An implementation of exact knapsack separation

Abstract

Cutting planes have been used with great success for solving mixed integer programs. In recent decades, many contributions have led to successive improvements in branch-and-cut methods which incorporate cutting planes in branch and bound algorithm. Using advances that have taken place over the years on 0-1 knapsack problem, we investigate an efficient approach for 0-1 programs with knapsack constraints as local structure. Our approach is based on an efficient implementation of knapsack separation problem which consists of the four phases: preprocessing, row generation, controlling numerical errors and sequential lifting. This approach can be used independently to improve formulations with cutting planes generated or incorporated in branch and cut to solve a problem. We show that this approach allows us to efficiently solve large-scale instances of generalized assignment problem, multilevel generalized assignment problem, capacitated p-median problem and capacitated network location problem to optimality.Keywords Knapsack problem · Cutting plane · Exact separation · Generalized assignment problem · Multilevel generalized assignment problem · Capacitated p-median problem · Capacitated network location problem B Igor Vasilyev

A scalable exact algorithm for the vertex p-center problem

“…The idea behind inequalities (24) is reminiscent of up-lifting and down-lifting in cutting plane theory. For instance, in a binary integer problem, a given inequality can be strengthened by considering the linear relaxation solution at a certain node of a branchand-bound tree and taking into account the previously performed branches to 0 or 1 (see, e.g., Kaparis andLetchford 2008, andVasilyev et al 2016).…”

An exact separation algorithm for unsplittable flow capacitated network design arc-set polyhedron

“…In order to test the effectiveness of the exact separation algorithm for solving the unsplittable capacitated network design problem, we implement it in C++ linked with IBM ILOG CPLEX optimizer 12.7.1 [16] library. Following [31], to avoid changing the problem structure, the presolving features are turned off in our experiments. Moreover, the dual simplex method is used to reoptimize the linear programming problem (27) after adding cutting planes.…”

“…To begin with, we note that since the number of constraints in problem ( 12) may be exponential, from a computational perspective, it is impractical to solve problem (12) when all the constraints are expressed explicitly. For this reason, we follow [31] to solve problem ( 12) by decomposing into the following four steps.…”

“…Vasilyev et al [31] suggested to use the enumeration of the multiplier θ from 2 to 10 4 and checked the integrality within the tolerance 10 −5 . To further avoid too much computational efforts, here we use the approach described in [1].…”

“…It is worth noting that the considered exact separation for the unsplittable flow arc-set polyhedron can be seen as an extension of the exact separation for the 0-1 knapsack polytope; see [5,6,9,10,11,12,20,31] and the references therein. The difference is that the exact separation for the 0-1 knapsack polytope cannot handle non-binary integer variables, whereas the approach in this paper takes the non-binary integer variables into consideration such that it can be customized to solve the unsplittable capacitated network design problem.…”