I have a program using PSO algorithm using penalty function for Constraint Satisfaction. But when I run the program for different iterations, the output of the algorithm would be :
"Iteration 1: Best Cost = Inf"
.
Does anyone know why I always get inf answer?
There could be many reasons for that, none of which will be accurate if you don't provide a MWE with the code you have already tried or a context of the function you are analysing.
For instance, while studying the PSO algorithm you might use it on functions which have analytical solutions first. By doing this you can study the behaviour of the algorithm before applying to a similar problem, and fine tune its parameters.
My guess is that you might not be providing either the right function (I have done that already, getting a signal wrong is easy!), the right constraints (same logic applies), your weights for the penalty function and velocity update are way off.
Related
I am very new to MatLab. Thus I am sorry if this is very basic.
I use a function called fmincon to do find a solution for minimizing a function. Why do I get different solutions for running fmincon?
I would like to know a satisfying or convincing mathematical or programming explanation for having different solutions using fmincon.
Check these limitations in the MATLAB documentation.
fmincon is a gradient-based method that is designed to work on problems where the objective and constraint functions are both continuous and have continuous first derivatives.
The function is very delicate and it is best if you can avoid it. It only works neatly on problems that are neatly defined to begin with. Any deviation can lead to local instead of global minima, and these can depend (among other things) on your initial solution estimate or starting point.
As fmincon is sensitive to initial point, If you set different start point for the fmincon, you might get a different solution in each apply. You can find one of the algorithms of fmincon here.
pow=fsolve(#eqns,pop);
This is the code I am using to solve a 2x2 non-linear system of equations, defined in the function eqns.m.
pop is a 2x1 initialisation vector pretty close to the solution. When I run it, the output says
No solution found.fsolve stopped because the relative size of the current step is less than the default value of the step size tolerance squared, but the vector of function values is not near zero as measured by the default value of the function tolerance.<stopping criteria details>
Any way out? I tried moving the initial point further away from the solution intentionally, still it is not working. How do I set the tolerance or some other parameter? Some posts gave me the impression that supplying the jacobian to matlab can be helpful, but how do I do that? Please note that I need the solution in the form of a code which I can put in a function file to be called repeatedly. I believe the interactive optimtool toolbox would not help here. Any help please?
Also from the documentation, the fsolve can employ three different algorithms. Is any of them more helpful than the others for certain problem structures? Where can I get a comparative study of them, suitable for some non-expert in optimisation?
When running the GlobalSearch solver on a nonlinear constrained optimization problem I have, I often get very different solutions each run. For the cases that I have an analytical solution, the numerical results are less dispersed than the non-analytical cases but are still different each run. It would be nice to get the same results at least for these analytical cases so that I know the optimization routine is working properly. Is there a good explanation of this in the Global Optimization Toolbox User Guide that I missed?
Also, why does GlobalSearch use a different number of local solver runs each run?
Thanks!
A full description of how the GlobalSearch algorithm works can be found Here.
In summary the GlobalSearch method iteratively performs convex optimization. Basically it starts out by using fmincon to search for a local minimum near the initial conditions you have provided. Then a bunch of "trial points", based on how good the initial result was, are generated using the "scatter search algorithm." Then there is some more convex optimization and rating of "how good" the minima around these points are.
There are a couple of things that can cause the algorithm give you different answers:
1. Changing the initial conditions you give it
2. The scatter search algorithm itself
The fact that you are getting different answers each time likely means that your function is highly non-convex. The best thing that I know of that you can do in this scenario is just to try the optimization algorithm at several different initial conditions and see what result you get back the most frequently.
It also looks like there is something called the 'PlotFcns' property which would allow you get a better idea what the functions the solver is generating for you look like.
You can use the ga or gamulti objective functions within the GlobalSearch api. I would recommend this. Convex optimizers wont be able to solve a non-linear problem. Even then Genetic Algorithms dont gaurantee the solution. If you run the ga and then use its final minimum as the start of your fmincon search then it should result in the same answer consistently. There may be better ones but if the search space is unknown you may never know.
Are there any faster and more efficient solvers other than fmincon? I'm using fmincon for a specific problem and I run out of memory for modest sized vector variable. I don't have any supercomputers or cloud computing options at my disposal, either. I know that any alternate solution will still run out of memory but I'm just trying to see where the problem is.
P.S. I don't want a solution that would change the way I'm approaching the actual problem. I know convex optimization is the way to go and I have already done enough work to get up until here.
P.P.S I saw the other question regarding the open source alternatives. That's not what I'm looking for. I'm looking for more efficient ones, if someone faced the same problem adn shifted to a better solver.
Hmmm...
Without further information, I'd guess that fmincon runs out of memory because it needs the Hessian (which, given that your decision variable is 10^4, will be 10^4 x numel(f(x1,x2,x3,....)) large).
It also takes a lot of time to determine the values of the Hessian, because fmincon normally uses finite differences for that if you don't specify derivatives explicitly.
There's a couple of things you can do to speed things up here.
If you know beforehand that there will be a lot of zeros in your Hessian, you can pass sparsity patterns of the Hessian matrix via HessPattern. This saves a lot of memory and computation time.
If it is fairly easy to come up with explicit formulae for the Hessian of your objective function, create a function that computes the Hessian and pass it on to fmincon via the HessFcn option in optimset.
The same holds for the gradients. The GradConstr (for your non-linear constraint functions) and/or GradObj (for your objective function) apply here.
There's probably a few options I forgot here, that could also help you. Just go through all the options in the optimization toolbox' optimset and see if they could help you.
If all this doesn't help, you'll really have to switch optimizers. Given that fmincon is the pride and joy of MATLAB's optimization toolbox, there really isn't anything much better readily available, and you'll have to search elsewhere.
TOMLAB is a very good commercial solution for MATLAB. If you don't mind going to C or C++...There's SNOPT (which is what TOMLAB/SNOPT is based on). And there's a bunch of things you could try in the GSL (although I haven't seen anything quite as advanced as SNOPT in there...).
I don't know on what version of MATLAB you have, but I know for a fact that in R2009b (and possibly also later), fmincon has a few real weaknesses for certain types of problems. I know this very well, because I once lost a very prestigious competition (the GTOC) because of it. Our approach turned out to be exactly the same as that of the winners, except that they had access to SNOPT which made their few-million variable optimization problem converge in a couple of iterations, whereas fmincon could not be brought to converge at all, whatever we tried (and trust me, WE TRIED). To this day I still don't know exactly why this happens, but I verified it myself when I had access to SNOPT. Once, when I have an infinite amount of time, I'll find this out and report this to the MathWorks. But until then...I lost a bit of trust in fmincon :)
I am trying to solve a system of non linear equations using fsolve; lets say
F(x;lambda) = 0, where lambda is a vector of parameters, and x the vector I want to solve for.
I am using Matlab's fsolve.
I have 2 values of the parameter lambda, that I want to solve the system for. For the one value of lambda I get a solution, which seems alright.
For the other value of lambda I get a solution again (matlab exits with a flag of 1. However I know this is not an actual solution For example I know that some of the dimensions of x have to be equal to each other, and this is not the case in the solution I get from fsolve.
I have tried both trust-region and the levenberg-marquardt algorithm, and I am not getting any better results. (explicitly enforcing those x's to be the same, still seems to give solutions that are not consistent with what I would be expecting from the properties of the system)
My question is: do the algorithms used by fsolve depend on any kind of stability of the system? Could it be that changing the parameter lambda in the second case I mention above, I make the system unstable, and could that make fsolve having a hard time to solve it correctly?
Thank you, George
fsolve isn't "failing" - as commented by jucestain, it's giving you a local minimum, which is not necessarily a global minimum. This is what it's designed to do.
To improve your chances of obtaining a global minimum you need to either:
Know that your initial guess is good
Run the optimisation several times with a grid of initial guesses, and pick the best result
Add constraints to prevent the solver straying into areas you know to have local minima
Modify your cost function to remove local minima
If you ever come across a non-linear solver that can guarantee a global minimum, do let us know!