I'd like to use an ILP solver (e.g. lp_solve) to find the solution for an optimization problem.
The challenge is that some constraints are too complex to be formalized as linear statements, but may be validated using a simulation framework.
So I'd need to run the solver, check the solution against my complex constraints, and in case they are not satisfied, continue with the second-best solution etc.
Is there a solver that does not only provide the optimal solution, but the best n solutions regarding the give objective function?
Yes, there is:
Either the solver supports it through a solution pool,
or you have to build this solution pool yourself through the callback functions provided by the API of the solver.
In the latter case, the corresponding callback function is called whenever a new, better solution is found than the best known solution up to that point. The second option is supported by all the notable solvers I know of.
More on this topic:
K Best Solutions
K Best Solutions in AMPL/CPLEX
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 got a job scheduling problem. We are given start time, time to
complete the order, deadline.
It is given that start time + time to
complete <= deadline.
I have also been given the loss that will occur if I am not able to
complete the job before the deadline. I have to design an algorithm to minimize the loss.
I have tried changing the standard algorithm of dynamic programming for maximizing the profit in job scheduling but to no success.
What algorithm can I use to solve the question?
Dynamic Programming isn't the right approach based on what you're aiming to optimize. You can find the optimized schedule by using a greedy approach.
Here's a thorough guide with sample code for your desire language (C++), in the guide it assumes each jobs takes only 1 unit of time, which you can easily modify by using time_to_complete instead.
Your problem is similar to the knapsack one. Using a greedy approach is convenient if you aren't actually looking for the best possible solution, but just a "good enough" one.
The big pro of the greedy approach is that the cost is rather lower than other "more thorough" approaches but, if you need the best solution to your problem, I would say that backtracking is the way to go.
Since the deadline can be violated, the problems looks like a Total Weighted Tardiness Scheduling Problem. There are many flavors of it, but most problems under this umbrella are computationally hard, therefore Dynamic Programming (DP) would not be my first choice. In my experience, DP also poses difficulties during modeling and implementation. Same comment for mathematical programming "as-is". Some approaches that can be implemented more quickly are:
constraint programming: very small learning curve, and there are many libraries out there, included very good open source ones (most have C++ API). Bonus: constraint programming can demonstrate optimality.
ad hoc heuristics: (1) start with a constructive algorithm (like the greedy approach suggested by Ling Zhong and Flavio Giobergia), then (2) use some local search approach to improve if and finally (3) embed the approach into a metaheuristic scheme. This way you can build on top of the previous step, and learn a lot about the problem. Note: in general, heuristics cannot demonstrate optimality
special mention: local solver, a hybrid approach between the two above: it lets you model the problem using a formalism similar to constraint programming and then it solves it using heuristics. It is very easy to learn, it usually lets you get started quickly and, in my tests, it provides remarkably good results.
What is the best online solution to solve a linear programming problem?
I heard about several like Gurobi.
One thing I especially want is the possibility to get an approximate solution when the exact resolution takes too long.
The most comprehensive online optimization system is NEOS. It takes models in a variety of input formats and has a wide range of solvers.
Many solvers have settings to allow them to terminate early, even before optimality is reached, if you want an approximate and quick solution. But often your best bet in that case is to use a heuristic algorithm designed specifically for your problem.
I have a set of problems (sets of equations and inequalities) for which I know that all variables have to be integers, and have finitely many solutions. I know that if I take any random objective function and let an lp or mip solver onto it, it finds a solution, however I want all solutions to the problem, and of course, as efficiently as possible. I don't really care about optimizing anything, but apparently most of the software that deals with it does. Is there any solver that can do that? If so, which one is the best/simplest one, or which one would you recommend? At best one that can be used as a C/C++ library.
There is a nice blog post by Paul Rubin on how to find K best solutions, which can be easily generalized to get all the solutions. As Ali suggested one of the approaches is to use a solution pool. Two other approaches are:
Use an incumbent callback to track and reject solutions.
Use an incumbent callback with solution injection.
See the blog post for details.
IBM ILOG CPLEX has a solution pool feature and it's free for academic purposes.
I guess you can probably get all solution if you set the maximum pool size sufficiently large. I don't know for sure, never tried.
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.