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

Question

Shuangfeng Jiang 2020-10-29

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/629933-the-fmincon-optimisation-doesn-t-stop-at-minimum-with-all-possible-algorithms

编辑： Matt J 2020-10-31

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);

2 个评论
显示无隐藏无

Walter Roberson 2020-10-29

Can you attach your code and data for testing purposes?

Shuangfeng Jiang 2020-10-29

Okay, I've just attached my codes, and it can be run diretly.

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

Matt J 2020-10-31

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/629933-the-fmincon-optimisation-doesn-t-stop-at-minimum-with-all-possible-algorithms#answer_530044

编辑：Matt J 2020-10-31

在 MATLAB Online 中打开

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

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

请先登录，再进行评论。

Answer 2

Walter Roberson 2020-10-29

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/629933-the-fmincon-optimisation-doesn-t-stop-at-minimum-with-all-possible-algorithms#answer_527543

在 MATLAB Online 中打开

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)

3 个评论
显示 1更早的评论隐藏 1更早的评论

Walter Roberson 2020-10-30

FiniteDifferenceStepSize only applies during the Finite Differences phase that is used to estimate the gradient.

Unfortunately for a few years now, the key implementing routines are compiled in, which perhaps is more efficient but means that I cannot dig into how the variables fit together.

Shuangfeng Jiang 2020-10-31

Well, do you mean there're different phases in an optimisation process where specfic options are called? Could you please tell me where to find the corresponding help page for the explanation of the different phases?

请先登录，再进行评论。