The issue of optimization (minimization) of the average relative error between experimental and calculated data

Question

roborrr 2024-8-22

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2147039-the-issue-of-optimization-minimization-of-the-average-relative-error-between-experimental-and-calc

评论： Matt J 2024-11-16，14:35

采纳的回答： Matt J

在 MATLAB Online 中打开

hello

I want to share the difficulties that I faced. Can someone help

problem statement:

there is a 'x' column where the values of the independent variable are written and there is a 'y' column where the experimental values of the dependent variable are written.

approximation model is considered:

y_calculate=A*x^B+C,

and based on this model, an objective function is created, which is equal to the average value of the relative deviations between y and y_calculate:

error_function = mean(abs(y - y_calculate)) / y)=mean(abs(y - =A*x^B+C)) / y);

Our goal is to select parameters A,B,C in such a way that 'error_function' takes the value of the global minimum.

I calculated the optimal values of A, B, C and got:

A = 85.5880, B = -0.0460, C = 4.8824,

at which error function value for optimized parameters: 0.0285.

but I know in advance the specific values of A, B, C:

A = 1005.6335852931, B = -1.59745963925582, C = 73.54149744754400,

at which error function value for specific parameters: 0.002680472178434,

which is much better than with optimization

Below is the code with visualization, which confirms the above.

clear
close all
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guesses for parameters A, B, C
initial_guess = [1, 1, 1];
% Error function
error_function = @(params) mean(abs(y - (params(1) * x.^params(2) + params(3))) ./ y);
% Optimization of parameters
optimized_params = fminsearch(error_function, initial_guess);
% Results of optimization
A_optimized = optimized_params(1);
B_optimized = optimized_params(2);
C_optimized = optimized_params(3);
% Calculation of the fitted function for optimized parameters
y_calculate_optimized = A_optimized * x.^B_optimized + C_optimized;
% Calculate and display the error function value for optimized parameters
value_error_optimized = error_function(optimized_params);
fprintf('Optimized parameters:\nA = %.4f\nB = %.4f\nC = %.4f\n', A_optimized, B_optimized, C_optimized);
fprintf(' error function value for optimized parameters: %.4f\n', value_error_optimized);
% Other specific parameters A, B, C
A_specific = 1005.63358529310;
B_specific = -1.59745963925582;
C_specific = 73.541497447544;
% Calculation of the fitted function for specific parameters
y_calculate_specific = A_specific * x.^B_specific + C_specific;
% Calculate and display the error function value for specific parameters
value_error_specific = error_function([A_specific, B_specific, C_specific]);
fprintf('Specific parameters:\nA = %.10f\nB = %.14f\nC = %.14f\n', A_specific, B_specific, C_specific);
fprintf(' error function value for specific parameters: %.4f\n', value_error_specific);
% Visualization
figure;
plot(x, y, 'bo-', 'DisplayName', 'Experimental data');
hold on;
plot(x, y_calculate_optimized, 'r--', 'DisplayName', 'Fitted model (Optimized)');
plot(x, y_calculate_specific, 'g-.', 'DisplayName', 'Fitted model (Specific)');
xlabel('x');
ylabel('y');
legend('Location', 'best');
title('Approximation of experimental data');
grid on;

Obviously, my optimization code does not lead to a global minimum of the objective function, since there is a better approximation for specific values of A,B,C. Maybe this is caused by a random selection of the initial values of the parameters A=1, B=1, c=1 and therefore my code is stuck in a local minimum?

who can write a code that will select the A,B,C parameters so as to achieve the global minimum of the target function 'error_function', for any initial iteration data of the variables A,B,C. Thoughts for testing: the value of the target function 'error_function' should not be worse (that is, more) than 0.002680472178434, which is obtained with the specific value of A,B,C: A = 1005.6335852931, B = -1.59745963925582, C = 73.54149744754400

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Answer 1

Matt J 2024-8-22

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2147039-the-issue-of-optimization-minimization-of-the-average-relative-error-between-experimental-and-calc#answer_1502929

编辑：Matt J 2024-9-5

在 MATLAB Online 中打开

If you download minL1lin from

https://www.mathworks.com/matlabcentral/fileexchange/52795-constrained-minimum-l1-norm-solutions-of-linear-equations

then you can do,

% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter B
initial_guess = 1;
% Error function
error_function = @(B) objFun(B, x,y);
% Optimization of parameters
B = fminsearch(error_function, initial_guess);
[fval,ABC]=error_function(B);
[A,B,C]=deal(ABC{:})
A = 1.0460e+03
B = -1.6184
C = 73.5535
fval
fval = 0.0025
function [fval,p]=objFun(theta, x,y)
arguments
  theta; x (:,1); y(:,1);
end
 B=theta;
 d=ones(size(y));
 Q=[x.^B, d]./y; 
 
 [AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 p={A, B, C};
end

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roborrr 2024-9-5

Can I assume "stunning" means that it did what you want? If so, please Accept-click the answer to indicate that it resolved your question.

Yes, "stunning" does mean that you did the initial task perfectly, which I did not even expect, for which I am very grateful. I simply forgot to click the "Accept" button, for which I apologize.

Here and elsewhere you have listed eta_inf twice, so I'm not sure what the model unknowns are.

I made a mistake in the letter, where eta_inf is repeated twice, while in place of the first eta_inf, there should be eta_0. I immediately edited the letter, apparently you read the old version of the letter. Please read the new version of the previous letter.

To see what went wrong, I would need to see what you tried.

I didn't try anything because I couldn't figure out how it all worked. How do you solve nonlinear problems with 'minL1lin', which solves the linear problem C*x=d. I tried to figure it out, but I couldn't. Is it possible to write similar codes, as you did for the model y=A*x^B+C, for the models I presented in the previous letter?

roborrr 2024-9-6

在 MATLAB Online 中打开

Following your tips I wrote a program for the model

y_calculate = A* (1 + (lambda * x).^m)+ C,

where it is necessary to calculate the best parameters

A,Lambda,m,C

or optimizing the objective functionЖ

mean( abs( y_calculate-y)/y) .

Here is the code that I wrote following your tips:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = 1;
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
theta = fminsearch(error_function, initial_guess);
[fval,AthetaC]=error_function(theta);
[A, theta,C]=deal(AthetaC{:})
fval
function [fval,p]=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:)';
  lambda=theta(1); m=theta(2);
  d=ones(size(y));
  Q=[ 1./(1 + (lambda * x).^m), d]./y;
 
 [AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 
 p={A, theta, C};
 
end

This code gives errors in my matlab. where am I making mistakes?

roborrr 2024-9-11

在 MATLAB Online 中打开

I corrected the code according to your tips and received:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
theta = fminsearch(error_function, initial_guess);
[fval,AthetaC]=error_function(theta);
[A, theta,C]=deal(AthetaC{:})
fval
function [fval,p]=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:)';
  lambda=theta(1); m=theta(2);
  d=ones(size(y));
  Q=[ 1./(1 + (lambda * x).^m), d]./y;
 
 [AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 
 p={A, theta, C};

for the values of the variables x,y specified in the code, the result is fine.

but I encountered values of the variables x,y for which the code gives an error. for example, for the following values of x,y:

 x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
 y = [21.58079135, 17.26809355, 15.3488609, 14.1074809, 13.81896005, 13.43865625, 13.49266905, 13.44, 13.44, 13.5349172, 13.4323965, 13.439328, 13.36635835, 13.365, 13.29, 13.29, 13.51, 13.365];

the code gives the following error:

Error using linprog (line 373)
A must be a real matrix.
Error in comment_3255504>minL1lin (line 147)
    [xz,~, varargout{3:nargout-1}]=linprog(f,A,b,Aeq,beq,LB,UB,args{:});
Error in comment_3255504>objFun (line 23)
 [AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
Error in comment_3255504>@(theta)objFun(theta,x,y) (line 8)
error_function = @(theta) objFun(theta, x,y);
Error in fminsearch (line 336)
    fxr = funfcn(x,varargin{:});

how to make the code work correctly for all values of the x,y variables?

Torsten 2024-9-13

编辑：Torsten 2024-9-13

在 MATLAB Online 中打开

The code requires "fminsearchbnd" from the file exchange:

https://uk.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd-fminsearchcon

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 = 0;
lambda1 = inf;
m0 = -inf;
m1 = inf;
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);
%theta = fminsearch(error_function, initial_guess);
[fval,AthetaC]=error_function(theta);
[A, theta,C]=deal(AthetaC{:})
A = 131.5017
theta = 1×2
    0.2036    1.9539
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
C = 73.9207
fval
fval = 0.0015
function [fval,p]=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:)';
  lambda=theta(1); m=theta(2);
  d=ones(size(y));
  Q=[ 1./(1 + (lambda * x).^m), d]./y;
  %Bounds on A and C
  A0 = 0;
  A1 = inf;
  C0 = 0.9*min(y);
  C1 = 1.1*min(y);
  lb=[A0,C0];
  ub=[A1,C1]; 
  %Call the optimizer
  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));
 %[AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 
 p={A, theta, C};
end

roborrr 2024-9-24

编辑：roborrr 2024-9-24

在 MATLAB Online 中打开

TThanks to Torsen for perfecting the code, it was trully amazing. Also thanks to Matt J who wrote the initial version of the code, that was also excellent.

The main beauty of this code is that it eliminates the occurrence of local minima during optimization, that could arise when using built-in Matlab functions.

In this code, the dependent variable 'y' depends only on one variable 'x' (one-dimensional problem). Is it possible to write code for a two-dimensional problem, when the dependent variable depends nonlinearly on two independent variables at once, using the scheme you used for the one-dimensional problem, which excludes the occurrence of local minima? Here is an example of such a problem:

The experiments were carried out on the following unique values of the variable x_un length of 5 and y_un length of 10 :

x_un = [0.4, 0.5, 0.6, 0.7, 0.8];
y_un = [283.15, 288.15, 290.65, 293.15, 295.15, 298.15, 300.65, 303.15, 313.15, 323.15];

based on x_un and y_un, new variables x and y length of 50 (5*10) are created so that the pairs (x(i),y(i)) i=1,50 exhaust all combinations of pairs (x_un, y_un):

x = [0.4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, ...
4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, ...
4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, 0.4, 0.5, 0.6, 0.7, 0.8, ...
4, 0.5, 0.6, 0.7, 0.8];
y = [283.15, 283.15, 283.15, 283.15, 283.15, 288.15, 288.15, 288.15, 288.15, 288.15, ...
65, 290.65, 290.65, 290.65, 290.65, 293.15, 293.15, 293.15, 293.15, 293.15, ...
15, 295.15, 295.15, 295.15, 295.15, 298.15, 298.15, 298.15, 298.15, 298.15, ...
65, 300.65, 300.65, 300.65, 300.65, 303.15, 303.15, 303.15, 303.15, 303.15, ...
15, 313.15, 313.15, 313.15, 313.15, 323.15, 323.15, 323.15, 323.15, 323.15];

variable z is also created, the length of which is also 50, in such a way that experimental data are entered into z(i), which correspond to (x(i),y(i)) i=1,50:

z = [12.61323, 15.96843, 22.20326, 30.15335, 60.71139, 10.97453, 14.54062, 18.84638, 24.82916, ...
02121, 9.57412, 13.34543, 17.50987, 22.78405, 46.60942, 8.74127, 11.59837, 14.62923, ...
16607, 42.31916, 7.93542, 10.73998, 13.71756, 19.05432, 39.08479, 7.33475, 10.12368, ...
67264, 18.14767, 35.87183, 6.52910, 9.07151, 11.60863, 16.87576, 32.95969, 5.43683, ...
19517, 10.54588, 15.03453, 29.99945, 4.25370, 6.86437, 9.20690, 12.29640, 24.78083, ...
99082, 5.05884, 6.68976, 9.54918, 18.76493];

approximation model is considered:

z_calculate = A * exp(E/ (8.314 * y)) * (1 + k * x^n);

and based on this model, an objective function is created, which is equal to the average value of the relative deviations between z and z_calculate:

error_function = mean(abs(z - z_calculate)) /z)

Our goal is to select parameters A,E,k,n in such a way that 'error_function' takes the value of the global minimum.

if possible, set lower and upper limits for parameters:

lb_A = 10^(-3); ub_A = 100;

lb_E = 2000; ub_E = 50000;

lb_k = 0.1; ub_k = 10;

lb_n = 1; ub_n = 3;

roborrr 2024-10-19

编辑：roborrr 2024-10-19

在 MATLAB Online 中打开

Thanks Torsten for the answer, but I'm again confused by the code syntax. Here is the code you created for the model, which contains linear coefficients:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 = 0;
lambda1 = inf;
m0 = -inf;
m1 = inf;
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);
%theta = fminsearch(error_function, initial_guess);
[fval,AthetaC]=error_function(theta);
[A, theta,C]=deal(AthetaC{:})
fval
function [fval,p]=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:)';
  lambda=theta(1); m=theta(2);
  d=ones(size(y));
  Q=[ 1./(1 + (lambda * x).^m), d]./y;
  %Bounds on A and C
  A0 = 0;
  A1 = inf;
  C0 = 0.9*min(y);
  C1 = 1.1*min(y);
  lb=[A0,C0];
  ub=[A1,C1]; 
  %Call the optimizer
  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));
 %[AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 
 p={A, theta, C};
end

as I understand, you suggest to replace "minL1lin" with "fminsearchbnd" in the "objFun" subroutine. but "fminsearchbnd" is already used in the main part of the code to determine the boundaries of the "lambda" and "m" parameters. I don't know the syntax of how to combine these two uses of "fminsearchbnd", how to output the optimized values of "lambda" and "m" and the error function "fval", for a model that does not have a linear part.

If it is not too much trouble, Please remake the code you sent which is written for the model y = A.* (1 + (lambda * x).^m)+ C, for the model y1= (lambda * x).^m)

roborrr 2024-10-28，17:40

移动：Torsten 2024-10-28，18:06

在 MATLAB Online 中打开

You guessed correctly: I wanted only lambda and m to be optimized, with A and C known, And you sent the code i wanted.

but I tested your code as follows:

first I solved the optimization problem for the model y=A.*(1+(lambda*x).^m)+C using minL1lin, assuming first that A and C are unknown and should be optimized jointly lambda and m:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 =0;
lambda1=inf;
m0 =0;
m1 =inf;
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);
%theta = fminsearch(error_function, initial_guess);
[fval,AthetaC]=error_function(theta);
[A, theta,C]=deal(AthetaC{:})
fval
function [fval,p]=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:)';
  lambda=theta(1); m=theta(2);
  d=ones(size(y));
  Q=[ 1./(1 + (lambda * x).^m), d]./y;
  %Bounds on A and C
  A0 = 0;
  A1 = inf;
  C0 =  0.2*min(y);
  C1 = 1.1*min(y);
  lb=[A0,C0];
  ub=[A1,C1]; 
  %Call the optimizer
  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));
 %[AC,fval]=minL1lin(Q,d,[],[],[],[],[],[],[],optimoptions('linprog','Display','off'));
 
 fval=fval/numel(y);
 
 A=AC(1); C=AC(2);
 
 p={A, theta, C};
end

and I got the following results:

 A = 131.5161945547369  
 theta = 0.2037    1.9538
C = 73.9206126555175
fval = 0.0015

after that the optimized values A= 131.5161945547369 and C= 73.9206126555175 (obtained using minL1lin) I used as previously known coefficients and optimized only lambda and m parameters using fminsearchbnd:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 = 0;
lambda1 = inf;
m0 = 0;
m1 = inf;
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);
fval=error_function(theta);
fval
function fval=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:);
  lambda=theta(1); m=theta(2);
  % Set A and B to appropriate values !
  A = 131.5161945547369;   
  C = 73.9206126555175;
  fval = sum(abs((y - (A.* (1 + (lambda * x).^m)+ C))./y));
  
  fval= fval/numel(y);
end

and got a much worse result for Fval:

fval = 3.3576

why the result of the first script - fval = 0.0015 and the result of the second script - fval = 3.3576 are not equal? where is the error in the second script?

roborrr 2024-11-1，14:08

在 MATLAB Online 中打开

thank you dear Torshen for correcting the error in my second script - instead of model function y=A.*(1+(lambda*x).^m)+C, there should indeed be function y=A./(1+(lambda*x).^m)+C:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 =0;
lambda1=inf;
m0 =0;
m1 =inf;
  A0 = 0;
  A1 = inf;
  C0 =  0.2*min(y);
  C1 = 1.1*min(y);
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);
fval=error_function(theta);
fval
function fval=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:);
  lambda=theta(1); m=theta(2);
 
  % Set A and B to appropriate values !
   A = 131.5161945547369;   
   C = 73.9206126555175;
  fval = sum(abs((y - (A./(1 + (lambda * x).^m)+ C))./y));
  
  fval= fval/numel(y);
end

This construct does give the best result - Fval = 0.015, when the best coefficients A and C are known in advance. But I tried to use this construct to optimize all coefficients A, C, lambda and m and got the worst result - Fval = 0.0240. Here is the script I wrote:

clear
% Data
x = [7.3392, 14.6784, 22.0176, 29.3436, 36.6828, 44.0088, 51.3348, 58.674, 66, 73.3392, 80.6652, 88.0044, 95.3304, 102.6696, 109.9956, 117.3348, 124.6608, 132];
y = [115.1079, 87.7698, 80.5755, 78.1611, 76.5743, 75.7074, 74.9375, 74.9453, 74.59, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990, 74.2990];
% Initial guess for parameter  theta
initial_guess = [1,1,1,1];
% Error function
error_function = @(theta) objFun(theta, x,y);
% Optimization of parameters
%Bounds on lambda and m 
lambda0 =0;
lambda1=inf;
m0 =0;
m1 =inf;
  A0 = 0;
  A1 = inf;
  C0 =  0.2*min(y);
  C1 = 1.1*min(y);
%Call the optimizer
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0,A0,C0],[lambda1,m1,A1,C1]);
fval=error_function(theta);
fval
function fval=objFun(theta, x,y)
  x=x(:);y=y(:); theta=theta(:);
  lambda=theta(1); m=theta(2);
  A=theta(3); 
  C=theta(4);
  % the best values ​​of A and C that should be obtained in this script
%    A = 131.5161945547369;   
%    C = 73.9206126555175;
  fval = sum(abs((y - (A./(1 + (lambda * x).^m)+ C))./y));
  
  fval= fval/numel(y);
end

why does the value - Fval = 0.015, obtained from the first script not match the value obtained from the second script - Fval = 0.0240? where am I making a mistake in the second script?

how universal is this construct? is it possible to use it for models of any complexity? or is the class of models limited?

roborrr 2024-11-16，9:28

在 MATLAB Online 中打开

I use construct:

  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));

matlab gives a message:

Exiting: Maximum number of function evaluations has been exceeded
         - increase MaxFunEvals option.
         Current function value: 0.012493 

What is the default maximum iteration count in aa and how to increase it?

and also: in some cases when using this construct matlab gives the following errors:

         Error using linprog (line 346)
LINPROG encountered an internal error that caused the solution to lose feasibility. We are sorry for the inconvenience.
Please contact technical support for assistance with your problem, quoting the code "-2_4".
Error in minL1lin (line 116)
    [xz,~, varargout{3:nargout-1}]=linprog(f,A,b,Aeq,beq,LB,UB,args{:});
Error in A_A_optimize_problem_A_B_C_lambda_m/objFun (line 42)
  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));
Error in A_A_optimize_problem_A_B_C_lambda_m>@(theta)objFun(theta,x,y) (line 9)
error_function = @(theta) objFun(theta, x,y);
Error in fminsearchbnd>intrafun (line 259)
fval = feval(params.fun,reshape(xtrans,params.xsize),params.args{:});
Error in fminsearch (line 324)
    x(:) = xr; fxr = funfcn(x,varargin{:});
Error in fminsearchbnd (line 229)
[xu,fval,exitflag,output] = fminsearch(@intrafun,x0u,options,params);
Error in A_A_optimize_problem_A_B_C_lambda_m (line 19)
theta = fminsearchbnd(error_function, initial_guess,[lambda0,m0],[lambda1,m1]);

Torsten 2024-11-16，10:42

编辑：Torsten 2024-11-16，10:43

Please include the full code to reproduce the error.

Matt J 2024-11-16，14:35

在 MATLAB Online 中打开

Error in A_A_optimize_problem_A_B_C_lambda_m/objFun (line 42)
  [AC,fval]=minL1lin(Q,d,[],[],[],[],lb,ub,[],optimoptions('linprog','Display','off'));

We need to examine Q,d, lb,ub when this happens.

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