I am working on Linear Programming Problem with 800K Constraints and the problem takes 20 mins to solve but if I solve the problem for half horizon it just takes 1 min. Is there a way in DoCPLEX where I can solve for partial horizon and then use the solution to solve for other half of the problem without using a for-loop
Three suggestions:
load your problem as LP or SAV into cplex interactive optimizer and run display problem stats. This might show (or rule out) precision issues (ill-conditioned problem). Also it will output number of nonzeros
set datacheck parameters to 2, this might detect numerical issues in data
have you tried different LP algorithms? Using the lpmethod parameter you could try primal, dual or barrier algorithm to see whether one runs faster on your problem.
Reference:
https://www.ibm.com/support/knowledgecenter/SSSA5P_12.10.0/ilog.odms.cplex.help/CPLEX/Parameters/topics/LPMETHOD.html
In DOcplex:
model.parameters.datacheck = 2
model.parameters.lpmethod = 4 # for barrier
From your answers, I can think of the following:
if you are in pure LP (is this true?) I see no point in rounding numbers (but yes, that would help in a MIP, try rounding coefficients whose fractional part is say less than 1e-7: 4.0000001 -> 4)
1e+14 conditioning denotes serious modeling issue: a common source is mixing different objectives with coefficients. Have you tried multi-objective to avoid that?
Another source is big_M formulations, to which you should prefer indicator constraints. If you are not in these two cases, then try to renormalize the data to keep in a smaller condition range...
Finally, you might try setting markowitz tolerance to 0.99, to add extra cautiouness in simplex factorizations, but behavior may vary from one dataset to the other...
Related
I am working on pure LP problem and using Multi-Objective to solve it. When I am using objective with weights:
solver1.minimize( A* 1000
+ B* 10000
+ C* 100
+ D* 1
)
Solver is able to get the optimal result successfully
But when I am using ObjectiveSense package with objective:
solver1.set_multi_objective(ObjectiveSense.Minimize, [A, B, C,D]
, priorities=[1, 2, 0, 1], weights=[1000,1,1,1])
Solver is getting infeasible after first hierarchy of solve and give: "Non-optimal status: infeasibleCPLEX Error 1300: Failure to solve multi-objective subproblem."
I am trying to figure out how to use parameters reltol and abstol to get a feasible solution. Any ideas or examples?
Also trying to understand why Multi-Objective is going infeasible though we have solution through weighted objective? Any input about same will help.
Thanks!
This page
https://www.ibm.com/docs/en/icos/12.10.0?topic=optimization-solving-multiple-objective-problems
explains in detail how tolerances are used to solve multi objective problems.
To summarize, two different algorithms are used for LP and MIP.
The LP algorithm uses LP bases, for which absolute tolerance is interpreted as a limit for reduced costs, and relative costs are not used.
The MIP algorithm uses an iterative solve approach, for which tolerances are interpreted as tolerances on each intermediate sub-objective.
We have seen cases where the LP algorithm may fail, when the model has difficult numerical properties. To check this case, run the following line on your model:
The range of coefficients should not exceed 1e+6.
print(mdl.cplex_matrix_stats)
As a workaround, you can try the Model.solve_with_goals method in Docplex, which runs the iterative multi objective algorithm for any kind of model.
Hello stackoverflow community,
I have troubles in understanding a least-square-error-problem in the c++ armadillo package.
I have a matrix A with many more rows than columns (5000 to 100 for example) so it is overdetermined.
I want to find x so that A*x=b gives me the least square error.
If i use the solve function of armadillo on my data like "x = Solve(A,b)" the error of "(A*x-b)^2" is sometimes way to high.
If on the other hand I solve for x with the analytical form by "x = (A^T * A)^-1 *A^T * b" the results are always right.
The results for x in both cases can differ by 10 magnitudes.
I had thought that armadillo would use this analytical form in the background if the system is overdetermined.
Now I would like to understand why these two methods give such different results.
I wanted to give a short example program, but i can't reproduce this behavior with a short program.
I thought about giving the Matrix here, but with 5000 times 100 it's also very big. I can deliver the values for which this happens though if needed.
So as a short background.
The matrix I get from my program is a numerically solved reaction of a nonlinear oscillator in which I put information inside by wiggeling a parameter of this system.
Because the influence of this parameter on the system is small, the values of my different rows are very similar but never the same, otherwise armadillo should throw an error.
I'm still thinking that this is the problem, but the solve function never threw any error.
Another thing that confuses me is that in a short example program with a random matrix, the analytical form is waaay slower than the solve function.
But on my program, both are nearly identically fast.
I guess this has something to do with the numerical convergence of the pseudo inverse and the special case of my matrix, but for that i don't know enough about how armadillo works.
I hope someone can help me with that problem and thanks a lot in advance.
Thanks for the replies. I think i figured the problem out and wanted to give some feedback for everybody who runs into the same problem.
The Armadillo solve function gives me the x that minimizes (A*x-b)^2.
I looked at the values of x and they are sometimes in the magnitude of 10^13.
This comes from the fact that the rows of my matrix only slightly change. (So nearly linear dependent but not exactly).
Because of that i was in the numerical precision of my doubles and as a result my error sometimes jumped around.
If i use the rearranged analytical formular (A^T * A)*x = *A^T * b with the solve function this problem doesn't occur anymore because the fitted values of x are in the magnitude of 10^4. The least square error is a little bit higher but that is okay, as i want to avoid overfitting.
I now additionally added Tikhonov regularization by solving (A^T * A + lambda*Identity_Matrix)*x = *A^T * b with the solve function of armadillo.
Now the weight vectors are in the order of around 1 and the error nearly doesn't change compared to the formular without regularization.
When optimizing a SVGP with Poisson Likelihood for a big data set I see what I think are exploding gradients.
After a few epochs I see a spiky drop of the ELBO, which then very slowly recovers after getting rid of all progress made before.
Roughly 21 iterations correspond to an Epoch.
This spike (at least the second one) resulted in a complete shift of the parameters (for vectors of parameters I just plotted the norm to see changes):
How can I deal with that? My first approach would be to clip the gradient, but that seems to require digging around the gpflow code.
My Setup:
Training works via Natural Gradients for the variational parameters and ADAM for the rest, with a slowly (linearly) increasing schedule for the Natural Gradient Gamma.
The batch and inducing point sizes are as large as possible for my setup
(both 2^12, with the data set consisting of ~88k samples). I include 1e-5 jitter and initialize the inducing points with kmeans.
I use a combined kernel, consisting of a combination of RBF, Matern52, a periodic and a linear kernel on a total of 95 features (a lot of them due to a one-hot encoding), all learnable.
The lengthscales are transformed with gpflow.transforms.
with gpflow.defer_build():
k1 = Matern52(input_dim=len(kernel_idxs["coords"]), active_dims=kernel_idxs["coords"], ARD=False)
k2 = Periodic(input_dim=len(kernel_idxs["wday"]), active_dims=kernel_idxs["wday"])
k3 = Linear(input_dim=len(kernel_idxs["onehot"]), active_dims=kernel_idxs["onehot"], ARD=True)
k4 = RBF(input_dim=len(kernel_idxs["rest"]), active_dims=kernel_idxs["rest"], ARD=True)
#
k1.lengthscales.transform = gpflow.transforms.Exp()
k2.lengthscales.transform = gpflow.transforms.Exp()
k3.variance.transform = gpflow.transforms.Exp()
k4.lengthscales.transform = gpflow.transforms.Exp()
m = gpflow.models.SVGP(X, Y, k1 + k2 + k3 + k4, gpflow.likelihoods.Poisson(), Z,
mean_function=gpflow.mean_functions.Constant(c=np.ones(1)),
minibatch_size=MB_SIZE, name=NAME)
m.mean_function.set_trainable(False)
m.compile()
UPDATE: Using only ADAM
Following the suggestion by Mark, I switched to ADAM only,
which helped me get rid of that sudden explosion. However, I still initialized with an epoch of natgrad only, which seems to save a lot of time.
In addition, the variational parameters seem to change a lot less abrupt (in terms of their norm at least). I guess they'll converge way slower now, but at least it's stable.
Just to add to Mark's answer above, when using nat grads in non-conjugate models it can take a bit of tuning to get the best performance, and instability is potentially a problem. As Mark points out, the large steps that provide potentially faster convergence can also lead to the parameters ending up in in bad regions of the parameter space. When the variational approximation is good (i.e. the true and approximate posterior are close) then there is good reason to expect that the nat grad will perform well, but unfortunately there is no silver bullet in the general case. See https://arxiv.org/abs/1903.02984 for some intuition.
This is very interesting. Perhaps trying to not use natgrads is a good idea as well. Clipping gradients indeed seems like a hack that could work. And yes, this would require digging around in the GPflow code a bit. One tip that can help towards this, is by not using the GPflow optimisers directly. The model._likelihood_tensor contains the TF tensor that should be optimised. Perhaps with some manual TensorFlow magic, you can do the gradient clipping on here before running an optimiser.
In general, I think this sounds like you've stumbled on an actual research problem. Usually these large gradients have a good reason in the model, which can be addressed with careful thought. Is it variance in some monte carlo estimate? Is the objective function behaving badly?
Regarding why not using natural gradients helps. Natural gradients use the Fisher matrix as a preconditioner to perform second order optimisation. Doing so can result in quite aggressive moves in parameter space. In certain cases (when there are usable conjugacy relations) these aggressive moves can make optimisation much faster. This case, with the Poisson likelihood, is not one where there are conjugacy relations that will necessarily help optimisation. In fact, the Fisher preconditioner can often be detrimental, particularly when variational parameters are not near the optimum.
Short version of the question: overflow or timeout in current settings when calculating large int64_t and double, anyway to avoid these?
Test case:
If only demand is 80,000,000,000, solved with correct result. But if it's 800,000,000,000, returned incorrect 0.
If input has two or more demands (means more inequalities need to be calculated), smaller value will also cause incorrectness. e.g., three equal demands of 20,000,000,000 will cause the problem.
I'm using COIN-OR CLP linear programming solver to solve some network flow problems. I use int64_t when representing the link bandwidth. But CLP uses double most of time and cannot transfer to other types easily.
When the values of the variables are not that large (typically smaller than 10,000,000,000) and the constraints (inequalities) are relatively few, it will give the solution I want it to. But if either of the above factors increases, the tool will stop and return a 0 value solution. I think the reason is the calculation complexity is over its maximum, so program breaks at some trivial point (it uses LP simplex method).
The inequality is some kind of:
totalFlowSum <= usePercentage * demand
I changed it to
totalFlowSum - usePercentage * demand <= 0
Since totalFLowSum and demand are very large int64_t, usePercentage is double, if the constraints like this are too many (several or even more), or if the demand is larger than 100,000,000,000, the returned solution will be wrong.
Is there any way to correct this, like increase the break threshold or avoid this level of calculation magnitude?
Decrease some accuracy is acceptable. I have a possible solution is that 1,000 times smaller on inputs and 1,000 time larger on outputs. But this is kind of naïve and may cause too much code modification in the program.
Update:
I have changed the formulation to
totalFlowSum / demand - usePercentage <= 0
but the problem still exists.
Update 2:
I divided usePercentage by 1000, making its coefficient from 1 to 0.001, it worked. But if I also divide totalFlowSum/demand by 1000 simultaneously, still no result. I don't know why...
I changed the rhs of equalities from 0 to 0.1, the problem is then solved! Since the inputs are very large, 0.1 offset won't impact the solution at all.
I think the reason is that previous coeffs are badly scaled, so the complier failed to find an exact answer.
Here's the problem:
I am currently trying to create a control system which is required to find a solution to a series of complex linear equations without a unique solution.
My problem arises because there will ever only be six equations, while there may be upwards of 20 unknowns (usually way more than six unknowns). Of course, this will not yield an exact solution through the standard Gaussian elimination or by changing them in a matrix to reduced row echelon form.
However, I think that I may be able to optimize things further and get a more accurate solution because I know that each of the unknowns cannot have a value smaller than zero or greater than one, but it is free to take on any value in between them.
Of course, I am trying to create code that would find a correct solution, but in the case that there are multiple combinations that yield satisfactory results, I would want to minimize Sum of (value of unknown * efficiency constant) over all unknowns, i.e. Sigma[xI*eI] from I=0 to n, but finding an accurate solution is of a greater priority.
Performance is also important, due to the fact that this algorithm may need to be run several times per second.
So, does anyone have any ideas to help me on implementing this?
Edit: You might just want to stick to linear programming with equality and inequality constraints, but here's an interesting exact solution that does not incorporate the constraint that your unknowns are between 0 and 1.
Here's a powerpoint discussing your problem: http://see.stanford.edu/materials/lsoeldsee263/08-min-norm.pdf
I'll translate your problem into math to make things a bit easier to figure out:
you have a 6x20 matrix A and a vector x with 20 elements. You want to minimize (x^T)e subject to Ax=y. According to the slides, if you were just minimizing the sum of x, then the answer is A^T(AA^T)^(-1)y. I'll take another look at this as soon as I get the chance and see what the solution is to minimizing (x^T)e (ie your specific problem).
Edit: I looked in the powerpoint some more and near the end there's a slide entitled "General norm minimization with equality constraints". I am going to switch the notation to match the slide's:
Your problem is that you want to minimize ||Ax-b||, where b = 0 and A is your e vector and x is the 20 unknowns. This is subject to Cx=d. Apparently the answer is:
x=(A^T A)^-1 (A^T b -C^T(C(A^T A)^-1 C^T)^-1 (C(A^T A)^-1 A^Tb - d))
it's not pretty, but it's not as bad as you might think. There's really aren't that many calculations. For example (A^TA)^-1 only needs to be calculated once and then you can reuse the answer. And your matrices aren't that big.
Note that I didn't incorporate the constraint that the elements of x are within [0,1].
It looks like the solution for what I am doing is with Linear Programming. It is starting to come back to me, but if I have other problems I will post them in their own dedicated questions instead of turning this into an encyclopedia.