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A Limited-Memory BFGS Algorithm Based on a Trust-Region Quadratic Model for Large-Scale Nonlinear Equations
Yong Li1, Gonglin Yuan2, Zengxin Wei2
1Department of Mathematics, Baise University, Baise, Guangxi, P. R. China.
Plos One
|May 8, 2015
Summary
A new trust-region algorithm uses limited-memory BFGS updates for large-scale nonlinear equations. This method demonstrates competitive performance and establishes global convergence for improved effectiveness in solving complex problems.
Area of Science:
- Numerical Analysis
- Optimization Theory
- Computational Mathematics
Background:
- Large-scale nonlinear equations pose significant computational challenges.
- Existing methods may lack efficiency or robust convergence for high-dimensional problems.
- The trust-region framework offers a structured approach to iterative optimization.
Purpose of the Study:
- To introduce a novel trust-region algorithm tailored for large-scale nonlinear equations.
- To enhance algorithmic effectiveness by incorporating limited-memory BFGS updates.
- To establish the theoretical convergence properties of the proposed method.
Main Methods:
- Development of a trust-region algorithm incorporating limited-memory BFGS (L-M-BFGS) matrix updates.
- Theoretical analysis to establish global convergence under specified conditions.
- Numerical experimentation on a suite of test problems to evaluate performance.
Main Results:
- The proposed L-M-BFGS trust-region algorithm effectively addresses large-scale nonlinear equations.
- Global convergence of the algorithm is proven theoretically.
- Numerical results indicate the method is competitive with established norm methods.
Conclusions:
- The L-M-BFGS trust-region algorithm provides an effective and robust approach for large-scale nonlinear equation solving.
- The integration of L-M-BFGS enhances the practical applicability of trust-region methods.
- The study contributes a valuable tool for computational mathematics and optimization.
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