I am having a non-linear optimization model with several variables and a certain function between three of them should be defined as a constraint. (Let us say, that the efficiency of a machine is dependent on the inlet and outlet temperatures). I have calculated some values in a table to visualize the dependency for T_inlets and T_outlets. It gives back a pretty ugly surface. A good fit would be something like a 5 degree polynomial equation if I wanted to define a function directly, but I do not think that would boost my computation speed... So instead I am considering simply having the created table and use it as a lookup table. Is a non-linear solver able to interpret this? I am using ipopt in Pyomo environment.
Another idea would be to limit my feasible temperature range and simplify the connection...maybe with using peace-wise linearization. Is it doable with 3d surfaces?
Thanks in advance!
Related
My problem is described in this picture(It's like a Pyramid structure):
The objective function is below:
In this problem, D is known, A is the object that I want to get. It is a layered structure, each block in the upper layer is divided into four sub-blocks in the layer below. And the value of the upper layer node is equal to the sum of the four child nodes of the lower layer. In above example, I used only 2 layers.
What I want to do is simulate the distribution of D with A, so in the objective function is the ratio of two adjacent squares in each row in A compared to the value in D. I do this comparison on each layer and sum them. Then it is all of my objective function. But in the finest layer, the value in A has a constrain A<=1, the value in A can be a number between 0 and 1. I have tried to solve it using Quadratic programming in python library CVXPY. However, it seems the speed is slow.
So I want to solve it in another way, because this is a convex optimization problem, which can guarantee the global optimal solution. What I think is whether it is possible to use the method of derivation. There are two unknown variables in each item, that is, the two items with A in the formula. Partial derivatives are obtained for them, and the restriction of A<=1 is added, then solve using gradient descent method. Is this mathematically feasible, because I don't know much about optimization, and if it is possible, how should I do it? If not possible, what other methods can I use?
I have a cost function and its gradient calculated with finite element discretization (values at integrations points) and I have the data in a text file.
The problem is the cost function and its gradient not mathematically explicit, calculated numerically at some points xi in volume V at each increment of time t using the finite element method. The results for the function and its gradient are stored in a text file.
How to minimize this function? any idea?
Thanks for your help
I don’t think you can do what you’re thinking to do. If your finite element simulations are already done then the only thing you can do is to create a proxy model of your results as a function of your parameters. One possibility is to model your results by interpolating them (using for example SciPy griddata https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.griddata.html) and use that as your proxy model.
Then you select your favorite optimization algorithm, specify your parameters (that have to be part of the griddata interpolant) and you’re ready to go. Depending on how many finite element simulations you have done you can expect very bad/meaningless outcomes (if you have too few) or very good ones (if your finite element simulations cover almost the entire optimization space).
Basically, I have a dataset that contains 'weights' for some (207) variables, some are more important than the others for determining the class variable (binary) and therefore they are bigger etc. at the end all weigths are summed up across all columns so that the resulting cumulative weight is obtained for each observation.
If this weight is higher then some number then class variable is 1 otherwise is 0. I do have true labels for a class variable so the problem is to minimize false positives.
The thing is, for me it looks like a OR problem as it's about finding optimal weights. However, I am not sure if there is an OR method for such problem, at least I have not heard about one. Question is: does anyone recognize this type of problems and can send some keywords for me to research?
Another thing of course is to predict that with machine learning rather then deterministic methods but I need to do it this way.
Thank you!
Are the variables discrete (integer numbers etc) or continuous (floating point numbers)?
If they are discrete, it sounds like the knapsack problem, which constraint solvers like OptaPlanner (see this training that builds a knapsack solver) excel at.
If they are continuous, look for an LP solver, like CPLEX.
Either way, you'll get much better results than machine learning approaches, because neural nets et al are great at pattern recognition use cases (image/voice recognition, prediction, catagorization, ...), but consistently inferior for constraint optimization problems (like this, I presume).
I am being asked to take a look at a scenario where a company has many projects that they wish to complete, but with any company budget comes into play. There is a Y value of a predefined score, with multiple X inputs. There are also 3 main constraints of Capital Costs, Expense Cost and Time for Completion in Months.
The ask is could an algorithmic approach be used to optimize which projects should be done for the year given the 3 constraints. The approach also should give different results if the constraint values change. The suggested method is multiple regression. Though I have looked into different approaches in detail. I would like to ask the wider community, if anyone has dealt with a similar problem, and what approaches have you used.
Fisrt thing we should understood, a conclution of something is not base on one argument.
this is from communication theory, that every human make a frame of knowledge (understanding conclution), where the frame construct from many piece of knowledge / information).
the concequence is we cannot use single linear regression in math to create a ML / DL system.
at least we should use two different variabel to make a sub conclution. if we push to use single variable with use linear regression (y=mx+c). it's similar to push computer predict something with low accuration. what ever optimization method that you pick...it's still low accuracy..., why...because linear regresion if you use in real life, it similar with predict 'habbit' base on data, not calculating the real condition.
that's means...., we should use multiple linear regression (y=m1x1+m2x2+ ... + c) to calculate anything in order to make computer understood / have conclution / create model of regression. but, not so simple like it. because of computer try to make a conclution from data that have multiple character / varians ... you must classified the data and the conclution.
for an example, try to make computer understood phitagoras.
we know that phitagoras formula is c=((a^2)+(b^2))^(1/2), and we want our computer can make prediction the phitagoras side (c) from two input values (a and b). so to do that, we should make a model or a mutiple linear regresion formula of phitagoras.
step 1 of course we should make a multi character data of phitagoras.
this is an example
a b c
3 4 5
8 6 10
3 14 etc..., try put 10 until 20 data
try to make a conclution of regression formula with multiple regression to predic the c base on a and b values.
you will found that some data have high accuration (higher than 98%) for some value and some value is not to accurate (under 90%). example a=3 and b=14 or b=15, will give low accuration result (under 90%).
so you must make and optimization....but how to do it...
I know many method to optimize, but i found in manual way, if I exclude the data that giving low accuracy result and put them in different group then, recalculate again to the data group that excluded, i will get more significant result. do again...until you reach the accuracy target that you want.
each group data, that have a new regression, is a new class.
means i will have several multiple regression base on data that i input (the regression come from each group of data / class) and the accuracy is really high, 99% - 99.99%.
and with the several class, the regresion have a fuction as a 'label' of the class, this is what happens in the backgroud of the automation computation. but with many module, the user of the module, feel put 'string' object as label, but the truth is, the string object binding to a regresion that constructed as label.
with some conditional parameter you can get the good ML with minimum number of data train.
try it on excel / libreoffice before step more further...
try to follow the tutorial from this video
and implement it in simple data that easy to construct in excel, like pythagoras.
so the answer is yes...the multiple regression is the best approach for optimization.
Minimally, I would like to know how to achieve what is stated in the title. Specifically, signal.lfilter seems like the only implementation of a difference equation filter in scipy, but it is 1D, as shown in the docs. I would like to know how to implement a 2D version as described by this difference equation. If that's as simple as "bro, use this function," please let me know, pardon my naiveté, and feel free to disregard the rest of the post.
I am new to DSP and acknowledging there might be a different approach to answering my question so I will explain the broader goal and give context for the question in the hopes someone knows how do want I want with Scipy, or perhaps a better way than what I explicitly asked for.
To get straight into it, broadly speaking I am using vectorized computation methods (Numpy/Scipy) to implement a Monte Carlo simulation to improve upon a naive for loop. I have successfully abstracted most of my operations to array computation / linear algebra, but a few specific ones (recursive computations) have eluded my intuition and I continually end up in the digital signal processing world when I go looking for how this type of thing has been done by others (that or machine learning but those "frameworks" are much opinionated). The reason most of my google searches end up on scipy.signal or scipy.ndimage library references is clear to me at this point, and subsequent to accepting the "signal" representation of my data, I have spent a considerable amount of time (about as much as reasonable for a field that is not my own) ramping up the learning curve to try and figure out what I need from these libraries.
My simulation entails updating a vector of data representing the state of a system each period for n periods, and then repeating that whole process a "Monte Carlo" amount of times. The updates in each of n periods are inherently recursive as the next depends on the state of the prior. It can be characterized as a difference equation as linked above. Additionally this vector is theoretically indexed on an grid of points with uneven stepsize. Here is an example vector y and its theoretical grid t:
y = np.r_[0.0024, 0.004, 0.0058, 0.0083, 0.0099, 0.0133, 0.0164]
t = np.r_[0.25, 0.5, 1, 2, 5, 10, 20]
I need to iteratively perform numerous operations to y for each of n "updates." Specifically, I am computing the curvature along the curve y(t) using finite difference approximations and using the result at each point to adjust the corresponding y(t) prior to the next update. In a loop this amounts to inplace variable reassignment with the desired update in each iteration.
y += some_function(y)
Not only does this seem inefficient, but vectorizing things seems intuitive given y is a vector to begin with. Furthermore I am interested in preserving each "updated" y(t) along the n updates, which would require a data structure of dimensions len(y) x n. At this point, why not perform the updates inplace in the array? This is wherein lies the question. Many of the update operations I have succesfully vectorized the "Numpy way" (such as adding random variates to each point), but some appear overly complex in the array world.
Specifically, as mentioned above the one involving computing curvature at each element using its neighbouring two elements, and then imediately using that result to update the next row of the array before performing its own curvature "update." I was able to implement a non-recursive version (each row fails to consider its "updated self" from the prior row) of the curvature operation using ndimage generic_filter. Given the uneven grid, I have unique coefficients (kernel weights) for each triplet in the kernel footprint (instead of always using [1,-2,1] for y'' if I had a uniform grid). This last part has already forced me to use a spatial filter from ndimage rather than a 1d convolution. I'll point out, something conceptually similar was discussed in this math.exchange post, and it seems to me only the third response saliently addressed the difference between mathematical notion of "convolution" which should be associative from general spatial filtering kernels that would require two sequential filtering operations or a cleverly merged kernel.
In any case this does not seem to actually address my concern as it is not about 2D recursion filtering but rather having a backwards looking kernel footprint. Additionally, I think I've concluded it is not applicable in that this only allows for "recursion" (backward looking kernel footprints in the spatial filtering world) in a manner directly proportional to the size of the recursion. Meaning if I wanted to filter each of n rows incorporating calculations on all prior rows, it would require a convolution kernel far too big (for my n anyways). If I'm understanding all this correctly, a recursive linear filter is algorithmically more efficient in that it returns (for use in computation) the result of itself applied over the previous n samples (up to a level where the stability of the algorithm is affected) using another companion vector (z). In my case, I would only need to look back one step at output signal y[n-1] to compute y[n] from curvature at x[n] as the rest works itself out like a cumsum. signal.lfilter works for this, but I can't used that to compute curvature, as that requires a kernel footprint that can "see" at least its left and right neighbors (pixels), which is how I ended up using generic_filter.
It seems to me I should be able to do both simultaneously with one filter namely spatial and recursive filtering; or somehow I've missed the maths of how this could be mathematically simplified/combined (convolution of multiples kernels?).
It seems like this should be a common problem, but perhaps it is rarely relevant to do both at once in signal processing and image filtering. Perhaps this is why you don't use signals libraries solely to implement a fast monte carlo simulation; though it seems less esoteric than using a tensor math library to implement a recursive neural network scan ... which I'm attempting to do right now.
EDIT: For those familiar with the theoretical side of DSP, I know that what I am describing, the process of designing a recursive filters with arbitrary impulse responses, is achieved by employing a mathematical technique called the z-transform which I understand is generally used for two things:
converting between the recursion coefficients and the frequency response
combining cascaded and parallel stages into a single filter
Both are exactly what I am trying to accomplish.
Also, reworded title away from FIR / IIR because those imply specific definitions of "recursion" and may be confusing / misnomer.