I am trying to use z3 to solve an linear programming problem and observing very poor performance relative to GLPK. There are reasons why I would prefer to use z3 so I'm wondering if there is something I'm missing.
The problem is essentially bin packing. I have on the order of 500 weighted items, and 5 bins. Every item must be placed in a bin. Objective is to minimize the total weight in the largest bin.
The problem was solved within 1-2 minutes by GLPK. I have yet to see z3 terminate.
The z3 optimization tutorial suggests both a boolean encoding and an integer encoding for a similar type of problem. Is one of these preferred for performance reasons? Basically I'm wondering if you need to follow a certain pattern for z3 to recognize it as linear programming.
How do I know what method it's applying to solve the problem?
Is there a way to configure logging on z3 so you can see what it's doing.
By the way, I'm using the C++ API.
Related
I have a set of data z(0), z(1), z(2)...,z(n) that I am currently fitting with a 2 variables polynomial of the kind p(x,y) = a(1)*x^2+a(2)*y^2+a(3)*x*y+a(4). I have i=1,...,n (x(i),y(i)) coordinates that I impose to be p(x(i),y(i))=z(i). In this way I have a Overdetermined System that I can solve using Eigen SVD . I am looking for a more sophisticated method that can take care of outliers, like a Least Median of Squares robust regression (as described here) but I haven't found a C++ implementation for 2 variables. I looked in GSL but it seems there is nothing for 2 variable functions. The only other solution I can think of is using a TGraph2D in ROOT. Do you know any other solution? Numerical recipes maybe? Since I am writing C++ code I would prefer C or C++ implementations.
Since non answer has been given yet, but I am still working on this problem, I will share my progresses here.
The class TLinearFitter has a fit method that allows you to select Robust fitting - Least Trimmed Squares regression (LTS):
https://root.cern.ch/root/html532/TLinearFitter.html
Another possible solution, more time consuming maybe, but maybe more efficient on the long run is to write my own function to be minimized, and the use:
https://projects.coin-or.org/Ipopt to minimize it. Although in this approach there is a bigger "step". I don't know how to use the library and I haven't (yet?) found a nice tutorial to understand it.
here: https://wis.kuleuven.be/stat/robust/software there is a Fortran implementation of the LMedS algorithm called PROGRESS. So another possible solution could be to port this software to C/C++ and make a library out of it.
I have a huge problem. I need to solve a non linear sistem of 3 equations in 3 variables with a C++ function or class. I thought about using Newton-Raphson method to perform the solution. Unlukily I didn't find a source code that can do that for me. There would be someone that knows a program like that? I'm near deciding to build it myself. Thanks
A 3x3 system is not huge; it's actually a very small problem. People routinely solve nonlinear systems of equations with thousands (and more) of variables and constraints.
Given that your system is 3x3 and possibly nasty, a more appropriate choice of method would be a line search method. You get global convergence to a local minimum of the residual this way; it's very easy to make straight Newton's method diverge.
Steepest descent with backtracking line search is the simplest line search method possible. You might try implementing it first.
First, see related questions What good libraries are there for solving a system of non-linear equations in C++? and https://stackoverflow.com/questions/4914967/could-you-explain-how-newton-raphson-for-a-set-of-equations-works-code-inside. Also, try to use boost.
Consider this cozy C++ library
I'm trying to solve a system of 4 second order polynomial equations using C++. What is the fastest method for solving the system, and if possible, could you link or write a little pseudocode to explain it? I'm aware of solutions involving a Groebners basis or QR decomposition, but I can't find a clear description of how they work and how to implement them. Maybe helpful info about the polynomials:
A solution(s) may exist or may not, but I am only interested in solutions in a certain range (e.g. x,y,z,t in [0,1])
The polynomials are of the form: a + bx + cy + d*x*y = e + fz + gt + h*z*t (solving for x,y,z,t). All coefficients are unique.
The polynomial equations come from bilinear interpolations.
I've tried finding an exact analytic solution, but as others have posted, solving large systems of polynomials in Mathematica and otherwise is time consuming
I would simply use the general-purpose solver IPOPT, written in C++. You can feed it with the [0, 1] bound constraints, it actually helps IPOPT and makes the solution procedure faster.
Does the sparsity pattern of the system change? If not, then you can probably save an initialization step. I am not 100% sure though. Either way, IPOPT is blazing fast compared to the analytic solution in Mathematica.
You can take a look at the Numerical Recipes book (chap. 9 in the c version) that describes solutions of non-linear systems of equations. There is an online version viewable from their web site http://www.nr.com/.
As their licensing is very restrictive, probably you can look at the method and then adapt it using a library such as gsl. I did not try but this page http://na-inet.jp/na/gslsample/nonlinear_system.html gives an example on how to do that with gsl.
I have an integer linear optimisation problem and I'm interested in feasible, good solutions. As far as I know, for example the Gnu Linear Programming Kit only returns the optimal solution (given it exists).
This takes endless time and is not exactly what I'm looking for: I would be happy with any good solution, not only the optimal one.
So a LP-Solver that e.g. stops after some time and returns the best solution he found so far, would do the job.
Is there any such software? It would be great if that software was open source or at least free as in beer.
Alternatively: Is there any other way that usually speeds up Integer LP problems?
Is this the right place to ask?
Many solvers provide a time limit parameter; if you set the time limit parameter, they will stop once the time limit is reached. If an integer feasible solution has been found, it will return the best feasible solution found to that point.
As you may know, integer programming is NP-hard, and there is a real art to finding optimal solutions as well as good feasible solutions quickly. To compare the different solvers, see Prof. Hans Mittelmann's Benchmarks for Optimization Software. The MILP benchmarks - particularly MIPLIB2010 and the Feasibility Benchmark should be most relevant.
In addition to selecting a good solver, there are many things that can be done to improve solve times including tuning the parameters of the solver and model reformulation. Many people in research and industry - including myself - spend our careers working on improving the solve times of MIP models, both in general and for specific models.
If you are an academic user, note that the top commercial systems like CPLEX and Gurobi are free for academic use. See the respective websites for details.
Finally, you may want to look at OR-Exchange, a sister site to Stack Overflow that focuses on the field of operations research.
(Disclaimer: I currently work for Gurobi Optimization and formerly worked for ILOG, which provided CPLEX).
If you would like to get a feasibel integer solution fast and if you don't need the optimal solution, you can try
Increase the relative or absolute Gap. Usually solvers have small gaps of say 0.0001% for relative gap. This means that the solver will continue searching for MIP solutions until it the MIP solution is not farther than 0.0001% away from the optimal solution. Increase this gab to e.g. 1%., So you get good solution, but the solver will not spent a long time in proving optimality.
Try different values for solver parameters concerning MIP heuristics.
CPLEX and GUROBI have parameters to control, MIP emphasis. This means that the solver will put more emphasis on looking for feasible solutions or on proving optimality. Set emphasis to feasible MIP solutions.
Most solvers like CPLEX, Gurobi, MOPS or GLPK have settings for gap and heuristics. MIP emphasis can be set - as far as I know - only in CPLEX and Gurobi.
A usual approach for solving ILP is branch-and-bound. This utilized the solution of many sub-LP (without-I). The finally optimal result is the best of all sub-LP. As at least one solution is found you could stop anytime and would have a best-so-far.
One package that could do it, is the free lpsolve. Look there at set_timeout for giving a time limit, and when it is ILP the solve function can return in SUPOPTIMAL the best_so_far value.
As far as I know CPLEX can. It can return the solution pool which contains primal feasible solutions in the search, and if you specify the search focus on feasibility rather on optimality, more faesible solutions can be generated. At the end you can just export the pool. You can use the pool to do a hot start so it's pretty up to you. CPlex is free now at least in my country as you can sign up as a researcher.
Could you take into account Microsoft Solver Foundation? The only restriction is technology stack that you prefer and here you should use, as you guess, Microsoft technologies: C#, vb.net, etc. Here is example how to use it with Excel: http://channel9.msdn.com/posts/Modeling-with-Solver-Foundation-30 .
Regarding to your question it is possible to have not a fully optimized solutions if you set efficiency (for example 85% or 0.85). In outcome you can see all possible solutions for such restriction.
I need to solve a few mathematical equations in my application. Here's a typical example of such an equation:
a + b * c - d / e = a
Additional rules:
b % 10 = 0
b >= 0
b <= 100
Each number must be integer
...
I would like to get the possible solution sets for a, b, c, d and e.
Are there any libraries out there, either open source or commercial, which I can use to solve such an equation? If yes, what kind of result do they provide?
Solving linear systems can generally be solved using linear programming. I'd recommend taking a look at Boost uBLAS for starters - it has a simple triangular solver. Then you might checkout libraries targeting more domain specific approaches, perhaps QSopt.
You're venturing into the world of numerical analysis, and here be dragons. Seemingly small differences in specification can make a huge difference in what is the right approach.
I hesitate to make specific suggestions without a fairly precise description of the problem domain. It sounds superficiall like you are solving constrained linear problems that are simple enough that there are a lot of ways to do it but "..." could be a problem.
A good resource for general solvers etc. would be GAMS. Much of the software there may be a bit heavy weight for what you are asking.
You want a computer algebra system.
See https://stackoverflow.com/questions/160911/symbolic-math-lib, the answers to which are mostly as relevant to c++ as to c.
I know it is not your real question, but you can simplify the given equation to:
d = b * c * e with e != 0
Pretty sure Numerical Recipes will have something
You're looking for a computer algebra system, and that's not a trivial thing.
Lot's of them are available, though, try this list at Wikipedia:
http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
-Adam
This looks like linear programming. Does this list help?
In addition to the other posts. Your constraint sets make this reminiscent of an integer programming problem, so you might want to check that kind of thing out as well. Perhaps your problem can be (re-)stated as one.
You must know, however that the integer programming problems tends to be one of the harder computational problems so you might end up using many clock cycles to crack it.
Looking only at the "additional rules" part it does look like linear programming, in which case LINDO or a similar program implementing the simplex algorithm should be fine.
However, if the first equation is really typical it shows yours is NOT a linear algebra problem - no 2 variables multiplying or dividing each other should appear on a linear equation!
So I'd say you definitely need either a computer algebra system or solve the problem using a genetic algorithm.
Since you have restrictions similar to those found in linear programming though you're not quite there, if you just want a solution to your specific problem I'd say pick up any of the libraries mentioned at the end of Wikipedia's article on genetic algorithms and develop an app to give you the result. If you want a more generalist approach, then you've got to simulate algebraic manipulations on your computer, no other way around.
The TI-89 Calculator has a 'solver' application.
It was built to solve problems like the one in your example.
I know its not a library. But there are several TI-89 emulators out there.