Achieving MILP feasibility quickly using general disjunctions

Abstract

Branch and bound algorithms for Mixed-Integer Linear Programming (MILP) almost universally branch on a single variable to create disjunctions. General linear expressions involving multiple variables are another option for branching disjunctions, but are not used for two main reasons: (i) descendent LPs tend to solve more slowly because of the added constraints, so the overall solution time is increased, and (ii) it is difficult to quickly find an effective general disjunction. We study the use of general disjunctions to reach the first MILP-feasible solution quickly, showing for the first time that general disjunctions can provide speed improvements for hard MILP models. The speed-up is due to new and efficient ways to (i) trigger the inclusion of a general disjunction only when it is likely to be beneficial, and (ii) construct effective general disjunctions very quickly. Our empirical results show performance improvements versus a state of the art commercial MILP solver.

Improved branching disjunctions for branch-and-bound: An analytic center approach

“…Recent work by Mahmoud and Chinneck (2013) showed that general branching disjunctions can help reach feasible solutions quickly without sacrificing computational efficiency. This is achieved by heuristically constructing general disjunctions fast, and including them only if they are likely to help.…”

On the Complexity of Finding Shortest Variable Disjunction Branch-and-Bound Proofs

“…Branching on general disjunctions lies at the core of Lenstra's algorithm for integer programming in fixed dimension [20]. It has also been used, for example, for special ordered sets [3], exploiting flatness [9], achieving feasibility [21], and symmetry handling [22].…”

Compressing Branch-and-Bound Trees

“…, e n }, in which case we say the algorithm uses variable disjunctions. However, many results explore the benefit of additional directions: various subsets of {−1, 0, 1} n are explored in [24,26,29]; directions derived from mixed integer Gomory cuts are explored in [8,18]; directions derived using basis reduction techniques are explored in [1,25]; Mahajan and Ralphs [22] solve a subproblem to find a disjunction that closes the duality gap by a certain amount. The largest set of directions is the set Z n , in which case the algorithm uses general disjunctions.…”