Numerical Optimization Techniques

本课题研究非线性优化的设计与实现算法。该研究侧重于解决大问题，并有两个广泛的目标。首先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。这项工作的动机是有效地解决大型和非稀疏问题的需要。新算法采用牛顿法和可变度量更新的思想，并在天气预报中出现的大问题上进行了测试。第二个目标是研究处理非线性规划中出现的不等式约束的有效技术。这项研究的重点是大问题，而技术的设计就是为了在这种情况下有用。探讨了椭球型方法的应用。这种方法的优点是可以很容易地推导出一系列等式约束子问题，这些子问题可以用现有的软件来求解。本文还探讨了另一种方法，即将线性规划的原对偶方法推广到非线性情况。该算法在信任域方法的框架下提出，充分利用了其与序列二次规划的关系。处理约束的两种新方法在一系列大问题上进行了测试，并且对不良条件的处理得到了仔细的考虑。

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.