Conflict-Driven Heuristics for Mixed Integer Programming

Abstract

Two essential ingredients of modern mixed-integer programming solvers are diving heuristics, which simulate a partial depth-first search in a branch-and-bound tree, and conflict analysis, which learns valid constraints from infeasible subproblems. So far, these techniques have mostly been studied independently: primal heuristics for finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. In this paper, we pose the question of whether and how the orthogonal goals of proving infeasibility and generating improving solutions can be pursued in a combined manner such that a state-of-the-art solver can benefit. To do so, we integrate both concepts in two different ways. First, we develop a diving heuristic that simultaneously targets the generation of valid conflict constraints from the Farkas dual and the generation of improving solutions. We show that, in the primal, this is equivalent to the optimistic strategy of diving toward the best bound with respect to the objective function. Second, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classic coefficient diving. In a detailed computational study, both methods are evaluated on the basis of an implementation in the source-open-solver SCIP. The experimental results underline the potential of combining both diving heuristics and conflict analysis. Summary of Contribution. This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed-integer linear programs. It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be combined successfully with diving heuristics that aim at finding primal solutions. For two newly designed diving heuristics, this paper features a thoroughly computational study regarding their impact on the overall performance of a state-of-the-art MIP solver.

Learning To Dive In Branch And Bound

“…We compared L2Dive against all other standard diving heuristics that are implemented in the open-source solver SCIP and do not require an incumbent solution. This includes coefficient, fractional, linesearch, pseudocost, distributional, vectorlength [6] and Farkas diving [47]. We briefly describe these baseline divers in Appendix A.…”

“…These are methods designed to quickly find good feasible solutions or an optimal solution x † ∈ X := {x ∈ P † | c x = z † }. Primal heuristics include problem-specific methods [e.g., 33,29], variants of large neighborhood search [15,13,7,37], rounding procedures [46,4] or diving heuristics [see e.g., 6,47].…”

Implications, conflicts, and reductions for Steiner trees

“…Any group [10 k , 86,400] contains each instance from Table 3 such that [37] or SCIP-Jack solves this instance in not less than 10 k , and at most 86,400 s. If an instance can be solved by only one solver within the time-limit, we consider the run-time of the other solver on this instance as 86,400 s. Such groupings are commonly used in computational mathematical optimization (also with the time lower bounds being powers of 10), see e.g. [28,49]. In addition to the shifted geometric mean, Table 4 also provides the arithmetic mean of the run-time for each group.…”

A Status Report on Conflict Analysis in Mixed Integer Nonlinear Programming

“…We plan to implement NLPbased conflict analysis into the academic constraint integer programming solver SCIP and to study its impact on solver behavior. As in the MIP case, infeasibility information might be used in several other contexts, consider hybrid branching [3], conflict-driven diving heuristics [54], and also rapid learning [7,9].…”