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

我们研究了求解无约束和约束优化问题的数值方法。具体而言，我们做了以下工作。(1)基于修正割线和多步割线条件，提出了求解大规模无约束优化问题的新的非线性共轭梯度法。我们证明了它们的全局收敛性。我们的数值实验表明，我们提出的方法表现良好。(2)提出了一种新的求解大规模无约束优化问题的记忆梯度法。我们证明了它们的全局收敛性。我们的数值实验表明，我们提出的方法表现良好。(3)将我们提出的有限记忆拟牛顿法与原对偶内点法结合起来求解非线性约束优化问题。(4)研究了退化非线性优化问题原对偶内点方法的局部性态。(5)我们提出了求解非线性二阶锥规划和非线性半定规划问题的原始-对偶内点方法。在线性搜索策略的框架下，利用原对偶价值函数证明了它们的全局收敛性。(6)本文证明了求解无约束优化问题的基于修正割线条件的Broyden族拟牛顿法的局部收敛性和q-超线性收敛性。(7)为了改善最速下降法的数值性能，我们研究了Barzilai-Borwein方法，并提出了扩展的Barzilai-Borwein方法。

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.