Research on Numerical Methods for Large-Scale Nonlinear Optimization Problems and their Applications to Software Codes

大规模非线性优化问题的数值方法及其在软件代码中的应用研究

基本信息

批准号：
16510123
负责人：
YABE Hiroshi
金额：
$ 1.98万
依托单位：
Tokyo University of Science
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

We have studied numerical methods for solving unconstrained and constrained optimization problems. Specifically, we have done the following.(1) We have proposed new nonlinear conjugate gradient methods based on the modified secant and the multi-step secant conditions for solving large-scale unconstrained optimization problems. We have proved their global convergence properties. Our numerical experiments show that our proposed methods perform well.(2) We have proposed a new memory gradient method for solving large-scale unconstrained optimization problems. We have proved their global convergence properties. Our numerical experiments show that our proposed method performs well.(3) We have combined the limited memory quasi-Newton method, which was proposed by us, and the primal-dual interior point method to solve nonlinearly constrained optimization problems.(4) We have analyzed local behavior of the primal-dual interior point method for degenerate nonlinear optimization problems.(5) We have proposed primal-dual interior point methods for solving nonlinear second-order cone programming and nonlinear semidefinite programming problems. We have proved their global convergence properties by using primal-dual merit function within the framework of the line search strategy.(6) We have proved the local and q-superlinear convergence of the quasi-Newton method with the Broyden family based on the modified secant condition for solving unconstrained optimization problems.(7) We have dealt with the Barzilai-Borwein method to improve numerical performance of the steepest descent method, and we have proposed the extended Barzilai-Borwein method.

我们研究了用于解决不受限制和约束优化问题的数值方法。具体而言，我们已经完成了以下操作。（1）我们根据修改后的SECANT和多步骤sest条件提出了新的非线性共轭梯度方法，以解决大规模的无约束优化问题。我们已经证明了它们的全球收敛属性。我们的数值实验表明我们提出的方法的性能都很好。（2）我们提出了一种新的记忆梯度方法来解决大规模的不受约束的优化问题。我们已经证明了它们的全球收敛属性。我们的数值实验表明，我们提出的方法的性能都很好。（3）我们将有限的记忆准牛顿方法（是我们提出的）结合在一起的，我们提出了有限的记忆，以及解决非线性约束优化问题的原始偶尔内点方法。（4）我们已经分析了针对不固定的互换的原始方法的原始方法的局部局部行为。二阶圆锥编程和非线性半决赛编程问题。我们已经通过在线路搜索策略的框架内使用原始的双重优点功能来证明其全球收敛属性。（6）我们证明了基于修改后的SECANT条件，我们证明了Quasi-Newton方法与Broyden家族的本地和Q-superlineareart融合，以解决稳定的优化问题。提出了扩展的Barzilai-Borwein方法。