Space Decomposition Methods in Nonsmooth Optimization

非光滑优化中的空间分解方法

• 批准号: 9703952
• 负责人: Robert Mifflin
• 金额: --
• 依托单位: Washington State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703952&HistoricalAwards=false
• 关键词: Space; Decomposition; Methods; Nonsmooth; Optimization

Robt. Mifflin Abstract for DMS-9703952 Abstract for Space Decomposition Methods in Nonsmooth Optimization The proposed research concerns developing theory and methods for implicitly combining polyhedral and quadratic approximation to produce a rapidly convergent algorithm for minimizing a nonsmooth function of many variables. The idea is to use a bundle method to determine a so-called VU-space decomposition such that on V-space the cutting-plane aspect of bundling works fast and on U-space a quasi-Newton approximation of a Hessian of a U-Lagrangian can be employed. Such methods can be used to solve large or complicated optimization models via separation of variables or constraints. For example, a water pressure control problem of finding values for flow rates and pump pressure gaps to minimize pumping cost subject to maintaining water pressures in allowable ranges at all points in a pipe network can be solved via separation of the variables. If the flow rates are fixed then the subproblem of finding optimal values for the pressure gaps is an easy-to- solve linear minimization problem. With this approach the outer problem of finding optimal values for the flow variables is a nonsmooth problem whose solution can be found efficiently via the techniques to be developed. The computational methods proposed for development are significant because they can be applied in many practical decision-making situations. These include decomposition of large or complicated models such as those occurring in allocation of scarce resources, traffic assignment, manufacturing, structural engineering, logistics and strategic planning. The pipe network problem in the first paragraph is an important example occurring in civil infrastructure design. It illustrates the need to make trade-off decisions when considering conflicting design criteria such as simultaneously having high reliability and low cost. Another area of applica tion is in power system planning subject to environmental protection constraints. These decision models will increase in importance as deregulation of the energy supply sector takes effect. These problems naturally decompose into higher level problems of deciding which types and sizes of power generation plants to build and lower level problems of deciding how best to operate the various units to meet daily energy demand. Additionally, this research fits in well with ongoing efforts in high-performance computing, because its methods will utilize parallel processing to solve larger or more complex decision- making problems than are currently possible to solve.

罗伯特. DMS-9703952 的 Mifflin 摘要 非光滑优化中的空间分解方法摘要 所提出的研究涉及开发隐式组合多面体和二次近似的理论和方法，以产生快速收敛的算法，以最小化多变量的非光滑函数。这个想法是使用捆绑方法来确定所谓的 VU 空间分解，以便在 V 空间上捆绑的切割平面方面可以快速工作，而在 U 空间上可以采用 U 拉格朗日的 Hessian 的拟牛顿近似。此类方法可用于通过变量或约束的分离来求解大型或复杂的优化模型。例如，在将管网中所有点的水压维持在允许范围内的情况下，找到流量和泵压间隙的值以最小化泵送成本的水压控制问题可以通过变量分离来解决。如果流量固定，那么寻找压力间隙最佳值的子问题就是一个易于解决的线性最小化问题。使用这种方法，寻找流量变量最优值的外部问题是一个非光滑问题，可以通过要开发的技术有效地找到其解决方案。为开发而提出的计算方法非常重要，因为它们可以应用于许多实际的决策情况。其中包括大型或复杂模型的分解，例如稀缺资源分配、交通分配、制造、结构工程、物流和战略规划中发生的模型。第一段中的管网问题是土木基础设施设计中发生的一个重要例子。它说明了在考虑相互冲突的设计标准（例如同时具有高可靠性和低成本）时需要做出权衡决策。另一个应用领域是受环境保护约束的电力系统规划。随着能源供应部门放松管制的生效，这些决策模型将变得更加重要。这些问题自然地分解为决定建造哪种类型和规模的发电厂的较高层次的问题和决定如何最好地运行各种装置以满足日常能源需求的较低层次的问题。此外，这项研究非常适合高性能计算领域的持续努力，因为其方法将利用并行处理来解决比当前可能解决的更大或更复杂的决策问题。