun_1 = 2*( rho * h .* Y .* sqrt((1 + X.^2)) .* Aref) - 0.5;

%Objective function for the displacement of truss beams

obj_fun_2 = P * h *(1+X.^2).^1.5 .*sqrt((1 + X.^4)) ./ ...

(2*sqrt(2)*E .*X.*Y*Aref) - 0.5;

%% PERFORMANCE CONSTRAINTS

% Constraint 1

con_fun_1 = (P * (1 + X).*sqrt((1 + X.^2)))./(2*(sqrt(2))*X.*Y*Aref) <= sigma_0;

% Constraint 2

con_fun_2 = (P * (1 - X).*sqrt((1 + X.^2)))./(2*(sqrt(2))*X.*Y*Aref) <= sigma_0;

%% Question 1a

figure('Name','Objective Function 1')

contour(X,Y, obj_fun_1,1)

hold on

contour(X,Y, con_fun_1,1)

contour(X,Y, con_fun_2,1)

hold off

%% Question 1b

figure('Name','Objective Function 2')

contour(X,Y, obj_fun_2,1)

hold on

contour(X,Y, con_fun_1,1)

contour(X,Y, con_fun_2,1)

hold off

%% Question 1c

obj_fun_combined =obj_fun_2 + obj_fun_2;

figure('Name','Objective Functions Combined')

contour(X,Y, obj_fun_combined,1)

hold on

contour(X,Y, con_fun_1,1)

contour(X,Y, con_fun_2,1)

hold off

`

the obj_fun_1 is supposed to plot the curve in blue, and when you change the constant, i.e -0.5 of the equation, it should make the mulitple plots. But i am getting something way off. And what i understand is that as change the value of the constant the position of the curve of the obj_fun_1 should also be changing but my code is not updating the position of the code even when I change the code. I really need help

Answer 1

[Sam Chak]

Hi @Donne

If this is indeed a static optimization problem, the minimization of the volume of the cone clutch can be achieved by first deriving an equation that describes the volume (V) as a cost function of the outer and inner radii, denoted as . For constrained nonlinear bi-variate function like your case, you can use fmincon() to solve the problem.

Vol = @(R) 100*(R(2) - R(1)^2)^2 + (1 - R(1))^2;    % <-- this is the Cost function
R0  = [7.5 5];      % initial guess: R(1) = 7.5, & R(2) =  5
lb  = [ 0  0];      % lower bound of R(1) =  0 and R(2) =  0
ub  = [15 10];      % upper bound of R(1) = 15 and R(2) = 10
opt	= optimset('Display', 'iter', 'PlotFcns', @optimplotfval);
[R, fval, exitflag, output] = fmincon(Vol, R0, [], [], [], [], lb, ub, [], opt)

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       3    2.626985e+05    0.000e+00    1.511e+05
    1       6    5.301462e+03    0.000e+00    2.751e+03    7.804e+00
    2       9    1.134062e+03    0.000e+00    2.642e+03    3.916e+00
    3      12    1.027922e+00    0.000e+00    1.951e+03    3.349e+00
    4      15    1.085873e+00    0.000e+00    1.174e+01    1.256e-02
    5      18    1.032356e+00    0.000e+00    2.268e+00    1.198e-02
    6      22    8.614146e-01    0.000e+00    3.390e+00    6.447e-02
    7      25    7.444495e-01    0.000e+00    5.582e+00    6.463e-02
    8      29    8.549904e-01    0.000e+00    1.558e+01    1.593e-01
    9      32    6.027884e-01    0.000e+00    3.348e+00    7.036e-02
   10      35    5.274331e-01    0.000e+00    2.398e+00    5.454e-02
   11      38    4.351649e-01    0.000e+00    6.432e+00    1.671e-01
   12      41    3.235134e-01    0.000e+00    3.622e+00    5.931e-02
   13      44    2.444178e-01    0.000e+00    5.693e+00    1.853e-01
   14      47    1.181306e-01    0.000e+00    3.392e+00    1.553e-01
   15      51    6.780342e-02    0.000e+00    2.489e+00    1.419e-01
   16      54    7.008936e-02    0.000e+00    6.057e+00    1.125e-01
   17      57    3.162683e-02    0.000e+00    2.629e+00    5.293e-02
   18      60    1.246636e-02    0.000e+00    2.824e+00    1.621e-01
   19      63    8.402327e-04    0.000e+00    4.321e-01    1.107e-01
   20      67    5.741135e-03    0.000e+00    2.906e+00    1.016e-01
   21      70    3.304907e-03    0.000e+00    1.244e+00    6.478e-02
   22      73    7.531599e-03    0.000e+00    1.044e+00    8.200e-02
   23      76    1.004160e-02    0.000e+00    8.230e-01    3.780e-02
   24      79    1.469296e-02    0.000e+00    4.367e-01    5.381e-02
   25      82    1.322302e-02    0.000e+00    8.969e-02    1.410e-02
   26      85    3.862072e-03    0.000e+00    9.063e-01    1.369e-01
   27      88    2.164421e-03    0.000e+00    6.908e-01    3.482e-02
   28      91    1.247162e-03    0.000e+00    2.624e-01    1.869e-02
   29      94    6.295441e-04    0.000e+00    9.372e-02    2.245e-02
   30      97    7.078191e-04    0.000e+00    1.948e-02    3.741e-03

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
   31     100    1.068763e-04    0.000e+00    1.841e-01    3.974e-02
   32     103    4.429148e-05    0.000e+00    4.264e-02    5.854e-03
   33     106    3.002993e-05    0.000e+00    3.978e-03    2.289e-03
   34     109    8.728886e-07    0.000e+00    2.631e-02    1.102e-02
   35     112    1.228581e-06    0.000e+00    1.990e-03    1.212e-03
   36     115    1.161267e-06    0.000e+00    7.991e-04    6.726e-05
   37     118    5.092558e-08    0.000e+00    6.126e-04    1.909e-03
   38     121    4.503264e-08    0.000e+00    1.600e-04    2.877e-05
   39     124    1.018006e-11    0.000e+00    4.570e-05    4.817e-04
   40     127    5.337221e-12    0.000e+00    1.600e-06    1.570e-06

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.

R = 1×2

1.0000 1.0000

fval = 5.3372e-12

exitflag = 2

output = struct with fields:
         iterations: 41
          funcCount: 148
    constrviolation: 0
           stepsize: 7.3289e-11
          algorithm: 'interior-point'
      firstorderopt: 1.6000e-06
       cgiterations: 25
            message: 'Local minimum possible. Constraints satisfied.↵↵fmincon stopped because the size of the current step is less than↵the value of the step size tolerance and constraints are ↵satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization stopped because the relative changes in all elements of x are↵less than options.StepTolerance = 1.000000e-10, and the relative maximum constraint↵violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.'
       bestfeasible: [1×1 struct]

For more info, please look up the examples shown in the fmincon() documentation:

https://www.mathworks.com/help/optim/ug/fmincon.html

[Donne]

Thank you