Linear Optimization: Theory and Applications

数据驱动的分析方法目前在许多行业处于高效决策和决策支持的前沿。这种先进的优化工具取得进展的一个突出例子是优化能源的产生、储存、传输和交付以及交易。这些应用从运营层面扩展到战略层面。仅举几例，优化与机器学习等其他方法相结合被成功地用于改进石油开采中的蒸汽辅助重力排水(SAGD)过程；优化模型和方法在确定智能电网的有效能量存储和调度策略以及帮助确定风电场和太阳能发电场的有效布局方面发挥了关键作用；定量建模和优化在交易(能源)金融衍生品时发挥了核心作用。许多数据驱动的问题可以表示或近似为线性优化问题。近年来，在理论公式和计算性能方面都取得了长足的进步，包括对线性优化算法和整数优化模型的新分析。例如，得到了对单纯形法的见解，证明了Hirsch猜想及其连续类比，证明了中心路径跟踪方法是非强多项式的。 然而，进一步发展线性优化理论和算法的工作仍然很少。该研究方案旨在通过对当前使用的先进算法的优点和局限性的调查，寻找新的想法和扩展。该方法基于新的构造和最坏情况的例子的组合，以及缩小目前建立的下界和上界之间差距的先进计算方法。最坏情况的实例由于其极端性质而出现在许多上下文中。例如，在确定凸拟阵优化的复杂性时，以及在计算量子场论中广义延迟函数的个数时，都会出现使晶格多面体的直径最大化的结构。组合和高维几何特性通常是意想不到的。因此，计算实验是确定和证明理论性质的关键因素。 这项研究提案的另一个关键重点是开发新的模型来处理在管理科学、供应链和交通运输中的应用问题。具体地说，该提案旨在进一步探索优化配方，以解决按订单组装(ATO)系统和共享电动汽车的问题。目标包括进一步分析组件通用性对定期审查ATO系统的影响，并通过在给定交通流量估计的情况下对充电站进行战略性定位来优化单向电动汽车共享计划的充电站位置。

Data-driven analytics methodologies are presently at the forefront of efficient decision making and decision support in many industries. One prominent set of examples of this state-of-the-art optimization tools that made headway is optimizing energy generation, storage, transmission and delivery, and trading. These applications spread from operational to strategic time horizons. To name a few, optimization combined with other methods such as machine-learning is successfully used to improve the steam assisted gravity drainage (SAGD) process in oil recovery; optimization models and methods play a key role in determining efficient energy storage and dispatch strategies for smart grids, as well as help determine effective layouts for wind and solar farms; quantitative modelling and optimization occupy a central role when trading (energy) financial derivatives. Many data-driven problems can be formulated or approximated as linear optimization problems. There has been substantial progress in recent years in both the theoretical formulations and computational performances, including novel analysis of linear optimization algorithms and models for integer optimization. For instance, insights into the simplex method were obtained, Hirsch conjecture and its continuous analogue were disproved, and central-path following methods were shown to be non-strongly polynomial. Still there remains a dearth of work to further advance linear optimization theory and algorithms. This research proposal aims at searching for new ideas and extensions via the investigation of the strengths and limitations of currently used advanced algorithms. The methodology is based on a combination of novel constructions and worst-case examples, and advanced computational approaches to close the gap between the currently established lower and upper bounds. Worst-case instances appear in many contexts due to their extremal properties. For instance, the structures conjectured to maximize the diameter of lattice polytopes arise in the determination of the complexity of convex matroid optimization, and in the computation of the number of generalized retarded functions in quantum field theory. Combinatorial and high dimensional geometric properties are often unexpected. Computational experiments are therefore a key factor for identifying and proving theoretical properties. Another key focus of this research proposal is to develop new models to handle questions with applications in management sciences, supply-chain and transportation. Specifically, the proposal aims at further exploring optimization formulations to tackle question dealing with assemble-to-order (ATO) system and with shared electric vehicles. The objectives includes to further analyze the impact of component commonality for periodic review ATO systems, and to optimize the locations for charging stations for one-way electric car sharing programs by strategically locating charging stations given estimates of traffic flow.