Automatically solving generic mixed integer linear programs by a Lagrangian metaheuristic

To solve many real world optimisation problems, mathematical programs are required to model the problem and then subsequently find the optimal solution. For many problems, the mathematical programs used are known as Mixed Integer Linear Programming (MIP) problems. Certain problem types, such as the Generalised Assignment Problem or the Maximum Edge Disjoint Path problem have decomposable structures, which can be exploited and solved by specialised decomposition based techniques such as Lagrangian Relaxation. As the Lagrangian Relaxation algorithm relies on constraint relaxation, solutions obtained are not guaranteed to be feasible, although they do provide dual bounds. Often, a repair heuristic is applied to obtain feasible solutions, forming what is known as a Lagrangian Heuristic. When a metaheuristic technique is used, this is also referred to as a Lagrangian Metaheuristic. This thesis presents a new framework called Generic Decompositions, or G-Dec for short, that is capable of automatically creating decomposition structures for generic MIPs that can then be solved by a Lagrangian Metaheuristic. G-Dec incorporates both metaheuristic and Machine Learning techniques to produce high quality decompositions. G-Dec relies on a Machine Learning model to predict the quality of candidate decompositions found during the search, as solving candidate decompositions by a Lagrangian Relaxation approach is computationally infeasible. The Machine Learning model used by G-Dec was developed and tested through a comprehensive analysis involving a significantly large dataset of decompositions, created in part by a specialised Non-Dominated Sorting Genetic Algorithm II based sampling algorithm developed in this thesis. The bulk of the G-Dec algorithm is based on a customised population Iterated Local metaheuristic, designed specifically to be computationally fast and effective for large scale problems. In addition, a Lagrangian Metaheuristic that hybridises Lagrangian Relaxation and Particle Swarm Optimisation, known as LaPSO is also explored in this thesis. Through the creation of a novel repair heuristic, known as Largest Violation Perturbation, we demonstrate the potential of a Lagrangian Metaheuristic by solving the Maximum Edge Disjoint Path problem.