Tuesday, June 19, 2018

Callback Cuts That Repeat

The following post is specific to the CPLEX integer programming solver. I have no idea whether it applies to other solvers, or even which other solver have cut callbacks.

Every so often, a user will discover that a callback routine they wrote has "rediscovered" a cut it previously generated. This can be a bit concerning at first, for a few reasons: it seems implausible, and therefore raises concerns of a bug someplace; it represents repeated work, and thus wasted effort; and it suggests at least the possibility of getting stuck in a loop, the programmatic equivalent of "Groundhog Day". (Sorry, couldn't resist.) As it happens, though, repeating the same cut can legitimately happen (though hopefully not too often).

First, I should probably define what I mean by callbacks here (although I'm tempted to assume that if you don't know what I mean, you've already stopped reading). If you want to get your geek on, feel free to wade through the Wikipedia explanation of a callback function. In the context of CPLEX, what I'm referring to is a user-written function that CPLEX will call at certain points in the search process. I will focus on functions called when CPLEX is generating cuts to tighten the bound at a given node of the search tree (a "user cut callback" in their terminology) or when CPLEX thinks it has identified an integer-feasible solution better than the current incumbent (a "lazy constraint callback"). That terminology pertains to versions of CPLEX prior to 12.8, when those would be two separate (now "legacy") callbacks. As of CPLEX version 12.8, there is a single ("generic") type of callback, but generic callbacks continue to be called from multiple "contexts", including those two (solving a node LP, checking a new candidate solution).

The purpose here is to let the user either generate a bound-tightening constraint ("user cut") using particular knowledge of the problem structure, or to vet a candidate solution and, if unsuitable, issue a new constraint ("lazy constraint") that cuts off that solution, again based on problem information not part of the original IP model. The latter is particularly common with decomposition techniques such as Benders decomposition.

So why would the same user cut or lazy constraint be generated more than once (other than due to a bug)? There are at least two, and possibly three, explanations.

Local versus Global

A user cut can be either local or global. The difference is whether the cut is valid in the original model ("global") or whether it is contingent on branching decisions that led to the current node ("local"). The user can specify in the callback whether a new cut is global or local. If the cut is specified as local, it will be enforced only at descendants of the current node.

That said, it is possible that a local cut might be valid at more than one node of the search tree, in which case it might be rediscovered when those other nodes are visited. In the worst case, if the cut is actually globally valid but for some reason added with the local flag set, it may be rediscovered quite a few times.

Parallel Threading

On a system with multiple cores (or multiple processors), using parallel threads can speed CPLEX up. Parallel threads, though, are probably the most common cause for cuts and lazy constraints repeating.

The issue is that threads operate somewhat independently and only synchronize periodically. So suppose that thread A triggers a callback that generates a globally valid user cut or lazy constraint, which is immediately added to the problem A is solving. Thread B, which is working on a somewhat different problem (from a different part of the search tree), is unaware of the new cut/constraint until it reaches a synchronization point (and finds out what A and any other sibling threads have been up to). Prior to that, B might stumble on the same cut/constraint. Since A and B are calling the same user callback function, if the user is tracking what has been generated inside that function, the function will register a repetition. This is normal (and harmless).

Cut Tables

This last explanation is one I am not entirely sure about. When cuts and lazy constraints are added, CPLEX stores them internally in some sort of table. I believe that it is possible in some circumstances for a callback function to be called before the table is (fully) checked, in which case the callback might generate a cut or constraint that is already in the table. Since this deals with the internal workings of CPLEX (the secret sauce), I don't know first-hand if this is true or not ... but if it is, it is again slightly wasteful of time but generally harmless.

User Error

Of course, none of this precludes the possibility of a bug in the user's code. If, for example, the user reacts to a candidate solution with a lazy constraint that is intended to cut off that solution but does not (due to incorrect formulation or construction), CPLEX will register the user constraint, notice that the solution is still apparently valid, and give the callback another crack at it. (At least that is how it worked with legacy callbacks, and I think that is how it works with generics.) Seeing the same solution, the user callback might generate the same (incorrect) lazy constraints, and off the code goes, chasing its own tail.

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