Algorithms for Large-Scale Nonlinearly Constrained Optimization

求解大规模非线性优化问题（最小化或最大化）是科学计算的核心。结构占据最小约束势能的位置，投资者的目标是在控制风险的同时实现利润最大化，公用事业运行输电网络以满足最低成本的需求，制药公司希望以最小的药物剂量靶向病原体。所有这些问题都很大，要么是因为数学模型涉及许多参数，要么是因为它们实际上是一些连续问题的有限离散，其中变量是函数。这项拨款申请的目的是支持设计、分析和开发新的非线性优化算法，特别是针对大规模的情况。我们将把重点放在试图改进对实际（复杂）问题的简化（廉价）近似的方法上。这样的过程可以递归地应用，这方面最成功的思想被称为顺序二次规划（SQP）。我们的研究是针对改进onSQP的方法，特别是当潜在的问题很大时，特别是在SQP本身可能太昂贵而无法考虑的情况下。我们研究的最终目标是生产高质量的、公开可用的软件，作为GALAHAD库的一部分。

The solution of large-scale nonlinear optimization-- minimization ormaximization - problems lies at the heart of scientificcomputation. Structures take up positions of minimal constrainedpotential energy, investors aim to maximize profit while controllingrisk, public utilities run transmission networks to satisfy demand atleast cost, and pharmaceutical companies desire minimal drug doses totarget pathogens. All of these problems are large either because themathematical model involves many parameters or because they are actuallyfinite discretisations of some continuous problem for which thevariables are functions.The purpose of this grant application is to support the design, analysisand development of new algorithms for nonlinear optimization that areparticularly aimed at the large-scale case.We shall focus on methods which attempt to improve simplified (cheaper) approximations of the actual (complicated) problem.Such a procedure may be applied recursively, and the mostsuccessful ideas in this vein are known as sequential quadraticprogramming (SQP). Our research is directed on ways to improve onSQP particularly when the underlying problem is large, and indeedparticularly in the case where SQP itself may be too expensive tocontemplate. The end goal of our research is to produce high-quality, publicly available software as part of the GALAHAD library.