Space Decomposition Methods in Nonsmooth Optimization

非光滑优化中的空间分解方法DMS 0071459提出了一种将多面体逼近和二次逼近相结合的理论和方法，以产生一种快速收敛的多变量非光滑函数的最小化算法。这个想法是使用一个束的方法来确定一个所谓的vu空间分解，这样在v空间上束的切割面方面工作得很快，在u空间上可以使用一个准牛顿近似的黑森量或拉格朗日量。这种方法可以通过分离变量或约束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保持在允许范围内的情况下，寻找流量和泵压力间隙值以最小化泵送成本的水压控制问题可以通过分离变量来解决。如果流量是固定的，那么寻找压力间隙最优值的子问题是一个易于解决的线性最小化问题。利用这种方法，寻找流动变量最优值的外部问题是一个非光滑问题，可以通过待开发的技术有效地找到其解。理论研究是如何适当地定义一类函数，使其足够特殊，使其成员具有u - hessin，又足够一般，使其包含多种应用的函数。为发展提出的计算方法意义重大，因为它们可以应用于许多实际的决策情况。这些包括确定参数值，以便将模型拟合到地形或地球物理模拟等数据中。它们还包括将分解技术应用于大型或复杂的模型，例如那些发生在稀缺资源分配、交通分配、制造、结构工程、物流和战略规划中的模型。第一段中的管网问题是土木基础设施设计中出现的一个重要例子。它说明了在考虑冲突的设计标准（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越来越重要。这些问题自然会分解为决定建造哪种类型和规模的发电厂的高级问题和决定如何最好地运行各种机组以满足日常能源需求的低级问题。此外，这项研究与正在进行的高性能计算工作非常吻合，因为它的方法将利用并行处理来解决非常大和/或非常复杂的决策问题。

Abstract for Space Decomposition Methods in Nonsmooth Optimization DMS 0071459 The proposed research concerns developing theory and methods for 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 theoretical research is concerned with properly defining a class of functions which is special enough for the members to have U-Hessians and general enough to include functions from many applications.The computational methods proposed for development are significant because they can be applied in many practical decision-making situations. These include the determination of values of parameters for fitting models to data such as in topographic or geophysical modeling. They also include application of decomposition techniques to 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 application 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 on-going efforts in high-performance computing, because its methods will utilize parallel processing for solving very large and/or very complex decision-makingproblems.