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

• Published in: Journal of optimization theory and applications Vol. 140; no. 2; pp. 265 - 286
• Main Authors: Luo, X.-L., Kelley, C. T., Liao, L.-Z., Tam, H. W.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-02-2009; Springer; Springer Nature B.V
• 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; Rosenbrock method; Ordinary differential equations; Unconstrained optimization; Gradient system; Trust-region methods; Differential equation; Stiff problem; Initial value problem; Runge Kutta method; Modeling; Global solution; Stationary condition; Confidence interval; Equilibrium point; Numerical convergence
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-008-9469-0

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.