# Using Multistart in an unconstrained optimization

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Reed on 19 Jul 2022
Commented: Alex Sha on 21 Jul 2022
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) 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')

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 CommentShowHide None
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.

### More Answers (1)

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') Alex Sha on 21 Jul 2022
Hi, Reed, it is ture that the results provided above are not from Matlab.
Matlab is a great and popular math software product, including me, widely used around the world. However, in some areas, for example, the global optimization algorithms or the tool of global optimization, the improvements are relatively minor and no longer have an advantage over some other products, this makes some of Matlab function or toolboxes (lsqcurvefit, gatool, fsolve, vpasolve, cftool...) to be difficult to get optimal results for some problems, for example, the curve fitting problem in this topic, even the result given by Alan Weiss is not correct.
Because of forum rules, it is not appropriate to discuss software other than matlab here, so only giving the right answer, and hope Mathworks will keep to improve Matlab more powerful.

R2022a

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