2021/10/14 by Tuyen Trung Truong, Truong, Tuyen Trung
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2110.07403
openalex publication_date 2021/10/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper we apply the ideas of New Q-Newton's method directly to a\nsystem of equations, utilising the specialties of the cost function\nf=||F||2, where F=(f1,\… ,fm).\n The first algorithm proposed here is a modification of Levenberg-Marquardt\nalgorithm, where we prove some new results on global convergence and avoidance\nof saddle points.\n The second algorithm proposed here is a modification of New Q-Newton's method\nBacktracking, where we use the operator \∇ 2f(x)+\δ ||F(x)||\τ\ninstead of \∇ 2f(x)+\δ ||\∇ f(x)||\τ. This new version is\nmore suitable than New Q-Newton's method Backtracking itself, while currently\nhas better avoidance of saddle points guarantee than Levenberg-Marquardt\nalgorithms.\n Also, a general scheme for second order methods for solving systems of\nequations is proposed. We will also discuss a way to avoid that the limit of\nthe constructed sequence is a solution of H(x) intercalF(x)=0 but not of\nF(x)=0.\n