Getting different values for an optimisation problem fmincon

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ABDULAZIZ ALTUN 2020-3-24

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/512679-getting-different-values-for-an-optimisation-problem-fmincon

I have the following function that I want to minimize

%% Liklihood: function description
function [outputs] = Liklihood_TCL(mu, sigma_epsilon,x_lf,c,N,C, S_inter)
	Omega=(mu.^abs(repmat((1:c),c,1)'-repmat((1:c),c,1)))*(sigma_epsilon^2)/(1-mu^2);
	% Compute the TCL
	phi=C'*inv(C*Omega*C');
	delta=inv(S_inter'*phi*C*S_inter)*S_inter'*phi;
	Log_likelihood = (-N/2)*log(2*pi) - (1/2)*log(det(Omega)) - (1/2)*x_lf'*inv(C*Omega*C')*(eye(N)-C*S_inter*delta)*x_lf;
	outputs=-Log_likelihood;
end

I wrote the following code assuming that x_lf, c, N, C, S_inter are given

loglik_TCL= @(a)Liklihood_TCL(a(1), a(2), x_lf*1000,c,N,C, S_inter);
	% Generate many potential input variables that may insure that the log-likelihood is finite and not complex 
	% Furthermore, that also ensures that the model is stationary
	
	v_x_lf=5;
	fd=[];
	fd1=[];
	while length(fd)<10
		% if we assume mu to be a number between -1 and 1 such that is is N(0,2)
		mu_init=2*trandn(0,1/2);
		sigma_epsilon_init=v_x_lf*trandn(0,1);
		b=[mu_init, sigma_epsilon_init];
		k=loglik_TCL(b);
		while isinf(k)||~isreal(k)||mu_init<0
			mu_init=2*trandn(-1/2,1/2);
			sigma_epsilon_init=v_x_lf*trandn(0,1);
			b=[mu_init, sigma_epsilon_init];
			k=loglik_TCL(b);
		end
		fd1=[fd1; k];
		fd=[fd; b];
	end
	% Set Initial values to the likelihood function
	[M,I] = min(fd1);
	input_1=fd(I,:);
 
	% Linear Constraint
	A = zeros(2,2);
	A(1,1)=1;
	A(2,1)=-1;
	b = [1,-0.8];
	% Constraint Optimization
	[b_min_TCL,fval,exitflag,output] = fmincon(loglik_TCL,input_1,A,b);
	% loglik_TCL(b_min_TCL)
	% Variables that need estimation using likelihood functions
	mu=b_min_TCL(1); %Error autoregressive parameter
	sigma_epsilon=b_min_TCL(2); %Error of the AR(1) error model

The outcome are mu and sigma_epsilon. I assumed for the same datasets and initial constants I should get the values that always minimize the function. However, this is not ture. As can be seen in the attached files.

In particular it is giving me a different maximume variable everytime I run it. for example once it gives me [0.8008 0.5442] as the optimal point however changing it gives me a lower likelihood function.

Results:

>> loglik_TCL= @(a)Liklihood_TCL(a(1), a(2), x_lf,c,N,C, S_inter);
>> loglik_TCL([0.9  0.5])
ans =
  -Inf
>> loglik_TCL([0.8008    0.5442])
ans =
 -340.6327