SCIP and Branch and Price

Question

I have a general question about SCIP. I need to use the SCIP as a Branch and Price framework for my problem, I code in c++ so I used the VRP example as a template. On some of the instances, the code stops at the fractional solution and returns that as a optimal solution, I think something is wrong, do I have to set some parameters in order to tell SCIP look for integer solution or I made a mistake, I believe it should not stop and instead branch on the fractional solution until it reaches the integer solution (without any other negative reduced cost column). I also solve the subproblem optimally! any commenets?!

Answer 1

If you define your variables to be continous and just add a pricer, SCIP will solve the master problem to optimality (ie, solve the restricted master, add improving columns, solve the updated restricted master, and so on, until no more improving columns were found).

There is no reason for SCIP to check if the solution is integral, because you explicitly said that you don't mind whether the values of the variables are integral or not (by defining them to be continuous). On the other hand, if you define the variables to be of integral (or binary) type, SCIP will do exactly as I described before, but at the end check whether all integral variables have an integral value and branch if this is not the case.

However, you should note that all branching rules in SCIP do branching on variables, ie, they take an integer variable with fractional value and split its domain; a binary variable would be fixed to 0 and 1 in the two child nodes. This is typically a bad idea for branch-and-price: first of all, it's quite unbalanced. You have a huge number of variables out of which only few will have value 1 in the end, most will be 0. Fixing a variable to 1 therefore has a high impact, while fixing it to 0 has almost no impact. But more importantly, you need to take the branching decision into account in your pricing problem. If you fixed a variable to 0, you have to keep the pricer from generating a copy of the forbidden column (which would probably improve the LP solution, because it was part of the former optimal solution). In order to to this, you might need to look for the 2nd (or later k)-best solution. Since you are solving the pricing problems as a MIP with SCIP, you might just add a constraint forbidding this solution (logicor (linear) for binary variables or bounddisjunction (not linear) for general integer variables).

I would recommend to implement your own branching rule, which takes into account that you are doing branch-and-price and branches in a way that is more balanced and does not harm your pricing too much. For an example, check out the Ryan&Foster branching rule, which is the standard for binary problems with a set-partitioning master structure. This rule is implemented in Binpacking as well as the Coloring example shipped with SCIP.

Please also check out the SCIP FAQ, where there is a whole section about branch-and-price which also covers the topic branching (in particular, how branching decisions can be stored and enforced by a constraint handler, which is something you need to do for Ryan&Foster branching): http://scip.zib.de/doc/html/FAQ.php

There were also a lot of questions about branch-and-price on the SCIP mailing list http://listserv.zib.de/mailman/listinfo/scip/ . If you want to search it, you can use google and search for "site:listserv.zib.de scip search-string"

Finally, I would like to recommend to have a look at the GCG project: http://www.or.rwth-aachen.de/gcg/ It is an extension of SCIP to a generic branch-cut-and-price solver, ie, you do not need to implement anything, you just put in an original formulation of your model, which is then reformulated by a Dantzig-Wolfe decomposition and solved via branch-cut-and-price. You can supply the structure for the reformulation, pricing problems are solved as a MIP (as you do it also), and there are also different branching rules. GCG is also part of the SCIP optimization suite and can be easily built within the suite.