Optimize model parameters given minimum sum of error square

I have a simple regression model for

y_model= b0+b1*x1+b2*x2+normrnd(0,x3)+x4

I know the values of b0, b1 and b2. I have some experimentally observed data, y_obs (stored in a vector). The sum of error squared is calculated as:

f= sum(y_obs-y_model)^2

Problem statement: I want to minimize the objective function, f, to obtain the values of x1, x2, x3 and x4. I know the upper and lower bounds of x1, x2, x3, x4. I do not have MATLAB toolbox. So I am trying using 'fmincon'. There might be some better method or function to solve this problem.

I tried to write the code. I see that the program ends with higher f value than f value of the first trial. But, MATLAB is reporting:

Local minimum possible. Constraints satisfied.

Can any of you please help me out in identify the problem? Am I missing the constraint?

Here is my code:

clear;close;clc;
format long;
% I want to calibrate alpha, beta, sigma and delta, given the sum of  squares of errors is minimum

% Experimentally observed data 
y_obs =[2708.7, 2937.8, 2934.9, 2877.8, 2823.1];

% The prediction model 
% y_model = (b0+b1*X(1)+ b2*X(2))
% Coefficients of the prediction model
b0 = 2670.5; b1 = -576.677; b2 = 138.65;

% Ranges of model paramters
X1 = [0.5;0.75]; X2 = [3.5;4.5]; X3=[80;110]; X4 =[-50;40];

% Bounds of model paramters
lb=[X1(1),X2(1),X3(1),X4(1)];
ub=[X1(2),X2(2),X3(2),X4(2)];
    
% Error term (normally distributed with mean 0 and std dev. X3)
% assumed_error ~ N(0, X(3))
% X4 ~ U(U_lb,U_ub)

% y_observed = (b0+b1*X(1)+ b2*X(2))+ normrnd(0,X(3))+ X(4);

% Sum of squares of errors
f=@(X) sum((y_obs- (b0+ b1*X(1)+ b2*X(2)+ normrnd(0,X(3))+ X(4))).^2);    % Objective function to be minimized

% Initial guess
x0 = (lb + ub)/2;

% Show initial objective
disp(['Initial Objective: ' num2str(f(x0))]);

% Linear constraints
A = []; b = []; Aeq = []; beq = [];
options = optimoptions('fmincon','Display','iter','Algorithm','sqp');

% Optimize with fmincon
[X] = fmincon(f,x0,A,b,Aeq,beq,lb,ub);

% Display final objective
disp(['Final Objective: ' num2str(f(X))]);

% Print solution
disp('Solution')
disp(['X1 = ' num2str(X(1))]);
disp(['X2 = ' num2str(X(2))]);
disp(['X3 = ' num2str(X(3))]);
disp(['X4 = ' num2str(X(4))]);

Answer 1

[Matt J]

It does not make sense that there are randomization operations like normrnd in your objective function. How can a function be "minimized" if its definition changes randomly every time you invoke it?

[Rudraprasad Bhattacharyya]

I do not expect a unique optimized value as I intentionally want to incorporate some uncertainty. I want to run this code for multiple samples, similar to Monte-Carlo simulation. Then I'll obtain the minimum value based on all the samples.

[Matt J]

You must run a separate optimization for each randomized sample.