Lagrangian Relax-and-Cut Algorithms

Attempts to allow exponentially many inequalities to be candidates to Lagrangian dualization date from the early 1980s. In the literature, the term Relax-and-Cut is being used to denote the whole class of Lagrangian Relaxation algorithms where Lagrangian bounds are attempted to be improved by dynamically strengthening relaxations with the introduction of valid constraints (possibly selected from families with exponentially many constraints). In this chapter, Relax-and-Cut algorithms are reviewed in their two flavors. Additionally, a general framework to obtain feasible integral solutions (that benefit from Lagrangian bounds) is also presented. Finally, the use of Relax-and-Cut is demonstrated through an application to a hard-to-solve instance of the Knapsack Problem. For that application, Gomory cuts are used, for the first time, within a Lagrangian relaxation framework.