A family of quasi-Newton methods for solving nonlinear least-squares, symmetric nonlinear equations and the nearest correlation matrix problems
黃德國先生國立中央大學數學系 12:00 – 13:00 鴻經館 M107Abstract：
This thesis investigates the unconstrained optimization problem and its special cases, such as nonlinear least-squares and nonlinear equations, which hold substantial importance in numerical optimization and have numerous practical uses.
Various methods are available to address these optimization challenges. This thesis investigates a family of quasi-Newton methods, a class of iterative techniques for solving these problems, and demonstrates their advantages over traditional optimization methods such as the steepest descent and Newton’s method. Quasi-Newton methods approximate the Hessian matrix, which encodes second-order information about the objective function, using only gradient evaluations. This approximation significantly reduces computational cost and complexity, making quasi-Newton methods particularly appealing for large-sized problems where calculating the exact Hessian is impractical or impossible.
By using secant-like diagonal matrix approximations, quasi-Newton methods provide efficient solutions for a wide range of optimization problems, demonstrating their effectiveness across diverse scenarios. These methods also exhibit global convergence properties under suitable conditions.
This thesis advances the understanding and application of quasi-Newton methods in solving nonlinear least squares and symmetric nonlinear equations. Additionally, by exploring their use in addressing the nearest correlation matrix problems through numerical experiments, this study establishes a foundation for further research and practical implementations in real-world applications.