Summary: How would I go about solving this problem?
Hi there,
I'm working on a mixture-style maximization problem where my variables are going to be bounded by minima and maxima. A representative example of my problem might be:
maximize: (2x-3y+4z)/(x^2+y^2+z^2+3x+4y+5z+10)
subj. to: x+y+z=1
1 < x < 2
-2 < y < 3
5 < z < 8
where numerical coefficients and the minima/maxima are given.
My final project is involving a more complicated problem similar to the one above. The structure of the problems won't change- only the coefficients and inputs will change. So with the example above, I would be looking for a set of functions that might allow a C# program to quickly determine x, then y, then z like:
x = f(given inputs)
y = f(given inputs,x)
z = f(given inputs,x,y)
Would love to hear your thoughts on this one!
Thanks!
The standard optimization approach for your type of problem, non-linear minimization, is the Levenberg-Marquardt algorithm:
Levenberg–Marquardt algorithm
but unfortunately it does not directly support the linear constraints you have added. Many different approaches have been tried to add linear constraints to Levenberg-Marquardt with varying success.
Another algorithm I can recommend in this situation is the Simplex algorithm:
Nelder–Mead method
Like the Levenberg-Marquardt, it also works with non-linear equations but handles linear constraints which act like discontinuities. This could work well for your case above.
In either case, this is not so much a programming problem as an algorithm selection problem. The literature is rife with algorithms and you can find C# implementations of either of the above with a little searching.
You can also combine algorithms. For example, you can do a preliminary search with Simplex with the constraints and the refine it with Levenberg-Marquardt without the constraints.
If your problem is that you want to solve linear programming problems efficiently, you can use Cassowary.net or NSolver.
If your problem is implementing a linear programming algorithm efficiently, you may want to read Combinatorial Optimization: Algorithms and Complexity which covers the Simplex algorithm in most of the detail provided in the short text An Illustrated Guide to Linear Programming but also includes information on the Ellipsoid algorithm, which can be more efficient for more complex constraint systems.
There's nothing inherently C#-specific about your question, but tagging it with that implies you're looking for a solution in C#; accordingly, reviewing the source code to the two toolkits above may serve you well.
Related
I've got two lists of points, let's call them L1( P1(x1, y1), ... Pn(xn, yn)) and L2(P'1(x'1, y'1), ... P'n(x'n, y'n)).
My task is to find the best match between their points for minimizing the sum of their distances.
Any clue on some algorithm? The two lists contain approx. 200-300 points.
Thanks and bests.
If the use case of your problem involves matching ever point present in list L1 with a point in list L2, then the Hungarian Algorithm would serve as a perfect fit.
The weights corresponding to your Hungarian matrix would be the distance between the point annotated for the row vs the column. The overall runtime for the optimized Hungarian algorithm is O(n3) which will comfortably fit for your given constraint of n = 300
A pretty nice tutorial covering the ideology and implementation of the Hungarian algorithm is https://www.topcoder.com/community/competitive-programming/tutorials/assignment-problem-and-hungarian-algorithm/
If not for the Hungarian algorithm, you can also morph the given problem into a max-flow-min-cost problem - the details of which I'll omit for now but can discuss if required.
I'm trying to reproduce same curve fitting (called "trending") in Excel but in C#: Exponential, Linear, Logarithmic, Polynomial and Power.
I found linear and polynomial as :
Tuple<double, double> line = Fit.Line(xdata, ydata);
double[] poly2 = Fit.Polynomial(xdata, ydata, 2);
I also found Exponential fit.
But I wonder how to do curve fitting for Power. Anybody has an idea?
I should be able to get both constants like shown into the Excel screen shot formula:
power
multiplier (before x)
Before anybody would be the fifth who vote to close this question...
I asked the question directly to the forum of mathdotnet (that I recently discovered). Christoph Ruegg, the main developper of the lib, answered me something excellent that I want to share in order to help other with the same problem:
Assuming with power you’re referring to a target function along the
lines of y : x -> a*x^b, then this is a simpler version of what I’ve
described in Linearizing non-linear models by transformation.
This seems to be used often enough so I’ve started to add a new
Fit.Power and Fit.Exponential locally for this case - not pushed yet
since it first needs more testing, but I expect it to be part of v4.1.
Alternatively, by now we also support non-linear optimization which
could also be used for use cases like this (FindMinimum module).
Link to my question: mathdonet - Curve fitting: Power
I'm trying to solve a problem statement using C# as programming language.
In the problem system for an input (double/decimal) say Hi, the output generated is a form of dataset containing number of parameters (Fi, Pi and Ti). I somehow have to filter out only those entries in the data set which would satisfy the following conditions.
Fi > Fmin, where Fmin is some constant
Pi > Pmin, where Pmin is some constant
Ti < Tmax, where Tmax is some constant
Is there an efficient algorithm I could use in such cases where I could zero in on an optimal set of values for Hi for which the output parameter values are well within the constraints. Also I thought using Genetic Algorithms in this case makes sense but somehow I'm not able to formulate and fit the problem specific to Genetic Algorithms.
Any pointers/ suggestions are truly appreciated.
you can use Linq query
var result = DataSet.Where(x=>x.Fi> Fmin && x.Pi>Pmin && Ti < Tmax);
Well, it's hard for me to guess. I don't know the properties of the function for Fi etc.
An log-Barrier Method could be something interesting here. Or the SQP Method. But it has to be differntiable.
Otherwise simulated annealing could be interesting.
But these are just some guesses. It really depends on the problem.
I doubt that a Genetic Algorithm makes sense, seeing as you have only one input variable (Hi) that determines the outputs (Fi, Pi, Ti). The power of a Genetic Algorithm is that it blends good solutions into new solutions. If your solution is only one number, blending two good solutions will probably mean that you're finding some Hi inbetween (such as the average -> 0.5Hi1 + 0.5Hi2 or some other linear combination aHi1 + (1-a)Hi2 with a between 0 and 1).
I would recommend looking into Multi-start Local Search heuristics, such as link. This is a pretty solid heuristic that allows you to explore the solution space for Hi.
In their simplest form, such heuristics calculate the performance for N random values of Hi, and then search for further improvements in the area of the best performing Hi values out of those N initial values.
This sort of stuff is also pretty straight-forward to code, assuming that you have a way to obtain the Fi, Ti, and Pi values from your Hi input, and that you have some way to figure out which of your solutions perform 'best' (for instance through a fitness function as mentioned in the comments).
I apologize if this has already been asked - I'm not certain of the right terms to use here, so if it has, hopefully it will help others like me find whatever this gets marked as a dupe of.
I'm looking to create a formula for a curve in code (C# or Javascript ideally) from 3 points - the formula should be of the form y = a/(t+b) + c where t is time - the horizontal axis - and y is the vertical axis. Obviously a, b, and c are just there for graph fit.
How would I go about this? Is there an existing library I should be using?
The source data has a lot more than 3 data points available, I'm just looking for the simplest way to fit a 1/x curve to the data - so if for example 4 points are required for accuracy that's easy to provide as input.
If you are looking to fit a function of the form
y(t) = a/(t + b) + c
to a set of data points you are faced with a nonlinear least-squares problem for which you can use Gauss-Newton or Levenberg-Marquardt methods. However, there is an old algorithm that goes by the name of Loeb's algorithm that can be used to generate good (but not best - it can be shown that it will not converge to the best approximation) approximations when your approximation is a ratio of polynomials. It works by linearising the least-squares problem and results in an iterative least-squares solution (although in practice you will get good results with a single iteration). I studied this algorithm for my doctorate and I would strongly recommend it for any practical problem in which you want to approximate data points using a polynomial ratio (of which your case is very simple example).
The downside is that this algorithm is very old and you may struggle to find decent documentation of it. If you can it is no more complicated to implement than a standard linear least-squares approximation. If you get no better answer here to your problem, consider googling for it. If you cant find and decent information, let me know and I will upload my thesis to my website (contains implementation details of the method) and you can download it.
As I say you may get a far simpler answer here but if not it will certainly be an option open to you.
I know there are quite some questions out there on generating combinations of elements, but I think this one has a certain twist to be worth a new question:
For a pet proejct of mine I've to pre-compute a lot of state to improve the runtime behavior of the application later. One of the steps I struggle with is this:
Given N tuples of two integers (lets call them points from here on, although they aren't in my use case. They roughly are X/Y related, though) I need to compute all valid combinations for a given rule.
The rule might be something like
"Every point included excludes every other point with the same X coordinate"
"Every point included excludes every other point with an odd X coordinate"
I hope and expect that this fact leads to an improvement in the selection process, but my math skills are just being resurrected as I type and I'm unable to come up with an elegant algorithm.
The set of points (N) starts small, but outgrows 64 soon (for the "use long as bitmask" solutions)
I'm doing this in C#, but solutions in any language should be fine if it explains the underlying idea
Thanks.
Update in response to Vlad's answer:
Maybe my idea to generalize the question was a bad one. My rules above were invented on the fly and just placeholders. One realistic rule would look like this:
"Every point included excludes every other point in the triagle above the chosen point"
By that rule and by choosing (2,1) I'd exclude
(2,2) - directly above
(1,3) (2,3) (3,3) - next line
and so on
So the rules are fixed, not general. They are unfortunately more complex than the X/Y samples I initially gave.
How about "the x coordinate of every point included is the exact sum of some subset of the y coordinates of the other included points". If you can come up with a fast algorithm for that simply-stated constraint problem then you will become very famous indeed.
My point being that the problem as stated is so vague as to admit NP-complete or NP-hard problems. Constraint optimization problems are incredibly hard; if you cannot put extremely tight bounds on the problem then it very rapidly becomes not analyzable by machines in polynomial time.
For some special rule types your task seems to be simple. For example, for your example rule #1 you need to choose a subset of all possible values of X, and than for each value from the subset assign an arbitrary Y.
For generic rules I doubt that it's possible to build an efficient algorithm without any AI.
My understanding of the problem is: Given a method bool property( Point x ) const, find all points the set for which property() is true. Is that reasonable?
The brute-force approach is to run all the points through property(), and store the ones which return true. The time complexity of this would be O( N ) where (a) N is the total number of points, and (b) the property() method is O( 1 ). I guess you are looking for improvements from O( N ). Is that right?
For certain kind of properties, it is possible to improve from O( N ) provided suitable data structure is used to store the points and suitable pre-computation (e.g. sorting) is done. However, this may not be true for any arbitrary property.