Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
A tool for the analysis of Quasi-Newton methods with application to unconstrained minimization
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Runge-Kutta methods for dissipative and gradient dynamical systems
SIAM Journal on Numerical Analysis
Using dynamical systems methods to solve minimization problems
NUMDIFF-7 Selected papers of the seventh conference on Numerical treatment of differential equations
Convergence Analysis of Pseudo-Transient Continuation
SIAM Journal on Numerical Analysis
Trust Region Algorithms and Timestep Selection
SIAM Journal on Numerical Analysis
Trust-region methods
A Revised Modified Cholesky Factorization Algorithm
SIAM Journal on Optimization
Stability Analysis of Gradient-Based Neural Networks for Optimization Problems
Journal of Global Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
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The Levenberg-Marquardt method is a popular method for both optimization problems and equilibrium problems in dynamical systems. In this article, we study the convergence properties of the Levenberg-Marquardt method with the standard matrix update scheme. In our global convergence proof, we relax the condition that update matrices be bounded, and only require that their norms increase at most linearly. Furthermore, we analyze its local convergence for the uniformly convex function. In this case, the Levenberg-Marquardt method has superlinear convergence, and the initial matrix can be chosen arbitrarily for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula.