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

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 = %.4fnB = %.4fnC = %.4fn’, A_optimized, B_optimized, C_optimized);
fprintf(‘ error function value for optimized parameters: %.4fn’, 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 = %.10fnB = %.14fnC = %.14fn’, A_specific, B_specific, C_specific);
fprintf(‘ error function value for specific parameters: %.4fn’, 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 = 73.54149744754400hello
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.

% 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;

% 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;

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 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.

% 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;

% 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;

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