Decomposition in Integer Linear Programming

Abstract

Both cutting plane methods and traditional decomposition methods are procedures that compute a bound on the optimal value of an integer linear program (ILP) by constructing an approximation to the convex hull of feasible solutions. This approximation is obtained by intersecting the polyhedron associated with the continuous relaxation, which has an explicit representation, with an implicitly defined polyhedron having a description of exponential size. In this paper, we first review these classical procedures and then introduce a new class of bounding methods called integrated decomposition methods, in which the bound yielded by traditional approaches is potentially improved by introducing a second implicitly defined polyhedron. We also discuss the concept of structured separation, which is related to the well-known template paradigm for dynamically generating valid inequalities and is central to our algorithmic framework. Finally, we briefly introduce a software framework for implementing the methods discussed in the paper and illustrate the concepts through the presentation of applications.

Data dependent input control for origin–destination demand estimation using observability analysis

“…We shall note that even though mixed-integer linear programming models appear widely in many network problems, solving these problems at large scale still present theoretical and computational challenges. Without deviating too much from the focus of this paper, we refer interested readers to a few popular MIP solvers such as CPLEX (2009) and Gurobi (2014) and review articles about decomposition methods for mixed-integer linear programs in Bixby and Rothberg (2007) and Galati (2010). Travel demand estimate vector t is obtained simultaneously with solving MIP for z, only if the optimal objective value is zero, i.e., the returned z indicates a feasible critical set.…”

Investment in electricity networks with transmission switching

“…In order to solve large instances, the Dantzig-Wolfe reformulation described above was implemented in a branch-and-price framework using the DIP software framework [17]. DIP (Decomposition for Integer Programming) is a general open source framework developed under COIN-OR for solving discrete optimization problems using various decomposition algorithms.…”

MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library

“…For this reason, several decomposition-based solvers have been developed that accept auxiliary input files indicating this structure when it is present. These include GCG [21], DIP [17][18][19], and DECOMP (a decomposition-based solver that is part of SAS/OR and SAS Optimization [44]). All three can exploit the identified structure by reformulating the instance and applying a branch-and-price algorithm to solve it.…”