Abstract
To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definite incomplete Hessian matrix. It also shows that the Wolfe conditions hold in IHN with a line search step length of one. As an important application, an effective IHN and a modified IHN, called the truncated-IHN method (T-IHN), are constructed for solving a large scale chemical database optimal projection mapping problem. T-IHN is shown to work well even with indefinite incomplete Hessian matrices. Numerical results confirm the theoretical results of IHN, and demonstrate the promising potential of T-IHN as an efficient minimization algorithm.
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This project was supported by the National Science Foundation (DMS-0241236, USA), the National Institutes of Health (PHS R01 EB005825-01, USA), the Natural Science Foundation of China (10471062), and Jiangsu Province (BK2006184, China).
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Xie, D., Ni, Q. An incomplete Hessian Newton minimization method and its application in a chemical database problem. Comput Optim Appl 44, 467–485 (2009). https://doi.org/10.1007/s10589-008-9164-y
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DOI: https://doi.org/10.1007/s10589-008-9164-y