Exploring Polyhedra Representing Large-Scale Data Sets

探索表示大规模数据集的多面体

• 批准号: RGPIN-2019-07134
• 负责人: Stephen, Tamon
• 金额: $ 1.89万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758584
• 关键词: Exploring; Polyhedra; Representing; Large; Scale

Linear programming models are valuable because they can be solved quickly, even for very large-scale problems, via either pivoting-based strategies, such as the simplex method, or by interior-point methods. Both of these are effective, but also have drawbacks: the pivoting algorithms do not have good worst-case complexity guarantees, while interior point methods are vulnerable to numerical issues. An intriguing idea is to move towards a hybrid procedure that retains a pivoting structure, as thus is effectively combinatorial, but also includes moves that pass through the interior of the polyhedron. The most natural implementation of this is to expand the set of available directions to contain all circuit (or elementary) directions, with moves in a direction continuing until a constraint is reached. These additional circuit directions give shorter routes through polytopes, in terms of the number of pivots. One aim of this research is to use circuit and other hybrid augmentation algorithms effectively in optimization. A strong incentive to do this is the possibility of expanding to discrete and non-linear contexts.  In these contexts, it will already be interesting to produce results that do not necessarily always attain the optimal solution, but provide approximation guarantees or simply work well on practical problems. In some applications, rather than working with a fixed objective, which may not be known in advance, it is better to develop a menu of potentially (or Pareto) optimal solutions.   An important special case is when the polyhedron represents a monotone Boolean function (MBF). Here the extreme points correspond to the minimal true settings of the function. Fredman and Khachiyan proposed an algorithm which generates all such extreme points in incremental quasi-polynomial time even when the MBF is only available as an oracle. A goal of this proposal is to improve our understanding of the Fredman-Khachiyan algorithm both in theory and in practice. This includes identifyng classes of MBFs that are particularly easy or difficult for the joint generation algorithm. This classification can then be used to improve implementations. A more ambitious target is determining if an output sensitive polynomial time algorithm exists for MBF generation. MBFs are a hidden mathematical structure underlying diverse complex systems, and we believe there are many applications where understanding could improve through awareness of this structure. We are motivated in particular by applications in metabolic networks. To work with these networks, it is helpful to understand their minimal functional subsystems, known as elementary modes as well as their minimal blocking (or knockout) sets.

线性编程模型是阀门，它们可以是单纯的，例如单纯形方法或通过内点方法：旋转算法没有良好的最差案例复杂性确保Guing Idea是移动移动到保留的混合过程一个通过方向的t（或基本）方向，直到达到约束的方式在这些情况下，有可能扩展到离散和非线性环境的可能性，它将产生必要的结果，而这些结果必然会获得Timal Solution ion保证或仅在某些应用中工作。固定目标，可能是未知的，最好开发陶器（或parette）的最佳解决方案。 Khachiyan提出了一种算法，即使在那时，该提案的目标是提高我们对Fredman-Khachiyan算法的理解，即使在理论上和实践中，也可以提高我们对弗雷德曼 - khachiyan算法的理解。识别MBF的类别可以用来改进，可以确定输出敏感的多项式时间算法是否存在MBF Neration。子系统，称为基本模式为-theil最小阻止（或淘汰）集。