Large-Scale Constrained Optimization

这个项目的目标是发展有效的数值方法来解决复杂的非线性优化问题。要开发和分析的算法遵循两种不同但相关的方法：-顺序二次规划（SQP）方法；-屏障方法（即内部方法）。本研究将集中于大规模问题、稀疏线性和非线性约束的直接处理以及超线性收敛的保证。SQP方法的研究将扩展基于线性约束子问题和增广拉格朗日价值函数的技术。要考虑的主题包括简化Hessian的使用，精确二阶导数的使用，接受子问题的不精确解的策略，以及处理不可行的子问题的技术。对于线性规划，屏障方法的工作已经证明是非常成功的。要提高它们的可靠性，并将其应用于大规模非线性（和凸）问题，还有很多研究要做。屏障方法的研究强调矩阵分解而不是矩阵更新，并且可以利用具有并行和矢量结构的机器的大量分解方法研究。我们提出了直接和迭代的方法来求解所涉及的不确定系统。

The goal of this project is to develop effective numerical methods for solving complex nonlinear optimization problems. The algorithms to be developed and analyzed follow two distinct but related approaches: - Sequential quadratic programming (SQP) methods; and - Barrier methods (i.e. interior methods). This research will concentrate on large-scale problems, direct treatment of sparse linear and nonlinear constraints, and the guarantee of superlinear convergence. The research on SQP methods will extend techniques based on linearly constrained subproblems and an augmented Lagrangian merit function. Topics to be considered include use of a reduced Hessian, use of exact second derivatives, strategies for accepting inexact solutions of subproblems, and techniques for treating infeasible subproblems. Work in barrier methods have already proved remarkably successful for linear programs. Much research remains to improve their reliability and to extend their application to large-scale nonlinear (and convex) problems. Research in barrier methods emphasizes matrix factorization rather than matrix updating, and can utilize the substantial body of research on factorization methods for machines with parallel and vector architectures. We propose both direct and iterative methods for solving the indefinite systems involved.