Novel Algorithms for Nonlinear Optimization

这个项目的两个研究目标是开发、分析和实现求解非线性优化问题的新的数值方法。第一个重点是非线性规划(NLP)方法的设计，与现有算法相比，该方法可以重复使用导数矩阵的因式分解来解决密切相关的问题实例。这种热启动方法有望显著提高混合整数非线性优化的分支定界算法的速度。第二个重点是开发基于广义Bders分解的高效并行算法来求解在不确定性或两阶段随机优化问题中出现的可分解NLP问题。数值优化的重点在于非凸问题局部解的快速计算，而现有的方法局限于凸实例或更耗时地搜索全局最优，数值优化已成为许多工业、经济和科学领域不可缺少的工具，回答了诸如“什么是设计和运行这座工厂的最佳方式”或“电网应该如何运行才能承受网络元件的故障”等问题。虽然有强大的计算方法可用于优化可由线性或受限于非离散决策的模型描述的系统，但解决实际中经常出现的非线性和离散问题往往太耗时。因此，拟议研究项目的第一部分旨在显著加速非线性离散优化算法中的一个关键组件。该项目的第二个研究目标涉及有效利用日益普遍的并行计算能力，以优化将许多潜在情景视为解决未来情况不确定性的一种方式的问题。

The two research objectives of this project concern the development, analysis, and implementation of novel numerical methods for the solution of nonlinear optimization problems. The first thrust addresses the design of nonlinear programming (NLP) methods that can, in contrast to existing algorithms, reuse the factorization of derivative matrices for the solution of closely related problem instances. Such hot-started methods are expected to lead to significant speedup of branch-and-bound algorithms for mixed-integer nonlinear optimization. The second focus is the development of efficient parallel algorithms based on Generalized Benders Decomposition for the solution of decomposable NLPs, as they arise in design under uncertainty or two-stage stochastic optimization problems. The emphasis lies in the fast computation of local solutions of nonconvex problems, whereas existing approaches are restricted to convex instances or limited to the much more time-consuming search for global optima.Numerical optimization has become an indispensable tool in many areas of industry, economy and science, answering questions such as "what is the best way to design and operate this plant" or "how should the electrical power grid be operated in order to be able to sustain failure of network components." While powerful computational methods are available for the optimization of systems that can be described by models that are either linear or restricted to non-discrete decisions, the solution of problems that are both nonlinear and discrete, as they frequently appear in practice, is often too time-consuming with current technology. Therefore, the first part of the proposed research project aims at significantly accelerating a crucial key component in algorithms for nonlinear discrete optimization. The second research objective of this project deals with the efficient exploitation of increasingly pervasive parallel computing power for the optimization of problems that consider many potential scenarios as a way of addressing the uncertainty of future circumstances.