Error: objective function (feval) is not zero or at optimum.

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Muna Tageldin
Muna Tageldin 2020-11-6
编辑： Muna Tageldin 2020-11-9
I am trying to use mle estimation to estimate parameters of customized pdf. I am using two methods: mle function and fmincon. The function I am trying to estimate has the following sotchastic distribution: f(x)=(1-c)/(sqrt(2*pi*)*op)*exp(-power(x-up,2)*0.5*op^-2) +(c)/(sqrt(2*pi*)*oz)*exp(-power(x-up-uz,2)*0.5*op^-2)
where c is a bionomial random variable.
the objective function using mle and fmincon does not progress to 0 (feval at the optimum is 4.7 e3). First optimality order is zero. This is the code that I have used for fmincon :
z=x(1,:);
options = optimoptions('fmincon','Display','iter-detailed','Algorithm','sqp','MaxIter',3e4,'MaxFunctionEvaluations',3e8,....
'StepTolerance',1e-10,'TolFun',1e-3,'OptimalityTolerance',1e-5,'ConstraintTolerance',1e-3 ,'FiniteDifferenceType','central')
problem.options = options;
% rng('default');
problem.solver = 'fmincon';%'TolX',10^-4,'TolFun',10^-4,
problem.objective = @(y)norm_likelhood_fun(y,z);%norm_likelhood_multi_fun(y,z);%
problem.x0 = rand(5,1);
problem.nonlcon = @(y) cdf_norm(y,z);
problem.A = [];
problem.b = [];
problem.Aeq = [];
problem.beq = [];
problem.lb = 0*ones(5,1);
problem.ub = 1*ones(5,1);%[0.99 0.99 0.99 0.99 0.99];
gs = GlobalSearch;
ms=MultiStart;
%problem.ub = [1,1,1,1,1];
%for j=1:1
problem.x0= [0.1 1e-2 0 0.05 1e-2];
%problem.x0 = [0;0.1;0;0.2;0.01];
[y,feval ,exitflag,output,lam]=fmincon(problem)
yy=[u1;o1;u2;o2;ep]
function l1=norm_likelhood_fun(y,z,cens,freq)
oz=sqrt(y(2).^2+y(4).^2);
x1=((1-y(5))/(sqrt(2*pi)*y(2)))*exp(-0.5*power(z(~cens)-y(1),2)/y(2)^2);
x2=(y(5)/(sqrt(2*pi)*oz))*exp(-0.5*power(z(~cens)-y(1)-y(3),2)/oz^2);
l1=-sum(log(x1+x2));
end
function [c,ceq]=cdf_norm(y,z)
oz=sqrt(y(2).^2+y(4).^2);
x1=((1-y(5))/(2))*(1+erf((z-y(1))/(y(2)*sqrt(2))));
x2=(y(5)/2)*(1+erf((z-y(1)-y(3))/(oz*sqrt(2))));
c=-1+x1+x2;
ceq=[];
end
I followed the recommendation on the below links:
https://www.mathworks.com/help/optim/ug/when-the-solver-might-have-succeeded.html
https://www.mathworks.com/help/optim/ug/when-the-solver-fails.html
However, the ouput that i got is this:
Iter Func-count Fval Feasibility Step Length Norm of First-order
step optimality
30 369 5.495456e+03 0.000e+00 1.000e+00 4.031e-03 2.805e+00
31 380 5.495456e+03 0.000e+00 1.000e+00 9.287e-03 3.505e+00
32 391 5.495455e+03 0.000e+00 1.000e+00 1.933e-02 3.550e+00
33 402 5.495455e+03 0.000e+00 1.000e+00 2.312e-02 3.031e+00
34 413 5.495455e+03 0.000e+00 1.000e+00 3.105e-03 1.810e-01
35 424 5.495455e+03 0.000e+00 1.000e+00 3.168e-03 5.573e-01
36 435 5.495455e+03 0.000e+00 1.000e+00 2.338e-03 7.707e-01
37 446 5.495455e+03 0.000e+00 1.000e+00 5.335e-03 9.047e-01
38 457 5.495455e+03 0.000e+00 1.000e+00 4.379e-03 6.183e-01
39 468 5.495455e+03 0.000e+00 1.000e+00 3.938e-03 1.542e-01
40 479 5.495455e+03 0.000e+00 1.000e+00 4.085e-03 1.323e-01
41 490 5.495455e+03 0.000e+00 1.000e+00 3.578e-03 9.856e-02
42 501 5.495455e+03 0.000e+00 1.000e+00 2.611e-03 4.071e-02
43 512 5.495455e+03 0.000e+00 1.000e+00 2.386e-03 5.575e-02
44 523 5.495455e+03 0.000e+00 1.000e+00 2.003e-03 1.303e-02
45 535 5.495455e+03 0.000e+00 7.000e-01 2.057e-03 5.140e-02
46 546 5.495455e+03 0.000e+00 1.000e+00 6.619e-05 1.113e-02
47 557 5.495455e+03 0.000e+00 1.000e+00 8.599e-04 5.128e-03
48 569 5.495455e+03 0.000e+00 7.000e-01 2.112e-03 2.619e-02
49 580 5.495455e+03 0.000e+00 1.000e+00 8.720e-04 1.873e-03
50 591 5.495455e+03 0.000e+00 1.000e+00 3.328e-04 8.088e-05
51 607 5.495455e+03 0.000e+00 1.681e-01 1.968e-05 4.356e-05
52 618 5.495455e+03 0.000e+00 1.000e+00 1.233e-05 5.707e-06
Optimization completed: The relative first-order optimality measure, 5.707383e-06,
is less than options.OptimalityTolerance = 1.000000e-05, and the relative maximum constraint
violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-03.
y =
0.3264 0.4192 0 0.0033 0.0304
feval =
5.4955e+03
exitflag =
1
output =
struct with fields:
iterations: 52
funcCount: 618
algorithm: 'sqp'
message: '↵Local minimum found that satisfies the constraints.↵↵Optimization completed because the objective function is non-decreasing in ↵feasible directions, to within the value of the optimality tolerance,↵and constraints are satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The relative first-order optimality measure, 5.707383e-06,↵is less than options.OptimalityTolerance = 1.000000e-05, and the relative maximum constraint↵violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-03.↵↵'
constrviolation: 0
stepsize: 1.2330e-05
lssteplength: 1
firstorderopt: 5.7074e-06

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