学术讲座 | 有限存储方法与置信域技术的有效结合
摘要：Limited-memory quasi-Newton methods and trust-region methods represent two efficient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited memory methods are usually combined with a line search. We show how to efficiently combine limited-memory and trust-region techniques. One of our approaches is based on the eigenvalue decomposition of the limited-memory quasi-Newton approximation of the Hessian matrix. The decomposition allows for finding a nearly-exact solution to the trust-region subproblem defined by the Euclidean norm with an insignificant computational overhead as compared with the cost of computing the quasi-Newton direction in line-search limited-memory methods. The other approach is based on two new eigenvalue-based norms. The advantage of the new norms is that the trust-region subproblem is separable and each of the smaller subproblems is easy to solve. We show that our eigenvalue-based limited-memory trust-region methods are globally convergent. Moreover, we propose improved versions of the existing limited-memory trust-region algorithms. The presented results of numerical experiments demonstrate the efficiency of our approach which is competitive with line-search versions of the L-BFGS method.
关于演讲人：Prof. YUAN Yaxiang graduated from the Department of Mathematics of Xiangtan University with his bachelor’s degree in 1981. After graduation, he came to Beijing as a graduate student at the Computing Center, Chinese Academy of Sciences. He stayed in Beijing for about 9 months before going to England studying at the Department of Applied Mathematics and Theoretical Physics of University of Cambridge in November 1982. Prof. YUAN Yaxiang finished his Ph.D. thesis by the end of 1985, and worked in Cambridge for three years as a Rutherford research fellow at Fitzwilliam College of Cambridge before returning to China in September 1988. He works on nonlinear optimization, mainly on trust region algorithms, quasi-Newton methods, etc. Prof. YUAN was the director of the Institute of Computational Mathematics and Scientific/Engineering Computing of the Chinese Academy of Sciences for about 12 years (from 1995 to 2006), the director of the State Key Laboratory of Scientific and Engineering Computing of China for 10 years (from 1996 to 2005), and one of the vice presidents of Academy of Mathematics and System Sciences of the Chinese Academy of Sciences for 8 years (from 1998 to 2006).