Mathematical Sciences: Quasi-Second-Order Methods for Nonsmooth Optimization

小行星9402018 建议的研究涉及开发一种方法，联合收割机多面体和二次逼近，以产生一个快速收敛的算法，最小化的非光滑函数的几个变量。 我们的想法是结合联合收割机捆绑和邻近点的方法，以确定一个空间分解，使在一个子空间上的切割平面方面的捆绑工作良好，并在其他子空间的拟牛顿近似的某种减少海森可以采用。 本研究可应用于许多实际决策情境。这些包括大型或复杂模型的分解，例如资源分配、制造、物流和民用基础设施设计中的模型。 例如，可以通过分解来解决水压控制问题，即找到流量和泵压间隙的值，以使泵送成本最小化，同时将管网中所有点处的水压保持在允许范围内。 如果流量是固定的，那么寻找压力间隙的最佳值的子问题是一个容易解决的线性最小化问题。 用这种分离变量的方法，找到最佳值的流量变量的外部问题是一个非光滑的问题，其解决方案可以有效地通过本建议的技术。

9402018 Mifflin The proposed research concerns developing a way to combine polyhedral and quadratic approximation to produce a rapidly convergent algorithm for minimizing a nonsmooth function of several variables. The idea is to combine bundle and proximal point methods to determine a space decomposition such that on one subspace the cutting-plane aspect of bundling works well and on the other subspace a quasi-Newton approximation of a certain reduced Hessian can be employed. This research can be applied in many practical decision-making situations. These include decomposition of large or complicated models such as those occurring in resource allocation, manufacturing, logistics and civil infrastructure design. 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 decomposition. 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 separation of variables 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 of this proposal.