Using Multistart in an unconstrained optimization

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Hello, suprisingly working with this model I have imposed nonlinear and upper/lower bounds in a separate script and have come up with a better fit. I am surprised to find that the unconstrained case, the code I have here, is not giving me correct solutions. I tried messing with ms.XTolerance and the functionTolerance but nothing helped (since I have only seen an exit flag of 2). It does not seem like multistart is minimizing my objective function, as the error for a solution is a massive number.
Is there anything you all see wrong, or an explanation as to why this is happening? (I would preferably use lsqnonlin, not lsqcurvefit just to stay consistent)
D = readmatrix('Treloar_Data.xlsx');
x = D(:,1);
y = D(:,2);
% Define Ogden Function
ogden_funct1 = @(c) (c(1).*(x.^(c(4)-1)-x.^(-1/2*c(4)-1)) + ...
c(2).*(x.^(c(5)-1)-x.^(-1/2*c(5)-1)) + ...
c(3).*(x.^(c(6)-1)-x.^(-1/2*c(6)-1))) - y;
% Initial_Guess
Initial_Guess1 = [1 1 1 1 1 1];
lb =[];
ub=[];
%Run multistart
problem1 = createOptimProblem('lsqnonlin', 'objective', ogden_funct1, 'x0', Initial_Guess1, 'lb', lb ,'ub', ub);
ms = MultiStart; ms.XTolerance =1*10.^(-13); ms.FunctionTolerance =1*10.^(-13);
[xmultinonlin,errormultinonlin] =run(ms,problem1, 1000)
% Plot Our Data
ogden_plot_func =@(c) c(1).*(x.^(c(4)-1)-x.^((-1)/2*c(4)-1)) + ...
c(2).*(x.^(c(5)-1)-x.^(-1/2*c(5)-1)) + ...
c(3).*(x.^(c(6)-1)-x.^((-1)/2*c(6)-1));
plot(x, y, 'ko', x, ...
ogden_plot_func(xmultinonlin), '-b')
legend('Data', 'lsqnonlin-Unconstrained')
title('Ogden Model Unconstrained')
xlabel('Stretch')
ylabel('Stress')

Accepted Answer

Alan Weiss
Alan Weiss on 20 Jul 2022
Edited: Alan Weiss on 20 Jul 2022
This type of problem is well-known to be numerically touchy, and to have multiple local minima. You really should give bounds, the tightest ones you can. Otherwise, MultiStart has way too large a space to search. And let MultiStart run for more iterations than I show (I gave 500 as a limit because of the limitations of online publishing, 50 second time limit).
x = [1.0000
1.0100
1.1200
1.2400
1.3900
1.6100
1.8900
2.1700
2.4200
3.0100
3.5800
4.0300
4.7600
5.3600
5.7600
6.1600
6.4000
6.6200
6.8700
7.0500
7.1600
7.2700
7.4300
7.5000
7.6100];
y = [ 0
0.0300
0.1400
0.2300
0.3200
0.4100
0.5000
0.5800
0.6700
0.8500
1.0400
1.2100
1.5800
1.9400
2.2900
2.6700
3.0200
3.3900
3.7500
4.1200
4.4700
4.8500
5.2100
5.5700
6.3000];
% Define Ogden Function
ogden_funct1 = @(c) (c(1).*(x.^(c(4)-1)-x.^(-1/2*c(4)-1)) + ...
c(2).*(x.^(c(5)-1)-x.^(-1/2*c(5)-1)) + ...
c(3).*(x.^(c(6)-1)-x.^(-1/2*c(6)-1))) - y;
% Initial_Guess
Initial_Guess1 = [1 1 1 1 1 1];
lb =-5*Initial_Guess1;
ub=5*Initial_Guess1;
rng(1)
%Run multistart
problem1 = createOptimProblem('lsqnonlin', 'objective', ogden_funct1, 'x0', Initial_Guess1, 'lb', lb ,'ub', ub);
ms = MultiStart;
% ms.XTolerance =1*10.^(-13); ms.FunctionTolerance =1*10.^(-13);
[xmultinonlin,errormultinonlin] =run(ms,problem1, 500)
MultiStart completed some of the runs from the start points. 135 out of 500 local solver runs converged with a positive local solver exit flag.
xmultinonlin = 1×6
0.9277 -0.2801 -0.9546 4.5049 -5.0000 4.4908
errormultinonlin = 0.3153
% Plot Our Data
ogden_plot_func =@(c) c(1).*(x.^(c(4)-1)-x.^((-1)/2*c(4)-1)) + ...
c(2).*(x.^(c(5)-1)-x.^(-1/2*c(5)-1)) + ...
c(3).*(x.^(c(6)-1)-x.^((-1)/2*c(6)-1));
plot(x, y, 'ko', x, ...
ogden_plot_func(xmultinonlin), '-b')
legend('Data', 'lsqnonlin+Bounds')
title('Ogden Model With Bounds')
xlabel('Stretch')
ylabel('Stress')
Alan Weiss
MATLAB mathematical toolbox documentation
  1 Comment
Reed
Reed on 20 Jul 2022
Hey thanks @Alan Weiss... that is what I was starting to think, just wanted some reassurance. I really would love to see this unconstrained but it seems like there are just too many points that steer lsqnonlin in the 'wrong direction'. Appreciate it.

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More Answers (1)

Torsten
Torsten on 19 Jul 2022
Edited: Torsten on 19 Jul 2022
D = [1.00 0.00
1.01 0.03
1.12 0.14
1.24 0.23
1.39 0.32
1.61 0.41
1.89 0.50
2.17 0.58
2.42 0.67
3.01 0.85
3.58 1.04
4.03 1.21
4.76 1.58
5.36 1.94
5.76 2.29
6.16 2.67
6.40 3.02
6.62 3.39
6.87 3.75
7.05 4.12
7.16 4.47
7.27 4.85
7.43 5.21
7.50 5.57
7.61 6.30];
x = D(:,1);
y = D(:,2);
% Define Ogden Function
ogden_funct1 = @(c) (c(1).*(x.^(c(4)-1)-x.^(-1/2*c(4)-1)) + ...
c(2).*(x.^(c(5)-1)-x.^(-1/2*c(5)-1)) + ...
c(3).*(x.^(c(6)-1)-x.^(-1/2*c(6)-1))) - y;
% Initial_Guess
Initial_Guess1 = [-1.7720e+01 4.7842e-02 6.7178e-01 -1.7202e-01 3.5402e+00 -3.7330e+00];
lb =[];
ub=[];
options = optimset('MaxFunEvals',10000);
c = lsqnonlin(ogden_funct1,Initial_Guess1,lb,ub,options)
Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance.
c = 1×6
-20.8780 0.0216 16.5974 -1.3422 3.8170 -1.6466
plot(x, y, 'ko', x, ...
ogden_funct1(c)+y, '-b')
  7 Comments
Reed
Reed on 27 Nov 2022 at 23:10
@Alex Sha hey alex... I know we are not allowed to discuss other software products on here but I was wondering if you could email me wink wink. I have a question about your answer reedshay00@gmail.com

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