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

9402018 Mifflin提出的研究涉及开发一种结合多面体和二次逼近的方法，以产生一种快速收敛的算法，用于最小化多个变量的非光滑函数。其思想是结合束法和近点法来确定一种空间分解，使得在一子空间上束法的切割面方面可以很好地工作，而在另一子空间上可以使用某个约简Hessian的准牛顿逼近。本研究可应用于许多实际的决策情境。这些包括分解大型或复杂的模型，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，在将管网中所有点的水压保持在允许范围内的情况下，找到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解来解决。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。利用这种分离变量的方法，寻找流动变量最优值的外部问题是一个非光滑问题，通过该方法可以有效地找到其解。

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.