Testing cut generators for mixed-integer linear programming

Abstract

In this paper, a methodology for testing the accuracy and strength of cut generators for mixed-integer linear programming is presented. The procedure amounts to random diving towards a feasible solution, recording several kinds of failures. This allows for a ranking of the accuracy of the generators. Then, for generators deemed to have similar accuracy, statistical tests are performed to compare their relative strength. An application on eight Gomory cut generators and six Reduce-and-Split cut generators is given. The problem of constructing benchmark instances for which feasible solutions can be obtained is also addressed.

On the safety of Gomory cut generators

“…However, when using a finite precision state-of-the-art solver, we observed no significant difference in the safety of the GMI cuts when turning on or off the safe rounding scheme in the code of [12]. Specifically, out of 5,240 experiments using the method of [23] on 29 MIPLIB3 instances, we observed 1,773 overall failures with the safe generator of [12] (38 failures of Type 1, 148 failures of Type 2 and the remaining ones of Type 3, as discussed in the previous section) and 1,737 failures when safe rounding was turned off (19 failures of Type 1, 128 failures of Type 2 and the rest of Type 3). The differences are not statistically significant but, clearly, the safe rounding scheme has little impact when used with a finite precision solver.…”

“…However, practitioners in integer programming know that numerical problems are not uncommon, and if the choice of the cut generation parameters is not careful, failures of the three types listed above can happen [23]. To decrease the occurrence of failures, several cut safety-enhancing steps have been devised empirically.…”

“…A step in the direction of testing safety of cutting plane generators is taken in [23]. The paper proposes a methodology for comparing the strength of cut generators.…”

“…Safety is a fundamental issue in this context, since only cut generators with a similar safety can be compared on equal ground with respect to strength (otherwise, the comparison would favor "unsafe" but more aggressive cut generators). [23] provides experimental evidence that existing cut generators in COIN-OR Cgl [10] may run into numerical problems on seemingly innocuous instances.…”

“…This scheme is similar to the method RandomDives proposed in [23], in the sense that the work horse is a large number of random dives to be able to perform meaningful statistical tests. However, Dive-and-cut improves over RandomDives on the three criteria mentioned in Section 1 (reasonable use of the generator, large and relevant set of instances).…”