Computational, Combinatorial, and Geometric Aspects of Linear Optimization

线性优化的计算、组合和几何方面

• 批准号: RGPIN-2015-06163
• 负责人: Deza, Antoine
• 金额: $ 2.04万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613982
• 关键词: Computational; Combinatorial; Geometric; Aspects; Linear

Rational decision-making through quantitative modelling and analysis is the guiding principle behind operations research, a field with several far-reaching applications across engineering, sciences, and industry. Finding optimal allocations of resources, scheduling tasks, and designing prototypes are a few of the areas operations research is concerned with. These problems can often be formulated, or approximated, as linear optimization problems, which involve maximizing or minimizing a linear function over a domain defined by a set of linear inequalities. The simplex and primal-dual interior point methods are currently the most computationally successful algorithms for linear optimization. The algorithmic issues are related to the combinatorial and geometric structure of the feasible region. In the last few years, there has been substantial progress in both the geometric analysis of linear programming algorithms and novel models for integer programming. The research proposal aims at consolidating and preserving the momentum in several research areas related to the computational, combinatorial, and geometric aspects of linear optimization with a focus on the analysis of worst-case constructions leading to computationally highly challenging instances. Developing new models to handle application driven questions forms another key focus of this research proposal. The anticipated outcome and significance include fostering cutting edge research and triggering novel approaches. Tightening of the bounds, deeper understanding of the interactions between the algorithmic performance and the structural properties of the input have the potential to stimulate novel approaches for solving linear optimization problems. The proposed methodology is based on a combination of novel constructions and worst-case examples and a tighter analysis of the current bounds and results such as a strengthening of the upper bound for the diameter of polytopes, a counterexample to the Hirsch conjecture, an exponential counterexample to the continuous analogue of the polynomial Hirsch conjecture, and continuous generalizations of the Klee-Minty construction. Supervision and training of highly qualified personnel is an essential part of my research proposal. As the head of the Advanced Optimization Laboratory (AdvOL), I will continue to seek top graduate students and further strengthen the reputation of AdvOL as one of the leading optimization research groups in Canada. I will nurture multifaceted, multidisciplinary training that produces highly marketable, qualified personnel for both industrial and academic positions. This will develop optimization models, algorithms, software and produce Highly Qualified Personnel to assist Canadian enterprises in strategic sectors of the economy, such as information technology, design, manufacturing, and transportation.

通过定量建模和分析进行理性决策是运筹学背后的指导原则，运筹学是一个在工程、科学分配和工业领域有着广泛应用的领域，其中包括寻找资源最优方案、调度任务和设计原型等。这些问题通常可以表述为或近似为线性优化问题，其中涉及在由一组线性不等式定义的域上最大化或最小化线性函数。目前是大多数计算上成功的线性优化算法的算法问题都与可行区域的组合和几何结构有关。 在过去的几年中，线性规划算法的几何分析和整数规划的新颖模型都取得了实质性进展，该研究提案旨在巩固和保持与计算、组合和几何相关的几个研究领域的势头。线性优化的重点是分析最坏情况的结构，从而导致计算上具有高度挑战性的实例，这是本研究提案的另一个重点，其预期成果和意义包括促进前沿研究和发展。触发新方法。对算法性能和输入结构特性之间相互作用的更深入理解有可能激发解决线性优化问题的新方法。​所提出的方法基于新颖结构的组合。和最坏情况的例子以及对当前界限和结果的更严格的分析，例如加强多胞体直径的上限，赫希猜想的反例，连续模拟的指数反例多项式赫希猜想，以及克利-明蒂构造的连续推广。 高素质人才的监督和培训是我研究计划的重要组成部分，作为高级优化实验室 (AdvOL) 的负责人，我将继续寻找顶尖的研究生，并进一步加强 AdvOL 作为领先的优化研究之一的声誉。我将培养多方面、多学科的培训，为工业和学术职位培养高素质的人才，这将开发优化模型、算法、软件并培养高素质人才，以协助加拿大企业在经济的战略领域，比如信息技术、设计、制造和运输。