How to define the objective function to solve an optimization problem with Matlab?

Dear all,

I have an optimization problem, but I don’t know how to define the minimization function properly and which optimization algorithm available in Maltab is the most appropriate.

I had previously used fminsearch to curve fitting a fractional derivative model to experimental data and it was OK. However, now I don’t have an analytical equation and so I don’t know how to define the minimization function of even if it is possible to use the Matlab Optimization toolbox for that.

I have the experimental vibration spectrum of hundreds of machines and their division by a pattern spectrum, which I would call Failure spectrum from now on. The aim of the optimization is to define a limit function that will lead to X % of defective machines. The limit is not constant in frequency, it is the following function:

Ylim, 1=< x <= xini

Ylim/ratio, xini< x <= xini+wide

Ylim, xini+wide< x <= 500

where the variables to be optimized are Ylim, ratio, xini and wide.

The code I have written does the following:

1. Initialize the variables and obtain the limit function.
2. Compare the first failure spectrum with the limit function, if the failure spectrum is bigger than the limit at any frequency count 1 and continue with the next failure spectrum.
3. Calculate the error between the define defective by the user and the defective obtain with the actual limit function. This error is the minimized function.

I have first tried with frmincon to restrict the value of the variables, but the value of the variables is all the time the same and nothing is optimized. I have then tried with fminsearch and patternsearch and the same happened. There is no a “Matlab error” but nothing is optimized. I know little about optimization and I am not sure if am defining in a wrong way the minimization function or this problem cannot be solved in this way. I would deeply appreciate any help or clarification.

This is part of the code where the error is calculate ( 2 and 3 steps of the code):

[r,c]=size(Failure_spec_matrix);

cont=cell(1,c);

for i=1:c

Filure=Failure_spec_matrix(:,i);

location=find(Filure>Limit_spec');

count(1,i)={location};

end

count_total=find(~cellfun(@isempty,cont));

error=(length(count_total)*100/c)-defective_defined;

Thank you very much in advance!!

Answer 1

[Alan Weiss]

If I understand you correctly, your objective function (the thing you are trying to minimize) is integer-valued. Small changes in the control variables (the things you can move to change the objective function) don't change the value of the objective function.

If I understand correctly, you might try patternsearch with a large initial mesh size.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Lia]

Thank you very much for your quick answer Alan!

You are right, in some cases small changes in the control variables don’t change the value of the objective function. I have tried to use Patternsearch with a large initial mesh size, but the program is not working. It seems like the poll at all interactions is unsuccessful and so the algorithm does not change the current point at next iterations. Is there any other optimization algorithm that can be better for my problem? Or is the problem in the objective function?

This is part of the code where I define the use of Patternsearch:

fun=@Defective_optm
lb = [12,10,0.5,1];
ub = [100,100,10,500]; 
A=[];
b=[];
Aeq=[];
beq=[];
nonlcon = [];
options = optimoptions('patternsearch','Display','iter','UseCompletePoll',true,'InitialMeshSize',100);
[X,fval,exitflag,output] = patternsearch(fun,X,A,b,Aeq,beq,lb,ub,options)

And this is the result I have obtained:

Iter     Func-count       f(x)      MeshSize     Method
    0           1        15.3846           100      
    1           3        15.3846            50     Refine Mesh
    2           7        15.3846            25     Refine Mesh
    3          11        15.3846          12.5     Refine Mesh
    4          15        15.3846          6.25     Refine Mesh
    5          22        15.3846         3.125     Refine Mesh
    6          29        15.3846         1.563     Refine Mesh
    7          36        15.3846        0.7813     Refine Mesh
    8          43        15.3846        0.3906     Refine Mesh
    9          50        15.3846        0.1953     Refine Mesh
   10          57        15.3846       0.09766     Refine Mesh
   11          64        15.3846       0.04883     Refine Mesh
   12          71        15.3846       0.02441     Refine Mesh
   13          78        15.3846       0.01221     Refine Mesh
   14          85        15.3846      0.006104     Refine Mesh
   15          92        15.3846      0.003052     Refine Mesh
   16          99        15.3846      0.001526     Refine Mesh
   17         106        15.3846     0.0007629     Refine Mesh
   18         113        15.3846     0.0003815     Refine Mesh
   19         120        15.3846     0.0001907     Refine Mesh
   20         127        15.3846     9.537e-05     Refine Mesh
   21         134        15.3846     4.768e-05     Refine Mesh
   22         141        15.3846     2.384e-05     Refine Mesh
   23         148        15.3846     1.192e-05     Refine Mesh
   24         155        15.3846      5.96e-06     Refine Mesh
   25         162        15.3846      2.98e-06     Refine Mesh
   26         169        15.3846      1.49e-06     Refine Mesh
   27         176        15.3846     7.451e-07     Refine Mesh
Optimization terminated: mesh size less than options.MeshTolerance.

X =
   20.0000   20.0000    0.5000  400.0000
fval =
   15.3846

exitflag =
     1
output = 
  struct with fields:
         function: @Ajustea
      problemtype: 'boundconstraints'
       pollmethod: 'gpspositivebasis2n'
    maxconstraint: 0
     searchmethod: []
       iterations: 27
        funccount: 176
         meshsize: 7.4506e-07
         rngstate: [1×1 struct]
          message: 'Optimization terminated: mesh size less than options.MeshTolerance.'

Thank you in advance,

Lia

[Alan Weiss]

If you have validated your objective function (seen that it gives correct values for various inputs), then the only other thing I can suggest is to try surrogateopt. This is a global solver that is time-consuming to run, but attempts to find a global minimum for small-dimensional problems like yours.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Lia]

I have tried with surrogateopt as you suggested and it works fine. I deeply appreciate your help, thank you!

Lia