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How to set the objective function when using linprog in Matlab to solve a system of linear inequalities? [closed]
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I want to solve a system of linear inequalities in Matlab, where the unknowns are x(1), x(2), x(3), x(4). I want the entire set of solutions x(1), x(2), x(3), x(4). Hence, I can't use linprog because it gives me just one feasible point.
Clarification: This question https://stackoverflow.com/questions/37258835/how-to-set-the-objective-function-when-using-linprog-in-matlab-to-solve-a-system was about linprogr which however gives only one possible solution. Here I'm asking how to find the entire set of solutions.
This is the set of inequalities. Any suggestion?
5x(1)+3x(2)+3x(3)+5x(4)<5
-5x(1)-3x(2)-3x(3)-5x(4)<-3
-x(2)-x(3)<0
x(2)+x(3)<1
-x(1)-x(4)<0
x(1)+x(4)<1
-3x(3)-5x(4)<-1
3x(3)+5x(4)>3
x(3)<1
-x(3)<0
x(4)<1
-x(4)<0
-5x(1)-3x(2)<0
5x(1)+3x(2)<2
x(2)<1
-x(2)<0
x(1)<1
-x(1)<0
With continuous variables we basically have zero, one or infinitely many solutions. Of course showing all the solutions is not possible for this. However, there is a concept of corner points in linear programming, and those points we can enumerate, although with much effort.
Here are some links for tools that can do this:
cdd
ppl
A different approach is to enumerate the optimal bases using extra binary variables. (You have a zero objective so this becomes effectively: enumerating all feasible LP bases). This approach makes the problem a MIP. We can enumerate this by an algorithm like:
solve mip
if infeasible: stop
add constraint to forbid current point
goto step 1
Here is a link that illustrates this approach to enumerate all optimal bases of a (continuous) LP problem.
Note that enumerating all integer solutions of a system of inequalities is easier. Many constraint programming tools will do that for you automatically. In addition we can use the "cutting plane" technique described above.
Related
I have been looking for a Matlab function that can do a nonlinear total least square fit, basically fit a custom function to data which has errors in all dimensions. The easiest case being x and y data-points with different given standard deviations in x and y, for every single point. This is a very common scenario in all natural sciences and just because most people only know how to do a least square fit with errors in y does not mean it wouldn't be extremely useful. I know the problem is far more complicated than a simple y-error, this is probably why most (not even physicists like myself) learned how to properly do this with multidimensional errors.
I would expect that a software like matlab could do it but unless I'm bad at reading the otherwise mostly useful help pages I think even a 'full' Matlab license doesn't provide such fitting functionality. Other tools like Origin, Igor, Scipy use the freely available fortran package "ODRPACK95", for instance. There are few contributions about total least square or deming fits on the file exchange, but they're for linear fits only, which is of little use to me.
I'd be happy for any hint that can help me out
kind regards
First I should point out that I haven't practiced MATLAB much since I graduated last year (also as a Physicist). That being said, I remember using
lsqcurvefit()
in MATLAB to perform non-linear curve fits. Now, this may, or may not work depending on what you mean by custom function? I'm assuming you want to fit some known expression similar to one of these,
y = A*sin(x)+B
y = A*e^(B*x) + C
It is extremely difficult to perform a fit without knowning the form, e.g. as above. Ultimately, all mathematical functions can be approximated by polynomials for small enough intervals. This is something you might want to consider, as MATLAB does have lots of tools for doing polynomial regression.
In the end, I would acutally reccomend you to write your own fit-function. There are tons of examples for this online. The idea is to know the true solution's form as above, and guess on the parameters, A,B,C.... Create an error- (or cost-) function, which produces an quantitative error (deviation) between your data and the guessed solution. The problem is then reduced to minimizing the error, for which MATLAB has lots of built-in functionality.
Is there a way to check if a rational function is a polynomial in Matlab?
I have a big rational function, call it R, that I am trying to show is a polynomial. I've tried the simplify and simplifyFraction functions and the following (not very effective) procedure:
Split it into denominator and numerator:
[num,den] = numden(R);
Calculate the roots of both polynomials:
r_num = roots(sym2poly(num));
r_den = roots(sym2poly(den));
Check if all the elements of r_den belong to r_num:
Because of numerical imprecision I haven't been able to come up with a reliable way of doing this.
This is a not-so-easy problem and finding greatest common divisor of polynomials is a very active area of research. There are tons of publications and you can find them online.
The main problem is that root finding is an ill-conditioned problem. And recently a few experts are trying to combine the numerical computations with symbolic representations. If you google for ERES method you will have an entry point together with thesis of Christou.
This problem is particularly important for signals and control people because of the transfer function representations and pole zero cancellations. Matlab goes out a long way to make sure that all is OK and a minimal neighborhood of each pole zero is accepted as a cancellation.
So as a quick remedy, convert your polynomial coefficients to 1D vectors, say a and b, and use minreal(tf(a,b)). Then you can extract num and den of that transfer representation.
Shameless plug: I am the author of a python3 library and I also implemented a system theoretical approach. Here and here is the full implementation details with citations about LCM and GCD operations.
This is a question that possibly borders on the intersection of the general usage of MATLAB and/or signal processing. Thought I would first ask the question in a MATLAB forum before trying signal processing.
So our lecturer read out his notes/paper and said the equation
could be implemented as a filter.
At first, it seemed difficult to follow the idea but when realizing that integration is same as finding areas under the curve which seems similar to applying a low pass filter so that only the portion of the signal under the threshold is allowed to pass through, it made a bit of sense. But how - meaning to say which function - can I use to implement the above equation? Do I need three filters or can I use just one? How do I use the terms preceding the integrals in the filter?
Thanks in advance
I've got a problem with my equation that I try to solve numerically using both MATLAB and Symbolic Toolbox. I'm after several source pages of MATLAB help, picked up a few tricks and tried most of them, still without satisfying result.
My goal is to solve set of three non-polynomial equations with q1, q2 and q3 angles. Those variables represent joint angles in my industrial manipulator and what I'm trying to achieve is to solve inverse kinematics of this model. My set of equations looks like this: http://imgur.com/bU6XjNP
I'm solving it with
numeric::solve([z1,z2,z3], [q1=x1..x2,q2=x3..x4,q3=x5..x6], MultiSolutions)
Changing the xn constant according to my needs. Yet I still get some odd results, the q1 var is off by approximately 0.1 rad, q2 and q3 being off by ~0.01 rad. I don't have much experience with numeric solve, so I just need information, should it supposed to look like that?
And, if not, what valid option do you suggest I should take next? Maybe transforming this equation to polynomial, maybe using a different toolbox?
Or, if trying to do this in Matlab, how can you limit your solutions when using solve()? I'm thinking of an equivalent to Symbolic Toolbox's assume() and assumeAlso.
I would be grateful for your help.
The numerical solution of a system of nonlinear equations is generally taken as an iterative minimization process involving the minimization (i.e., finding the global minimum) of the norm of the difference of left and right hand sides of the equations. For example fsolve essentially uses Newton iterations. Those methods perform a "deterministic" optimization: they start from an initial guess and then move in the unknowns space essentially according to the opposite of the gradient until the solution is not found.
You then have two kinds of issues:
Local minima: the stopping rule of the iteration is related to the gradient of the functional. When the gradient becomes small, the iterations are stopped. But the gradient can become small in correspondence to local minima, besides the desired global one. When the initial guess is far from the actual solution, then you are stucked in a false solution.
Ill-conditioning: large variations of the unknowns can be reflected into large variations of the data. So, small numerical errors on data (for example, machine rounding) can lead to large variations of the unknowns.
Due to the above problems, the solution found by your numerical algorithm will be likely to differ (even relevantly) from the actual one.
I recommend that you make a consistency test by choosing a starting guess, for example when using fsolve, very close to the actual solution and verify that your final result is accurate. Then you will discover that, by making the initial guess more far away from the actual solution, your result will be likely to show some (even large) errors. Of course, the entity of the errors depend on the nature of the system of equations. In some lucky cases, those errors could keep also very small.
I am willing to compute a Frechet/Gateaux derivative of a function which is not entirely explicit and my question is : What would be the most efficient way to do it ? Which language would you recommend me to use ?
Precisely, my problem is that I have a function, say F, which is the square of the euclidean norm of the sum of products of pairs of multidimensional functions (i.e. from R^n to R^k).
AFAIK, If I use Maple or Maxima, they will ask me to explicit the functions involved in the formula whereas I would like to keep them abstract. Then, I necessarly need to compute a Frechet/Gateaux derivative so as to keep the expressions simple. Indeed, when I proceed the standard way, I start to develop the square of the euclidean norm as a sum of squares and there is a lot of indexes. My goal being to make a Taylor developpement with integral remainder to the third order, the expression becomes, according to me, humanly infeasible (the formula is more than one A4 page long).
So I would prefer to use a Frechet/Gateaux derivative, which would allow me, among other, to keep scalar products instead of sums.
As the functions invloved have some similarities with their derivatives (due to the presence of exponentials) there is just a small amount of rules to know. So I thought that I might make such a one-purposed computer algebra system by myself.
And I started to learn LISP, as I read that it would be efficient for my problem, but I am a little bit lost now, since this language is very different and I am still used to think in terms of C/Python/Perl...
Here is another question : would you have some links to courses or articles about how an algebra system for symbolic computations is made (preferably in LISP) ? Any suggestions are welcome.
Thank you very much for your answers.
My advice is to use Maxima. Maxima is inspired by Lisp, and implemented in Lisp, so using Maxima will save you a tremendous amount of time and trouble. If Lisp is suitable for your problem, Maxima is even more so.
Maxima will allow you to use undefined terms in an expression; it is not necessary to define all terms.
Post a message to the Maxima mailing list (maxima#math.utexas.edu) to ask for specific advice. Please explain in detail about what you are trying to accomplish.