I'm writing code, that executes MLE. At each step, I get gradient at one point and then move along it to another point. But I have problem with determination of magnitude of the move. How to determine the best magnitude for good convergence? Can you give me an advice how to avoid other pitfalls, such as presence of several maximums?
Regarding the presence of several maxima: this issue will occur when dealing with a function that is not convex. It can be partially solved by multi-start optimization, which essentially means that you run the simulation multiple times in order to find as many maxima as possible and then selecting the 'highest' maximum from among them. Note that this does not guarantee global optimality, as the global optimum might be hard to reach (i.e. the local optima have a larger domain of attraction).
Regarding the optimal step size for convergence: you might want to look at back-tracking linesearch. A short explanation of it can be found in the answer to this question
We might be able to give you more specific help if you could give us some code to look at, as jkalden already pointed out.
Related
I am using Gurobi to run a MIQP (Mixed Integer Quadratic Programming) with linear constraints in Matlab. The solver is very slow and I would like your help to understand whether I can do something about it.
These are the lines which I use to launch the problem
clear model;
clear params;
model.A=[Aineq; Aeq];
model.rhs=[bineq; beq];
model.sense=[repmat('<', size(Aineq,1),1); repmat('=', size(Aeq,1),1)];
model.Q=Q;
model.obj=c;
model.vtype=type;
model.lb=total_lb;
model.ub=total_ub;
params.MIPGap=10^(-1);
result=gurobi(model,params);
This is a screenshot of the output in the Matlab window.
Question 1: It is the first time I am trying to run a MIQP and I would like to have your advice to understand what I can do to improve performance. Let me tell what I have tried so far:
I cheated by imposing params.MIPGap=10^(-1). In this way the phase of node exploration is made shorter. What are the cons of doing this?
I have big-M coefficients and I have tied them to the smallest possible values.
I have tried setting params.ScaleFlag=2; params.ObjScale=2 but it makes things slower
I have changed params.method but it does not seem to help (unless you have some specific recommendation)
I have increase params.Threads but it does not seem to help
Question 2 (minor): Why do I get a negative objective in the root simplex log? How can the objective function be negative?
Without having the full model here, there is not much on advise to give. Tight Big-M formulations are important, but you said, you checked them already. Sometimes splitting them up might help, but this is a complex field.
What might give great benefits for some problems is using the Gurobi parameter tuning tool. So try to export your model and feed the tuning tool with it. It automatically tries different of the hundreds of tuning parameters and might give some nice results.
Regarding the question about negative objectives in the simplex logs, I can think of a couple of possible explanations. First, note that the negative objective values occur in the presence of dual infeasibilities in the dual simplex run. In such a case, I'm not sure exactly what the primal objective values correspond to. Second, if you have a MIQP with products of binaries in the objective, Gurobi may convexify the objective in a way that makes it possible for a negative objective to appear in the reformulated model even when the original model must have a nonnegative objective in any feasible solution.
Matlab offers multiple algorithms for solving Linear Programs.
For example Matlab R2012b offers: 'active-set', 'trust-region-reflective', 'interior-point', 'interior-point-convex', 'levenberg-marquardt', 'trust-region-dogleg', 'lm-line-search', or 'sqp'.
But other versions of Matlab support different algorithms.
I would like to run a loop over all algorithms that are supported by the users Matlab-Version. And I would like them to be ordered like the recommendation order of Matlab.
I would like to implement something like this:
i=1;
x=[];
while (isempty(x))
options=optimset(options,'Algorithm',Here_I_need_a_list_of_Algorithms(i))
x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options);
end
In 99% this code should be equivalent to
x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options);
but sometimes the algorithm gives back an empty array because of numerical problems (exitflag -4). If there is a chance that one of the other algorithms can find a solution I would like to try them too.
So my question is:
Is there a possibility to automatically get a list of all linprog-algorithms that are supported by the installed Matlab-version ordered like Matlab recommends them.
I think looping through all algorithms can make sense in other scenarios too. For example when you need very precise data and have a lot of time, you could run them all and than evaluate which gives the best results.
Or one would like to loop through all algorithms, if one wants to find which algorithms is the best for LPs with a certain structure.
There's no automatic way to do this as far as I know. If you really want to do it, the easiest thing to do would be to go to the online documentation, and check through previous versions (online documentation is available for old versions, not just the most recent release), and construct some variables like this:
r2012balgos = {'active-set', 'trust-region-reflective', 'interior-point', 'interior-point-convex', 'levenberg-marquardt', 'trust-region-dogleg', 'lm-line-search', 'sqp'};
...
r2017aalgos = {...};
v = ver('matlab');
switch v.Release
case '(R2012b)'
algos = r2012balgos;
....
case '(R2017a)'
algos = r2017aalgos;
end
% loop through each of the algorithms
Seems boring, but it should only take you about 30 minutes.
There's a reason MathWorks aren't making this as easy as you might hope, though, because what you're asking for isn't a great idea.
It is possible to construct artificial problems where one algorithm finds a solution and the others don't. But in practice, typically if the recommended algorithm doesn't find a solution this doesn't indicate that you should switch algorithms, it indicates that your problem wasn't well-formulated, and you should consider modifying it, perhaps by modifying some constraints, or reformulating the objective function.
And after all, why stop with just looping through the alternative algorithms? Why not also loop through lots of values for other options such as constraint tolerances, optimality tolerances, maximum number of function evaluations, etc.? These may have just as much likelihood of affecting things as a choice of algorithm. And soon you're running an optimisation algorithm to search through the space of meta-parameters for your original optimisation.
That's not a great plan - probably better to just choose one of the recommended algorithms, stick to that, and if things don't work out then focus on improving your formulation of the problems rather than over-tweaking the optimisation itself.
I am solving a stiff PDE in MATLAB using ode15, and it often freezes depending on the initial conditions. I never actually get an error, it just won't finish even after 10 hours when it should take around 30 seconds to run. I am experimenting with different spatial and time node intervals, but it is hard, because I don't get feedback.
Is there some sort of equivalent to diagnostic for fsolve? stats is not useful because it only displays an output after fsolve is finished.
Check out the documentation on odeset, and specifically the stats option. I think you basically just want to set stats to on and you will get some feedback.
Also, depending on your ODE, you may need a different solver. About half way down the page on this page there is a list of most of the solvers available in MATLAB. Depending on whether your function is stiff or non-stiff, and how accurate you need to get, one of those might work better for you. Sometimes I just code them all in and comment out all but one until I find the one that runs the best for me, but check out the documentation on each if you want to find the "right" one for your application.
Your question is confusing because you refer to both ode15s and fsolve locking up. These are two completely different functions. One does numerical integration and the other solves for roots. Also, fsolve has no option called 'Stats' (see doc fsolve). If you want continuous output from fsolve use:
options = optimist('Display','iter');
[x,fval,exitflag] = fsolve(myfun,x0,options)
This will display the iteration count, number of function evaluations, the function value, and other stuff depending on what algorithm you use (the alorithm can be adjusted via the 'Algorithm' option). Again see doc fsolve for full details.
As far as the 'Stats' option with ode15s goes, it's not going to give you very much information. I doubt that it will you figure out why your system is halting (if it even is ode15s that you have a problem with). What you can try is using an output function via the 'OutputFcn' option of odeset. You can try the simple odeprint first:
options = odeset('OutputFcn',#odeprint)
which will print your state after each integration step. Type edit odeprint to see the code and how you might write your own output function if you need to do more.
I used Matlab-fminsearch for a negativ max likelihood model for a binomial distributed function. I don't get any error notice, but the parameter which I want to estimate, take always the start value. Apparently, there is a mistake. I know that I ask a totally general question. But is it possible that anybody had the same mistake and know how to deal with it?
Thanks a lot,
#woodchips, thank you a lot. Step by step, I've tried to do what you advised me. First of all, I actually maximized (-log(likelihood)) and this is not the problem. I think I found out the problem but I still have some questions, if I don't bother you. I have a model(param) to maximize in paramstart=p1. This model is built for (-log(likelihood(F))) and my F is a vectorized function like F(t,Z,X,T,param,m2,m3,k,l). I have a data like (tdata,kdata,ldata),X,T are grids and Z is a function on this grid and (m1,m2,m3) are given parameters.When I want to see the value of F(tdata,Z,X,T,m1,m2,m3,kdata,ldata), I get a good output. But I think fminsearch accept that F(tdata,Z,X,T,p,m2,m3,kdata,ldata) like a constant and thatswhy I always have as estimated parameter the start value. I will be happy, if you have any advise to tweak that.
You have some options you can try to tweak. I'd start with algorithm.
When the function value practically doesn't change around your startpoint it's also problematic. Maybe switching to log-likelyhood helps.
I always use fminunc or fmincon. They allow also providing the Hessian (typically better than "estimated") or 'typical values' so the algorithm doesn't spend time in unfeasible regions.
It is virtually always true that you should NEVER maximize a likelihood function, but ALWAYS maximize the log of that function. Floating point issues will almost always corrupt the problem otherwise. That your optimization starts and stops at the same point is a good indicator this is the problem.
You may well need to dig a little deeper than the above, but even so, this next test is the test I recommend that all users of optimization tools do for every one of their problems, BEFORE they throw a function into an optimizer. Evaluate your objective for several points in the vicinity. Does it yield significantly different values? If not, then look to see why not. Are you creating a non-smooth objective to optimize, or a zero objective? I.e., zero to within the supplied tolerances?
If it does yield different values but still not converge, then make sure you know how to call the optimizer correctly. Yeah, right, like nobody has ever made this mistake before. This is actually a very common cause of failure of optimizers.
If it does yield good values that vary, and you ARE calling the optimizer correctly, then think if there are regions into which the optimizer is trying to diverge that yield garbage results. Is the objective generating complex or imaginary results?
I'm currently writing an optimization algorithm in MATLAB, at which I completely suck, therefore I could really use your help. I'm really struggling to find a good way of representing a graph (or well more like a tree with several roots) which would look more or less like this:
alt text http://img100.imageshack.us/img100/3232/graphe.png
Basically 11/12/13 are our roots (stage 0), 2x is stage1, 3x stage2 and 4x stage3. As you can see nodes from stageX are only connected to several nodes from stage(X+1) (so they don't have to be connected to all of them).
Important: each node has to hold several values (at least 3-4), one will be it's number and at least two other variables (which will be used to optimize the decisions).
I do have a simple representation using matrices but it's really hard to maintain, so I was wondering is there a good way to do it?
Second question: when I'm done with that representation I need to calculate how good each route (from roots to the end) is (like let's say I need to compare is 11-21-31-41 the best or is 11-21-31-42 better) to do that I will be using the variables that each node holds. But the values will have to be calculated recursively, let's say we start at 11 but to calcultate how good 11-21-31-41 is we first need to go to 41, do some calculations, go to 31, do some calculations, go to 21 do some calculations and then we can calculate 11 using all the previous calculations. Same with 11-21-31-42 (we start with 42 then 31->21->11). I need to check all the possible routes that way. And here's the question, how to do it? Maybe a BFS/DFS? But I'm not quite sure how to store all the results.
Those are some lengthy questions, but I hope I'm not asking you for doing my homework (as I got all the algorithms, it's just that I'm not really good at matlab and my teacher wouldn't let me to do it in java).
Granted, it may not be the most efficient solution, but if you have access to Matlab 2008+, you can define a node class to represent your graph.
The Matlab documentation has a nice example on linked lists, which you can use as a template.
Basically, a node would have a property 'linksTo', which points to the index of the node it links to, and a method to calculate the cost of each of the links (possibly with some additional property that describe each link). Then, all you need is a function that moves down each link, and brings the cost(s) with it when it moves back up.