Developing and Understanding Methods for Nonlinear Optimization

开发和理解非线性优化方法

• 批准号: 9101795
• 负责人: Richard Byrd
• 金额: $ 13.59万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101795&HistoricalAwards=false
• 关键词: Developing; Understanding; Methods; Nonlinear; Optimization

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.

该项目延续了过去在无约束和约束优化方面的研究计划。 该项目分为三个主要部分。 首先，将针对大边界约束和非线性约束优化问题开发有限记忆方法。 早期开发的有限内存更新的新代数表示将极大地促进使用有限内存进行约束优化，并且应该有助于为大型约束问题产生非常有效的方法。 其次，张量方法将被开发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。 这种方法显示出产生对非奇异和奇异问题非常稳健和高效的方法的巨大前景，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，将针对具有不等式和等式约束的非线性约束优化问题开发和分析新的信任域方法。 即使存在线性相关约束梯度，这些方法也有望具有强大的全局收敛特性，并且在实践中稳健且高效地执行。 此外，还将研究用于大型稀疏约束优化的改进 Cholesky 分解，以及隐式非线性最小二乘问题的有效方法。