Numerical Optimization Techniques

本项目研究了 非线性优化算法 的研究热点 解决大问题，有两个广泛的目标。 一是 开发一种新的算法来解决大型和非结构化的无约束 问题 这项工作的动机是需要解决 大型和非稀疏问题。 新算法使用了 牛顿方法和变尺度更新，并进行了测试，在大 天气预报中出现的问题。 第二 目的是研究处理不等式约束的有效技术 发生在非线性规划中。 这项研究侧重于大问题， 技术被设计成在这种情况下有用。 使用 的椭球型方法进行了探讨。 这种方法有 容易引入等式约束序列的优点 这些问题可以通过现有的软件解决。 另一种方法， 它包括扩展线性规划的原始-对偶方法， 非线性的情况下，也进行了探讨。 该算法在框架中提出， 的信赖域方法，以及它与序列 二次规划得到充分利用。 两种新的处理方法 在一组大问题上测试约束， 对病态的治疗要认真考虑。

This project investigates the design and implementation algorithms for nonlinear optimization. The research focuses on the solution of large problems, and has two broad objectives. The first is to develop a new algorithm for solving large and unstructured unconstrained problems. This work is motivated by the need of solving large and nonsparse problems efficiently. The new algorithm uses ideas from both Newton methods and variable metric updating, and is tested on large problems arising in weather forecasting. The second objective is to study efficient techniques for handling inequality constraints occurring in nonlinear programming. This research focuses on large problems, and techniques are designed to be useful in that case. The use of an ellipsoidal-type method is explored. This approach has the advantage of leading readily into a sequence of equality constrained subproblems which can be solved by existing software. An alternative approach, which consists of extending the primal-dual method of linear programming to the nonlinear case, is also explored. The algorithm is posed in the framework of trust region methods, and its relationship with sequential quadratic programming is fully exploited. The two new approaches for handling constraints are tested on a set of large problems, and the treatment of ill-conditioning receives careful consideration.