I have a large MIP problem, and I use GLPSOL in GLPK to solve it. However, solving the LP relaxation problem takes many iterations, and each iteration the obj and infeas value are all the same. I think it has found the optimal solution, but it won't stop and has continued to run for many hours. Will this happen for every large-scale MIP/LP problem? How can I deal with such cases? Can anyone give me any suggestions about this? Thanks!
The problem of solving MIPs is NP-complete in general, which means that there are instances which can't be solved efficiently. But often our problems have enough structure, so that heuristics can help to solve these models. This allowed huge gains in solving-capabilities in the last decades (overview).
For understanding the basic-approach and understanding what exactly is the problem in your case (no progress in upper-bound, no progress in lower-bound, ...), read Practical Guidelines for Solving Difficult Mixed Integer Linear
Programs.
Keep in mind, that there are huge gaps between commercial solvers like Gurobi / Cplex and non-commercial ones in general (especially in MIP-solving). There is a huge amount of benchmarks here.
There are also a lot of parameters to tune. Gurobi for example has different parameter-templates: one targets fast findings of feasible solution; one targets to proof the bounds.
My personal opinion: compared to cbc (open-source) & scip (open-source but non-free for commercial usage), glpk is quite bad.
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I have a large MILP that I build with cvxpy and want to solve with GUROBI. When I give use the solve() function of cvxpy it take a really really really long time to setup and does not start solving for hours. Whilest doing that only 1 core of my cluster is being used. It is used for 100%. I would like to use multiple cores to build the model so that the process of building the model does not take so long. Running grbprobe also shows that gurobi knows about the other cores and for solving the problem it uses multiple cores.
I have tried to run with different flags i.e. turning presolve off and on or giving the number of Threads to be used (this seemed like i didn't even for the solving.
I also have reduce the number of constraints in the problem and it start solving much faster which means that this is definitively not a problem of the model itself.
The problem in it's normal state should have 2200 constraints i reduce it to 150 and it took a couple of seconds until it started to search for a solution.
The problem is that I don't see anything since it takes so long to get the ""set username parameters"" flag and I don't get any information on what the computer does in the mean time.
Is there a way to tell GUROBI or CVXPY that it can take more cpus for the build-up?
Is there another way to solve this problem?
Sorry. The first part of the solve (cvxpy model generation, setup, presolving, scaling, solving the root, preprocessing) is almost completely serial. The parallel part is when it really starts working on the branch-and-bound tree. For many problems, the parallel part is by far the most expensive, but not for all.
This is not only the case for Gurobi. Other high-end solvers have the same behavior.
There are options to do less presolving and preprocessing. That may get you earlier in the B&B. However, usually, it is better not to touch these options.
Running things with verbose=True may give you more information. If you have more detailed questions, you may want to share the log.
Which of the following optimisation methods can't be done in an optimisation software such as CPLEX? Why not?
Dynamic programming
Integer programming
Combinatorial optimisation
Nonlinear programming
Graph theory
Precedence diagram method
Simulation
Queueing theory
Can anyone point me in the right direction? I didn't find too much information regarding the limitations of CPLEX on the IBM website.
Thank you!
That's kind-of a big shopping list, and most of the things on it are not optimisation methods.
For sure CPLEX does integer programming, non-linear programming (just quadratic, SOCP, and similar but not general non-linear) and combinatoric optimisation out of the box.
It is usually possible to re-cast things like DP as MILP models, but will obviously require a bit of work. Lots of MILP models are also based on graphs, so yes it is certainly possible to solve a lot of graph problems using a MILP solver such as CPLEX.
Looking wider at topics like simulation, then that is quite a different approach. Simulation really is NOT an optimisation method, but it can be used alongside optimisation to get extra insights which may be useful in a business context. Might be used for example to discover some empirical relationships that could be used in an optimisation model by CPLEX.
The same can probably also be said for things like queuing theory, precedence, etc. Basically, use CPLEX as an optimisation tool to solve part or all of your problem once you have structured and analysed it via one of these other approaches.
Hope that helps.
To optimize a production system by planning ~1000 timesteps ahead I try to solve an optimization problem with around 20000 dimensions containing binary and continuous variables and several complex constraints.
I know the provided information is little, but can someone give a hint which approach would be suitable for such big problems? Would you recommend some metaheuristic or a commercial solver?
I have an optimization problem that is subjected to linear constraints.
How to know which method is better for modelling and solving the problem.
I am generally asking about solving a problem as a satisfiability problem (SAT or SMT) vs. Solving as a linear programming problem (ILP OR MILP).
I don't have much knowledge in both. So, please simplify your answer if you have any.
Generally speaking, the difference is that SAT is only trying for feasible solutions, while ILP is trying to optimize something subject to constraints. I believe some ILP solvers actually use SAT solvers to get an initial feasible solution. The sensor array problem you describe in a comment is formulated as an ILP: "minimize this subject to that." A SAT version of that would instead pick a maximum acceptable number of sensors and use that as a constraint. Now, this is a satisfiability problem, but not one that's easily expressed in conjunctive normal form. I'd recommend using a solver with a theory of integers. My favorite is Z3.
However, before you give up on optimizing, you should try GMPL / GLPK. You might be surprised by how tractable your problem is. If you're not so lucky, turn it into a satisfiability problem and bring out Z3.
I know that column generation gives an optimal solution and it can be used with other heuristics. But does that make it an exact algorithm? Thanks in advance.
Traditional CG operates on the relaxed problem. Although it finds the optimal LP solution, this may not translate directly into an optimal MIP solution. For some problems (e.g. 1d cutting stock) there is evidence this gap is small, and we just apply the set of columns found for the relaxed problem to a final MIP knowing this is a good solution but necessarily optimal. So it is a heuristic.
With some effort you can use column generation inside a branch-and-bound algorithm (this is called branch-and-price). This gives proven optimal solutions.
An exact algorithm means that the algorithm can solve the optimization problem globally i.e it has given the global optima.
Column generation technique is conventionally applied to relaxed LP problem and tries to optimize the relaxed LP problem by constantly improving the current solution with the help of dual multipliers. It gives an exact LP solution for the relaxed LP problem. But sometimes in real-world problems, the exact solution of the relaxed Lp problem is not feasible to use, it needs to be translated to an integer solution in order to use it. Now if the problem scale is small, then there are many exact MIP algorithms (such as Branch and Bound) which can solve it exactly and give an integer solution. But if the problem is large-scale, even the exact MIP algorithms can take longer runtimes, hence, we use some special/intelligent heuristics to lower the difficulty of the MIP problem.
Summary: Column generation is an exact technique for solving the relaxed LP problem, not the original IP problem.
First, strictly speaking, all algorithms are heuristic, including Simplex Method.
Second, I think Column generation is a heuristic algorithm, because it solves the LP relaxation of the master problem. It does not guarantee IP optimal. Actually CG does not always converge very well.