Solving nonlinear equation using lsqnonlin
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Hi
I am trying to solve a set of non-linear equations using lsqnonlin. Initially I was using 'levenberg-marquardt', but I was getting arbitrary results than what I was supposed to get. Then I tried solving using 'lsqnonlin' by defining some upper and lower bounds for the variables. However, the variables are now clinging to either the upper or the lower bound value. Therefore, the reults once again are not correct. What might be the reason for such behaviour?
Here is the code that I am using: (There are 36 to 56 variables. I am showing just 2 for example)
lb=[0.5,-0.5];
ub=[1.5,0.5];
x0=[1,0];
options=optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
[x,res]=lsqnonlin(fun,x0,lb,ub,options);
2 个评论
Ameer Hamza
2020-11-28
编辑：Ameer Hamza
2020-11-28
This is likely caused by your objective function 'fun'. It may be unbounded, so the optimal solution is clinging to the extreme limits.
Anuj Kumar Sahoo
2020-11-30
采纳的回答
Walter Roberson
2020-11-30
编辑：Walter Roberson
2020-12-2
0 个投票
When I use those values in my objective function I get good results. However, when I am trying to findout the values of those unknown coefficients/variables of the same ideal system using 'lsqnonlin' they are coming far from the expected results.
There is no possible algorithm that can find the global minimum of arbitrary "black box" functions (a "black box" function is one that has to just be executed but cannot be effectively examined.) Literally not possible -- it has been proven mathematically.
Different minimizer algorithms have different kinds of functions that they do a good job on. And, conversely, different kinds of situations that they tend to get stuck in.
lsqnonlin often does much better than I expect, but there are some situations it does not do well on.
One example of a situation lsqnonlin does not do well on, is the sum of two gaussians that are slightly different formulas, and which are not well separated. In such a situation, lsqnonlin tends to drive one of the two gaussians to effectively take over all of the fitting, with the other one being driven indefinitely wide -- effectively the second one gets driven to be a constant term. If you start too close to one side the first gaussian takes over; if you start too close to the other side, the other gaussian takes over. You might need to start quite close to the true positions for lsqnonlin to find the correct values. lsqnonlin is not able to find the correct positions itself because every time it moves away from the "one gaussian plus effectively constant" situation, the fit seems to get much much worse.
Another case that optimizers have trouble with is a central region that is narrow and deep, with asymptoptic descent on either side of that: if the optimizer happens to land outside the edge of the central region, then the slope is always decreasing away from the central region, so the optimizers tend to chase that out until infinity or the bounds. The optimizer would have to deliberately "go uphill" hoping that there might just happen to be a well to be discovered.
These kinds of problems are often difficult to deal with unless you can find an optimizer that is designed for the particular shape of function you are dealing with. There are algorithms for sum of gaussians, for example (but I have never researched to find out how they work.)
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Ameer Hamza
2020-11-30
0 个投票
As Walter already explained that there is no guaranteed way to get a globally optimum solution for an arbitrary problem using a numerical method. Because of the way they are formulated, gradient-based methods can at best reach a locally optimal solution depending on the initial guess. Some metaheuristic optimizers, such as genetic algorithm ga(), or particle swarm optimizer particleswarm() have a higher chance of reaching a global solution, but still, the globally optimum solution can never be guaranteed. However, you can increase the probability in several ways. Global Optimization toolbox gives necessary tools for that.
For example, see GlobalSearch(): https://www.mathworks.com/help/gads/globalsearch.html or multistart(): https://www.mathworks.com/help/gads/multistart.html functions. They provide a systematic way to run the common optimizers, such as fmincon(), with several different starting points hoping that this will lead to a global solution. Similarly, check ga(): https://www.mathworks.com/help/gads/ga.html and particleswarm(): https://www.mathworks.com/help/gads/particleswarm.html.
8 个评论
Anuj Kumar Sahoo
2020-12-2
Ameer Hamza
2020-12-2
Without being able to run this example, it is difficult to say anything. Can you attach a small example, which we can run. It will make it easier to suggest anything.
Anuj Kumar Sahoo
2020-12-2
编辑：Walter Roberson
2020-12-2
Walter Roberson
2020-12-2
You are using abs() . However, the derivative of abs() is discontinuous, which is a problem for fsolve(). Especially with you using abs()^2 you might be able to rewrite the calculations without using abs() -- in particular, abs(a)^2 == a.*conj(a)
Anuj Kumar Sahoo
2020-12-3
Anuj Kumar Sahoo
2020-12-3
Walter Roberson
2020-12-5
The editor points out that you assign to gblce and gblco but you do not use those variables. But on the line after you assign to gblce you use blce and on the line after you assign to gblco you use blco . Is it possible that you should have been using the g* version of the variables on those lines?
Anuj Kumar Sahoo
2020-12-5
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