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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=649191
• 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、科学和工业领域有着几个深远应用的领域。运筹学关注的几个领域是寻找资源的最优分配、调度任务和设计原型。这些问题通常可以表述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区域上最大化或最小化线性函数。单纯形法和原对偶内点法是目前计算最成功的线性最优化算法。算法问题与最优可行域的组合和几何结构有关。*在过去几年中，线性规划算法的几何分析和用于整数规划的新模型都取得了实质性进展。该研究提案旨在巩固和保持与线性优化的计算、组合和几何方面相关的几个研究领域的势头，重点是分析导致计算高度挑战的实例的最坏情况结构。开发新的模型来处理应用程序驱动的问题形成了本研究提案的另一个关键重点。预期的结果和意义包括促进尖端研究和触发新的方法。收紧边界，更深入地理解算法性能和输入的结构属性之间的相互作用，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的结构和最坏情况的例子的组合，以及对当前边界和结果的更紧密的分析，例如，加强多面体直径的上限，对Hirsch猜想的反例，对多项式Hirsch猜想的连续模拟的指数反例，以及对Klee-Minty结构的连续推广。*对高素质人员的监督和培训是我研究建议的重要部分。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作为加拿大领先的优化研究小组之一的声誉。我将培养多方面、多学科的培训，为工业和学术职位培养高度市场化、合格的人才。这将开发优化模型、算法、软件并培养高素质的人员，以帮助加拿大企业在经济的战略部门，如信息技术、设计、制造和运输。**