Decomposition and Duality: New Approaches to Integer and Stochastic Integer Programming

整数规划由于其丰富的建模能力，在工业决策和规划中得到了广泛的应用。然而，它们的解算法不像凸优化那样成熟，在凸优化中，强对偶理论无处不在。不确定条件下凸优化的有效方法不适用于高度非凸的整数情况。此外，整数模型通常假设数据是确定的，而在现实世界中通常不是这样。该项目将开发新的理论和算法，以加强对整数模型的分析，包括那些包含不确定性的模型，同时也使并行计算范式的使用成为可能。

Because of their rich modelling capabilities, integer programs are widely used in industry for decision making and planning. However their solution algorithms do not have the maturity of their cousins in convex optimisation, where the theory of strong duality is ubiquitous. Efficient methods for convex optimisation under uncertainty do not apply to the integer case, which is highly non-convex. Furthermore, integer models usually assume the data is known with certainty, which is often not the case in the real world. This project will develop new theory and algorithms to enhance the analysis of integer models, including those that incorporating uncertainty, while also enabling the use of parallel computing paradigms.