Integral Simplex Methods for the Set Partitioning Problem: Globalisation and Anti-Cycling

Abstract

The set partitioning problem is a generic optimisation model with many applications, especially within scheduling and routing. It is common in the context of column generation, and its importance has grown due to the strong developments in this field. The set partitioning problem has the quasi-integrality property, which means that every edge of the convex hull of the integer feasible solutions is also an edge of the polytope of the linear programming relaxation. This property enables, in principle, the use of solution methods that find improved integer solutions through simplex pivots that preserve integrality; pivoting rules with this effect can be designed in a few different ways. Although seemingly promising, the application of these approaches involves inherent challenges. Firstly, they can get be trapped at local optima, with respect to the pivoting options available, so that global optimality can be guaranteed only by resorting to an enumeration principle. Secondly, set partitioning problems are typically massively degenerate and a big hurdle to overcome is therefore to establish anti-cycling rules for the pivoting options available. The purpose of this chapter is to lay a foundation for research on these topics.