the 'fmincon' optimisation doesn't stop at minimum with all possible algorithms

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Shuangfeng Jiang on 29 Oct 2020
Edited: Matt J on 31 Oct 2020
I've got an constrained optimisation problem, thus I adopted the function ‘fmincon’.
The curve of the f(x) is like the figure below (This is just 1 specific case as the coefficients of f(x) would vary from case to case but the form is the same) , and we can see the minimum point is around the labelled one at x=7.1e-12. zoom in → However, the optimisation would stop at x=5e-12 (with initial x=1e-11) because the ‘step tolerance’ is satisfied with the default algorithm 'interior-point’. After that, no matter how I decrease the step tolerance (shown in the figure below), the function just yields the same results (x=5e-12). Then I tried algorithm ’sqp’, the step length became too small at the 3rd row, and the function just cannot get out from ‘fval=1.017079’ after that (don't forget the f_min=0.09793 approximately at x=7.1e-12), and it seems the optimisation would just never stop (shown below). Finally, I tried 'active-set’ algorithm, it also gives me x=5e-12. Could anyone please tell me how to push the optimisation closer to the optimal point (x=7.1e-12)? Peronally the initial point is OK. Did I do something wrong in the optimisation?
%% 29.10.2020 optimisation codes
clear
syms x;
size=1e-11;
C_Gmm1=[441400000,298900000000.000];
C_Gmm2=[4140000,34650000000.0000];
fun_2Gmm=@(t) C_Gmm1(1).*t.*exp(-C_Gmm1(2).*t) + C_Gmm2(1).*t.*exp(-C_Gmm2(2).*t);% double gamma function
term_1=@(x) 2.1e-6 + 2.433093685001734e+06.*x;
term_2=@(x) integral(fun_2Gmm,0,x)./size;
f_x=@(x) 2.*term_1(x)./(4.*term_1(x)+term_2(x));% function to be optimised w.r.t. x
%% plot the function w.r.t. the x
fig_1=figure();
fplot(@(x) f_x(x),[0 5.*size],'b');% see where the minimum is before optimisation
%% fmincon optimisation
x0_con=size;
A_fmin=[];
b=[];Aeq=[];beq=[];
lb=0; % x should be bigger than 0
ub=[];nonlcon=[];
options_con = optimoptions('fmincon','Algorithm','interior-point','Display','iter-detailed','OptimalityTolerance', 1e-26,'ConstraintTolerance', 1e-7, 'StepTolerance', 1e-34, 'FunctionTolerance',1e-15,'MaxFunctionEvaluations', 1e10, 'MaxIterations', 1e11);
[x_optmzd_con,Fval_con,exitflag_con,output_con] =fmincon(f_x,x0_con,A_fmin,b,Aeq,beq,lb,ub,nonlcon,options_con);
Shuangfeng Jiang on 29 Oct 2020
Okay, I've just attached my codes, and it can be run diretly.

Matt J on 31 Oct 2020
Edited: Matt J on 31 Oct 2020
Change the units of your unknowns to a less extreme order of magnitude. Also, fmincon is way overkill for a 1D problem. Use fminbnd instead,
size=1e-11;
C_Gmm1=[441400000,298900000000.000];
C_Gmm2=[4140000,34650000000.0000];
fun_2Gmm=@(t) C_Gmm1(1).*t.*exp(-C_Gmm1(2).*t) + C_Gmm2(1).*t.*exp(-C_Gmm2(2).*t);% double gamma function
term_1=@(x) 2.1e-6 + 2.433093685001734e+06.*x;
term_2=@(x) integral(fun_2Gmm,0,x)./size;
f_x=@(x) 2.*term_1(x)./(4.*term_1(x)+term_2(x));% function to be optimised w.r.t. x
s=1e11; %unit-changing scale factor
[x_optmzd_con,Fval_con,exitflag_con,output_con] =fminbnd( @(x)f_x(x/s), 0,5);
x_optmzd_con=x_optmzd_con/s
x_optmzd_con = 7.0782e-12

Walter Roberson on 29 Oct 2020
Change the FiniteDifferenceStepSize
%% 29.10.2020 optimisation codes
close all
syms x;
size=1e-11;
C_Gmm1=[441400000,298900000000.000];
C_Gmm2=[4140000,34650000000.0000];
fun_2Gmm=@(t) C_Gmm1(1).*t.*exp(-C_Gmm1(2).*t) + C_Gmm2(1).*t.*exp(-C_Gmm2(2).*t);% double gamma function
term_1=@(x) 2.1e-6 + 2.433093685001734e+06.*x;
%fplot is for some reason invoking the first time with [-5 0 5]
%and then it complains about the nan that shows up for negative values
term_2=@(x) arrayfun(@(X)integral(fun_2Gmm,0,X),max(0,x))./size;
f_x=@(x) 2.*term_1(x)./(4.*term_1(x)+term_2(x));% function to be optimised w.r.t. x
%% plot the function w.r.t. the x
fig_1=figure();
fplot(@(x) f_x(x),[0 5.*size],'b');% see where the minimum is before optimisation
%% fmincon optimisation
x0_con=size;
A_fmin=[];
b=[];Aeq=[];beq=[];
lb=0; % x should be bigger than 0
ub=5e-11;nonlcon=[];
options_con = optimoptions('fmincon', ...
'Algorithm', 'interior-point', ...
'Display', 'iter-detailed', ...
'OptimalityTolerance', 1e-16, ...
'ConstraintTolerance', 1e-8, ...
'StepTolerance', 1e-20, ...
'FunctionTolerance',1e-17, ...
'MaxFunctionEvaluations', 1e10, ...
'MaxIterations', 1e11, ...
'FiniteDifferenceStepSize', 1e-13); ... 'Diagnostics', 'off');
[x_optmzd_con,Fval_con,exitflag_con,output_con] =fmincon(f_x,x0_con,A_fmin,b,Aeq,beq,lb,ub,nonlcon,options_con);
disp(x_optmzd_con)
disp(Fval_con)