I am pretty new to OR-TOOLS in Python. I have made several tutorial examples, but I am facing issues trying to model my problem.
Let's see we have a bin packing problem, in which I need to find the fewest bins that will hold all the items in function of their weight. In this typical problem we would want to minimize the number of bins used. But let's say we have an additional objective: to maximize the "quality" of the bin. Here's the problem: to evaluate the quality of that bin, we need to call a non-linear function that takes the items in that bin and returns a quality. I guess I cannot use a multi-objective approach with CP/SAT, so we could model it weighting both objectives.
The problem I am facing is thus the following:
I cannot set the 'quality' as variable because it depends on the
current solution (the items associated to a bin)
How can I do that? assigning a callback? is it possible?
Depending on the "current" solution is not a problem. You could add a "quality" variable, which depends on the values of the variables representing the bins and their contents, and uses the solver's primitives to calculate the desired quantity.
This might not be possible for just any function, but the solver's primitives do allow some forms of non-linear calculations (just as an example, you can calculate abs(x), or x^2, (ref)).
So, for instance, you could have a quality variable which calculates the (number of bins used)^2.
Once you get to a form of quality calculation which works within the solver, you can go back to use one of the approaches for solving for more than a single objective, like weighted sum.
Related
I am working with a programming task that includes a randomly given parameter that I want to optimize. The tricky part is, that this parameter is not part of a function and can not be expressed as such. The task:
I am given values of 4 sets of functions, but I don't have f(x)=... but just the x and y values of the function, which have some noise. I can't fit a curve to find the function because the four functions are too different from each other to construct an algorithm to detect the data model.
I am then given test points (not part of a function), which have to be assigned to one of the 4 functions depending on their distance to the function. The criteria depend on the random factor, i.e. if a test point is within a corridor of sqrt(2)*std of a function, it can be assigned to this function. In doing so, some points can not be assigned to any of the functions.
I now want to find the ideal factor for the distance so that we can assign as many points as possible to one of the functions, with the constraint that none of the points will be assigned to more than one function.
I am a beginner, so I tried to sovle it with marginal increase over a wile loop:
(m is a dictionary)
while m["multiple_matches"]==0:
factor += 0.001
m=map_with_factor(factor)
print(factor)
if m["multiple_matches"]==1:
factor -= 0.0001
m_opt=map_with_factor(factor)
I get decent results, but I feel like there has to be a better way to solve this!
Is there a way that I can tell Python to marginally increase or to turn this into an optimization problem?
As I said, I can not turn the code in the map_with_factor() function into a mathematical function...
The Problem
I've been doing a bit of research on Particle Swarm Optimization, so I said I'd put it to the test.
The problem I'm trying to solve is the Balanced Partition Problem - or reduced simply to the Subset Sum Problem (where the sum is half of all the numbers).
It seems the generic formula for updating velocities for particles is
but I won't go into too much detail for this question.
Since there's no PSO attempt online for the Subset Sum Problem, I looked at the Travelling Salesman Problem instead.
They're approach for updating velocities involved taking sets of visited towns, subtracting one from another and doing some manipulation on that.
I saw no relation between that and the formula above.
My Approach
So I scrapped the formula and tried my own approach to the Subset Sum Problem.
I basically used gbest and pbest to determine the probability of removing or adding a particular element to the subset.
i.e - if my problem space is [1,2,3,4,5] (target is 7 or 8), and my current particle (subset) has [1,None,3,None,None], and the gbest is [None,2,3,None,None] then there is a higher probability of keeping 3, adding 2 and removing 1, based on gbest
I can post code but don't think it's necessary, you get the idea (I'm using python btw - hence None).
So basically, this worked to an extent, I got decent solutions out but it was very slow on larger data sets and values.
My Question
Am I encoding the problem and updating the particle "velocities" in a smart way?
Is there a way to determine if this will converge correctly?
Is there a resource I can use to learn how to create convergent "update" formulas for specific problem spaces?
Thanks a lot in advance!
Encoding
Yes, you're encoding this correctly: each of your bit-maps (that's effectively what your 5-element lists are) is a particle.
Concept
Your conceptual problem with the equation is because your problem space is a discrete lattice graph, which doesn't lend itself immediately to the update step. For instance, if you want to get a finer granularity by adjusting your learning rate, you'd generally reduce it by some small factor (say, 3). In this space, what does it mean to take steps only 1/3 as large? That's why you have problems.
The main possibility I see is to create 3x as many particles, but then have the transition probabilities all divided by 3. This still doesn't satisfy very well, but it does simulate the process somewhat decently.
Discrete Steps
If you have a very large graph, where a high velocity could give you dozens of transitions in one step, you can utilize a smoother distance (loss or error) function to guide your model. With something this small, where you have no more than 5 steps between any two positions, it's hard to work with such a concept.
Instead, you utilize an error function based on the estimated distance to the solution. The easy one is to subtract the particle's total from the nearer of 7 or 8. A harder one is to estimate distance based on that difference and the particle elements "in play".
Proof of Convergence
Yes, there is a way to do it, but it requires some functional analysis. In general, you want to demonstrate that the error function is convex over the particle space. In other words, you'd have to prove that your error function is a reliable distance metric, at least as far as relative placement goes (i.e. prove that a lower error does imply you're closer to a solution).
Creating update formulae
No, this is a heuristic field, based on shape of the problem space as defined by the particle coordinates, the error function, and the movement characteristics.
Extra recommendation
Your current allowable transitions are "add" and "delete" element.
Include "swap elements" to this: trade one present member for an absent one. This will allow the trivial error function to define a convex space for you, and you'll converge in very little time.
I have written python (2.7.3) code wherein I aim to create a weighted sum of 16 data sets, and compare the result to some expected value. My problem is to find the weighting coefficients which will produce the best fit to the model. To do this, I have been experimenting with scipy's optimize.minimize routines, but have had mixed results.
Each of my individual data sets is stored as a 15x15 ndarray, so their weighted sum is also a 15x15 array. I define my own 'model' of what the sum should look like (also a 15x15 array), and quantify the goodness of fit between my result and the model using a basic least squares calculation.
R=np.sum(np.abs(model/np.max(model)-myresult)**2)
'myresult' is produced as a function of some set of parameters 'wts'. I want to find the set of parameters 'wts' which will minimise R.
To do so, I have been trying this:
res = minimize(get_best_weightings,wts,bounds=bnds,method='SLSQP',options={'disp':True,'eps':100})
Where my objective function is:
def get_best_weightings(wts):
wts_tr=wts[0:16]
wts_ti=wts[16:32]
for i,j in enumerate(portlist):
originalwtsr[j]=wts_tr[i]
originalwtsi[j]=wts_ti[i]
realwts=originalwtsr
imagwts=originalwtsi
myresult=make_weighted_beam(realwts,imagwts,1)
R=np.sum((np.abs(modelbeam/np.max(modelbeam)-myresult))**2)
return R
The input (wts) is an ndarray of shape (32,), and the output, R, is just some scalar, which should get smaller as my fit gets better. By my understanding, this is exactly the sort of problem ("Minimization of scalar function of one or more variables.") which scipy.optimize.minimize is designed to optimize (http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.optimize.minimize.html ).
However, when I run the code, although the optimization routine seems to iterate over different values of all the elements of wts, only a few of them seem to 'stick'. Ie, all but four of the values are returned with the same values as my initial guess. To illustrate, I plot the values of my initial guess for wts (in blue), and the optimized values in red. You can see that for most elements, the two lines overlap.
Image:
http://imgur.com/p1hQuz7
Changing just these few parameters is not enough to get a good answer, and I can't understand why the other parameters aren't also being optimised. I suspect that maybe I'm not understanding the nature of my minimization problem, so I'm hoping someone here can point out where I'm going wrong.
I have experimented with a variety of minimize's inbuilt methods (I am by no means committed to SLSQP, or certain that it's the most appropriate choice), and with a variety of 'step sizes' eps. The bounds I am using for my parameters are all (-4000,4000). I only have scipy version .11, so I haven't tested a basinhopping routine to get the global minimum (this needs .12). I have looked at minimize.brute, but haven't tried implementing it yet - thought I'd check if anyone can steer me in a better direction first.
Any advice appreciated! Sorry for the wall of text and the possibly (probably?) idiotic question. I can post more of my code if necessary, but it's pretty long and unpolished.
I'm trying to model a monotonic function, that is bounded by y_min and y_max values, satisfies two value pairs (x0,y0), (x1,y1) within that range.
Is there some kind of package in python that might help in solving for the parameters for such a function?
Alternatively, if someone knows of a good source or paper on modeling such a function, I'd be much obliged...
(for anyone interested, i'm actually trying to model a market respons curve of a number of buyers vs. the price of a product, this is bounded by zero and the maximal demand)
well it's still not clear to me what you want, but the classic function of that form is the sigmoid (it's used in neural nets, for example). in a more flexible form that becomes the general logistic function which will fit your (x,y) constraints with suitable parameters. there's also the gompertz curve, which is similar.
however, those are all defined over an open domain and i doubt you have negative numbers of buyers. if that's an issue (you may not care as they get very close to zero) you could try transforming the number of buyers (taking the log kind-of works, but only if you can have a fraction of a buyer...).
I have a list of many float numbers, representing the length of an operation made several times.
For each type of operation, I have a different trend in numbers.
I'm aware of many random generators presented in some python modules, like in numpy.random
For example, I have binomial, exponencial, normal, weibul, and so on...
I'd like to know if there's a way to find the best random generator, given a list of values, that best fit each list of numbers that I have.
I.e, the generator (with its params) that best fit the trend of the numbers on the list
That's because I'd like to automatize the generation of time lengths, of each operation, so that I can simulate it during n years, without having to find by hand what method fits best what list of numbers.
EDIT: In other words, trying to clarify the problem:
I have a list of numbers. I'm trying to find the probability distribution that best fit the array of numbers I already have. The only problem I see is that each probability distribution has input params that may interfer on the result. So I'll have to figure out how to enter this params automatically, trying to best fit the list.
Any idea?
You might find it better to think about this in terms of probability distributions, rather than thinking about random number generators. You can then think in terms of testing goodness of fit for your different distributions.
As a starting point, you might try constructing probability plots for your samples. Probably the easiest in terms of the math behind it would be to consider a Q-Q plot. Using the random number generators, create a sample of the same size as your data. Sort both of these, and plot them against one another. If the distributions are the same, then you should get a straight line.
Edit: To find appropriate parameters for a statistical model, maximum likelihood estimation is a standard approach. Depending on how many samples of numbers you have and the precision you require, you may well find that just playing with the parameters by hand will give you a "good enough" solution.
Why using random numbers for this is a bad idea has already been explained. It seems to me that what you really need is to fit the distributions you mentioned to your points (for example, with a least squares fit), then check which one fits the points best (for example, with a chi-squared test).
EDIT Adding reference to numpy least squares fitting example
Given a parameterized univariate distirbution (e.g. exponential depends on lambda, or gamma depends on theta and k), the way to find the parameter values that best fit a given sample of numbers is called the Maximum Likelyhood procedure. It is not a least squares procedure, which would require binning and thus loosing information! Some Wikipedia distribution articles give expressions for the maximum likelyhood estimates of parameters, but many do not, and even the ones that do are missing expressions for error bars and covarainces. If you know calculus, you can derive these results by expressing the log likeyhood of your data set in terms of the parameters, setting the second derivative to zero to maximize it, and using the inverse of the curvature matrix at the minimum as the covariance matrix of your parameters.
Given two different fits to two different parameterized distributions, the way to compare them is called the likelyhood ratio test. Basically, you just pick the one with the larger log likelyhood.
Gabriel, if you have access to Mathematica, parameter estimation is built in:
In[43]:= data = RandomReal[ExponentialDistribution[1], 10]
Out[43]= {1.55598, 0.375999, 0.0878202, 1.58705, 0.874423, 2.17905, \
0.247473, 0.599993, 0.404341, 0.31505}
In[44]:= EstimatedDistribution[data, ExponentialDistribution[la],
ParameterEstimator -> "MaximumLikelihood"]
Out[44]= ExponentialDistribution[1.21548]
In[45]:= EstimatedDistribution[data, ExponentialDistribution[la],
ParameterEstimator -> "MethodOfMoments"]
Out[45]= ExponentialDistribution[1.21548]
However, it might be easy to figure what maximum likelihood method commands the parameter to be.
In[48]:= Simplify[
D[LogLikelihood[ExponentialDistribution[la], {x}], la], x > 0]
Out[48]= 1/la - x
Hence the estimated parameter for exponential distribution is sum (1/la -x_i) from where la = 1/Mean[data]. Similar equations can be worked out for other distribution families and coded in the language of your choice.