## Friday, December 2, 2016

### Support for Benders Decomposition in CPLEX

As of version 12.7, CPLEX now has built-in support for Benders decomposition. For details on that (and other changes to CPLEX), I suggest you look at this post on J-F Puget's blog and Xavier Nodet's related slide show. [Update 12/7/16: There is additional information about the Benders support in a presentation by IBM's Andrea Tramontani at the 2016 INFORMS national meeting, "

I've previously posted Java code for a simple example of Benders decomposition (a fixed-cost warehousing/distribution problem). To get a feel for the new features related to Benders, I rewrote that example. The code for the revised example is in this Git repository, and is free for use under a Creative Commons 3.0 license. There is also an issue tracker, if you bump into any bugs.

Going through the code here would be pretty boring, but there are a few bits and pieces I think are worth explaining, and I'll show some far from definitive timing results.

### Modeling Approaches

I have five modeling approaches in the code.
• STANDARD: This is just a single MIP model, using no decomposition. It's present partly for benchmark purposes and partly because a few of the newer approaches build on it.
• MANUAL: This is essentially a reprise of my earlier code. I write two separate models, a master problem (MIP) and a subproblem (LP), just as one would do prior to CPLEX 12.7.
• ANNOTATED: This is the first of three methods exploiting the new features of CPLEX 12.7 I create a single model (by duplicating the STANDARD code) and then annotate it (using the annotatiion method added to CPLEX 12.7) to tell CPLEX how to split it into a master problem and a single subproblem. The annotation method would also let me create multiple disjoint subproblems, but the example we are using only needs one subproblem.
• WORKERS: This is the ANNOTATED method again, but with a parameter setting giving CPLEX the option to split the single subproblem into two or more subproblems if it sees a structure in the subproblem that suggests partitioning is possible.
• AUTOMATIC: Here I create a single model (identical to the STANDARD method) and, via a parameter setting, let CPLEX decide how to split it into a master problem and one or more subproblems.

### Benders Strategy Parameter

The key to all this is a new parameter, whose Java name is IloCplex.Benders.Strategy. It is integer valued, but for some reason is not part of the IloCplex.IntParam class hierarchy (which threw me off at first when I tried to use it in my code). The values defined for it are:
• -1 = OFF: Use standard branch and cut, doing no Benders decomposition (even if annotations are present). Note that this will not stop manually coded Benders from working (as in my MANUAL method); it just stops CPLEX for creating a decomposition. This is the default value.
• 0 = AUTO: If no annotations are present, do standard branch and cut (akin to the OFF value). If the user supplies annotations, set up a Benders decomposition using those annotations, but partition the subproblems further if possible (akin to the WORKERS behavior below). If the user supplies incorrect annotations, throw an exception. This is the default value.
• 1 = USER: Decompose the problem, adhering strictly to the user's annotations (with no additional decomposition of subproblems). If the user fails to annotate the model, or annotates it incorrectly, throw an exception.
• 2 = WORKERS: This is similar to USER, but gives CPLEX permission to partition subproblems into smaller subproblems if it identifies a way to do so.
• 3 = FULL: CPLEX ignores any user annotations and attempts to decompose the model. If either all the variables are integer (no LP subproblem) or none of them are (no MIP master problem), CPLEX throws an exception.
There are also two other Benders-related parameters, in Java IloCplex.Param.Benders.Tolerances.feasibilitycut and IloCplex.Param.Benders.Tolerances.optimalitycut, that set tolerances for feasibility and optimality cuts. (In my code I leave those at default values. If it ain't broke, don't fix it.)

#### Syntax

Coding the strategy parameter looks like the following.
IloCplex cplex;      // declare a model object
// ... build the model ...
int strategy = ...;  // pick the strategy value to use
cplex.setParam(IloCplex.Param.Benders.Strategy, strategy);


### Annotations

Annotations can be specified in a separate file (if you are reading in a model) or added in code (which is what I do in my demo program). IBM apparently intends annotations to be used more generally than just for Benders decomposition. To use them for Benders, what you do is annotate each model variable with index number of the problem into which it should be placed (where problem 0 is the master problem and problems 1, 2, ... are subproblems). Only variables are annotated, not constraints or objective functions. If you fail to annotate a variable, it is given a default annotation (discussed further below). If you assign a negative integer as an annotation, the universe will implode spectacularly, and CPLEX with throw an exception just before it does.

You start by creating a single MIP model, as if you were not going to use Benders decomposition. Assuming again that your model exists in a variable named cplex, you begin the decomposition process with a line like the following:
IloCplex.LongAnnotation benders =
cplex.newLongAnnotation("cpxBendersPartition");
The name "benders" for the variable in which the annotation is stored is arbitrary, but the name cpxBendersPartition for the annotation must be used verbatim. This version of the annotation constructor sets the default value to 0, so that variables lacking annotations are assumed to belong to the master problem. An alternate version of newLongAnnotation uses a second argument to set the default value.

Next, you annotate each variable with the index of the problem into which it should be placed. Suppose that we have two variables defined as follows.
IloIntVar x = cplex.boolVar("Use1");
// open warehouse 1?
IloNumVar y = cplex.numVar("Ship12")
// amount shipped from warehouse 1 to customer 2

Assuming that we want x to belong to the master problem (index 0) and y to belong to the first and only subproblem (index 1), we would add the following code:
cplex.setAnnotation(benders, x, 0);
cplex.setAnnotation(benders, y, 1);
where benders is the variable holding our annotation.

There are versions of setAnnotation that take vector arguments, so that you can annotate a vector of variables in a single call. If the model in question has a fairly small number of integer variables and a rather large number of continuous variables (which is pretty common), you might want to use the two-argument version of newLongAnnotation to set the default value at 1, and then annotate only the integer variables, letting the continuous variables take the default annotation (i.e., be assigned to subproblem 1).

That's all there is to creating a Benders decomposition from a standard MIP model. Note, in particular, that you do not need to create callbacks. CPLEX handles all that internally. You can still add things like lazy constraint callbacks if you have a reason to do so; but for a "typical" Benders decomposition (dare I use that phrase?), you just need to create a single MIP model, annotate it, and set the Benders strategy parameter. What could be easier than that? Well, actually, one thing. If you set the Benders strategy to FULL, you don't even have to annotate the model! You can let CPLEX figure out the decomposition on its own.

### Performance

Test runs of a single model (especially one as simple as the fixed-charge warehouse problem) on a single computer (especially one with a measly four cores) written by a single programmer (who happens not to be terribly good at it) don't prove anything. I'll leave it to someone else to do serious benchmarking. That said, I was curious to see if there were any gross differences in performance, and I also wanted to see if all methods led to correct answers. (I'm not what you would call a trusting soul.) So I ran 25 replications of each method, using a different problem instance (and a different random seed for CPLEX) each time. In an attempt to level the playing field, I forced Java to collect garbage after each model was solved (and timing stopped), to minimize the impact of garbage collection on run times. All problem instances were the same size: 50 warehouses serving 4,000 customers. The basic model has these dimensions: 4,050 rows; 200,050 columns (of which 50 are binary variables, the rest continuous); and 400,050 nonzero constraint coefficients.

The first plot below shows the wall-clock time spent setting up (and, where relevant, decomposing) each model.
None of the methods strikes me as being overly slow, ignoring a couple of outliers. Using annotations (the "Annotated" and "Workers" methods) does seem to impose a nontrivial burden on model construction.

The second plot shows solution times. For each problem instance, all five methods achieved identical objective values (and proved optimality), and all used the same number of warehouses. (I did not check whether the values of the flow variables were identical.) So any concerns I might have had about validity were assuaged.
I'm not sure how much to read into this, but "manual" decomposition (the old fashioned approach, using explicit callbacks) seems to have the highest variance in solution time. The three approaches using new features of CPLEX 12.7 ("Annotated", "Automatic" and "Workers") had very similar run times. I'm fairly certain that the "Workers" method, wherein CPLEX tries to further partition the single subproblem I specified, wound up sticking with a single subproblem (due to the structure of the model).

### Which Is Better?

Assume that you have a problem that is amenable to "standard" Benders decomposition (as opposed to some of the funky variants, such as combinatorial Benders, Benders with MIP subproblems, Benders with subproblems that are not necessarily optimization problems, ...). The easiest approach from the user perspective is clearly the automatic option (Benders strategy = FULL), in which CPLEX literally does all the work. Runner up is the annotation approach, which is both much easier and much less prone to coding errors than the "manual" approach (defining separate problems and writing a lazy constraint callback).

On the performance side, things are a bit less clear. If you dig through Xavier Nodet's slides, you'll see that CPLEX use somewhat sophisticated techniques, based on recent research, to generate Benders cuts. I suspect that, on the test problem, their cuts would be a bit stronger than the ones I generate in the "manual" approach, which may account for some of the difference (in their favor) in median run times seen in the second plot. Also, since they can access the subproblems and add cuts to the master directly, rather than having to go through callbacks, using the annotation feature may result in a bit less "drag".

With other cases, I suspect that if you have a particular insight into the model structure that lets you generate problem-specific Benders cuts that are tighter than the generic cuts, you may do better sticking to the manual approach. Fortunately, since you can turn on automatic Benders decomposition just by tweaking a single parameter, it should be easy to tell whether your cuts really do improve performance.