Developing and Understanding Methods for Nonlinear Optimization

该项目延续了以往的研究计划， 约束优化 该项目有三个主要部分。 第一、 有限内存方法将被开发用于大范围约束 和非线性约束优化问题。 一种新的代数 先前开发的有限内存更新的表示将 极大地方便了有限的内存的使用， 约束优化，并应有助于产生非常有效的 大约束问题的方法。 其次，张量方法将 开发用于大型、稀疏的非线性问题，包括非线性问题 方程、非线性最小二乘法和无约束优化， 也适用于非线性约束优化问题。 这 这种方法显示出产生非常稳健的方法的巨大希望 和有效的非奇异和奇异的问题，并在 稀疏情况下，可以有效地使用任何所需的直接或迭代 解算器 第三，将开发新的信任区域方法， 分析了非线性约束优化问题， 不平等和平等的限制。 预计这些方法将 具有强的全局收敛性，即使在存在 线性相关的约束梯度，并执行鲁棒性和 在实践中有效。 此外，还将研究 大型稀疏约束问题的一种改进的Cholesky分解 最优化，并在隐式非线性最小的有效方法 广场问题

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.