Developing and Understanding Methods for Nonlinear Optimization

这个项目延续了过去在无约束和约束优化方面的研究计划。该项目由三个主要部分组成。首先，对于大界约束优化问题和非线性约束优化问题，将发展有限内存方法。早先开发的有限内存更新的新的代数表示将极大地促进用于约束优化的有限内存的使用，并且应该有助于产生用于大型约束问题的非常有效的方法。其次，张量方法将被发展用于大型稀疏非线性问题，包括非线性方程、非线性最小二乘和无约束优化，以及非线性约束优化问题。这种方法很有希望产生对非奇异和奇异问题非常健壮和有效的方法，并且在稀疏情况下，可以有效地使用任何所需的直接或迭代求解器。第三，发展和分析了具有等式和等式约束的非线性约束最优化问题的信赖域方法。即使在线性相关约束梯度的情况下，这些方法也有望具有很强的全局收敛性质，并在实践中表现出稳健和高效的性能。此外，还将研究大型稀疏约束优化问题的修正Cholesky分解和隐式非线性最小二乘问题的有效方法。

This project continues the past research program in unconstrained and constrained optimization. The project has three main parts. First, limited memory methods will be developed for large bound constrained and nonlinearly constrained optimization problems. A new algebraic representation of the limited memory update developed earlier will greatly facilitate the use of the limited memory approached for constrained optimization, and should help produce very efficient methods for large constrained problems. Second, tensor methods will be developed for large, sparse nonlinear problems, including nonlinear equations, nonlinear least squares, and unconstrained optimization, and also for nonlinearly constrained optimization problems. This approach shows great promise of producing methods that are very robust and efficient on nonsingular and singular problems and that, in the sparse case, can efficiently use any desired direct or iterative solver. Third, new trust region methods will be developed and analyzed for nonlinearly constrained optimization problems with inequality and equality constraints. These methods are expected to have strong global convergence properties even in the presence of linearly dependent constraint gradients, and to perform robustly and efficiently in practice. In addition, research will be carried out on a modified Cholesky factorization for large sparse constrained optimization, and on efficient methods for implicit nonlinear least squares problems.