Nonlinear Optimization

将进行三个非线性优化研究项目;它们的范围从约束优化算法的理论研究到开发一个能够通过互联网自动解决问题的基于网络的优化系统。 第一个项目致力于大规模非线性规划的有限内存方法的设计。 有限记忆方法已被证明是非常有用的解决许多类大型无约束问题，但其经济性和简单性尚未充分利用在非线性约束的情况下。在这项工作中的基本指导原则是只采用具有适度计算要求的算法，并以紧凑的形式表示有限的内存矩阵，因此迭代的总成本仅为标准序列二次规划迭代所需的一小部分。 新的算法将允许用户选择直接和迭代的方法来解决迭代中出现的线性子问题。 第二个项目致力于设计和分析非线性约束（非凸）优化问题的边界点方法。这是一个长期的研究工作，首先旨在确定算法的关键成分，如模型的选择，功能和数值技术解决（病态）子问题。 新的方法将制定在一个信赖域框架，它允许一个统一的治疗凸和非凸的情况下。研究将结合数值试验和收敛性分析。 西北大学和阿贡大学的一组研究人员最近开发了一个名为近地天体的因特网优化系统。近地天体观测系统的目标之一是使用户能够使用现代网络浏览器以最小的努力远程解决优化问题。该建议的第三个项目考虑扩展近地天体的能力，使其能够与现代建模语言进行互动。还将开发一个新的约束优化服务器。 ***

Three research projects in nonlinear optimization will be undertaken; they range from theoretical studies of constrained optimization algorithms to the development of a network-based optimization system capable of solving problems automatically over the Internet. The first project is devoted to the design of limited memory approaches for large-scale nonlinear programming. Limited memory methods have proved to be quite useful for solving many classes of large unconstrained problems, but their economy and simplicity have not been fully exploited in the nonlinearly constrained case. The basic guiding principle in this work is to employ only algorithms with modest computational requirements and to represent limited memory matrices in compact form, so that the overall cost of the iteration is only a fraction of that required by the standard Sequential Quadratic Programming iteration. The new algorithms will allow the user to choose between direct and iterative methods for solving the linear subproblems arising in the iteration. The second project is devoted to the design and analysis of interior-point methods for nonlinearly constrained (non-convex) optimization problems. This is a long term research effort that first aims at identifying the crucial ingredients of the algorithms, such as the choice of model, the function, and the numerical techniques for solving the (ill-conditioned) subproblems. The new methods will be formulated in a trust region framework, which allows a uniform treatment of the convex and non-convex cases. The research will be guided by a combination of numerical testing and convergence analysis. An Internet-based optimization system called NEOS has recently been developed by a group of investigators at Northwestern and Argonne. One of the objectives of NEOS is to allow users to solve optimization problems remotely and with minimal effort using modern web browsers. The third project in this proposal considers extending the capabilities of NEOS so that it can interact with modern modeling languages. A new constrained optimization server will also be developed. ***