Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Abstract

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes