Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta m...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2009-02, Vol.140 (2), p.265-286
Main Authors: Luo, X.-L., Kelley, C. T., Liao, L.-Z., Tam, H. W.
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Applied sciences
Calculus of Variations and Optimal Control
Optimization
Convergence
Differential equations
Engineering
Equilibrium
Exact sciences and technology
Implicit methods
Information industry
Iterative methods
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Methods
Nonlinear equations
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimization
Ordinary differential equations
Runge-Kutta method
Studies
Theory of Computation
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Summary:Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-008-9469-0