Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver

We consider a variant of inexact Newton Method, called Newton-MR, in which the least-squares sub-problems are solved approximately using Minimum Residual method. By construction, Newton-MR can be readily applied for unconstrained optimization of a class of non-convex problems known as invex, which subsumes convexity as a sub-class. For invex optimization, instead of the classical Lipschitz continuity assumptions on gradient and Hessian, Newton-MR's global convergence can be guaranteed under a weaker notion of joint regularity of Hessian and gradient. We also obtain Newton-MR's problem-independent local convergence to the set of minima. We show that fast local/global convergence can be guaranteed under a novel inexactness condition, which, to our knowledge, is much weaker than the prior related works. Numerical results demonstrate the performance of Newton-MR as compared with several other Newton-type alternatives on a few machine learning problems.Comment: 35 page

Basins of attraction for several third order methods to find multiple roots of nonlinear equations

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.06.068There are several third order methods for solving a nonlinear algebraic equation having roots of a given multiplicity m. Here we compare a recent family of methods of order three to Euler-Cauchy's method which is found to be the best in the previous work. There are fewer fourth order methods for multiple roots but we will not include them here.Basic Science Research Program through the National Reserach Foundation of Korea (NRF)Ministry of Education (NRF-2013R1A1A2005012

A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.08.039Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. Wit the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction.National Research Foundation of KoreaMinistry of Education, Science and Technology under the research grant (Project Number: 2015-R1D1A3A-0102080

A Probability-one Homotopy Algoithm for Non-Smooth Equations and Mixed Complementarity Problems

A probability-one homotopy algorithm for solving nonsmooth equations is described. This algorithm is able to solve problems involving highly nonlinear equations,where the norm of the residual has non-global local minima.The algorithm is based on constructing homotopy mappings that are smooth in the interior of their domains.The algorithm is specialized to solve mixed complementarity problems through the use of MCP functions and associated smoothers.This specialized algorithm includes an option to ensure that all iterates remain feasible.Easily satisfiable sufficient conditions are given to ensure that the homotopy zero curve remains feasible,and global convergence properties for the MCP algorithm are developed.Computational results on the MCPLIB test library demonstrate the effectiveness of the algorithm

A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2016.02.029A class of three-point sixth-order multiple-root finders and the dynamics behind their extraneous fixed points are investigated by extending modified Newton-like methods with the introduction of the multivariate weight functions in the intermediate steps. The multivariate weight functions dependent on function-to-function ratios play a key role in constructing higher-order iterative methods. Extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction. Numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper. Relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials.Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project Number: 2015-R1D1A3A-01020808)Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project Number: 2015-R1D1A3A-01020808

pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual

pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path for elliptic systems of PDEs over bounded 2D domains, based on the Matlab pdetoolbox. The new features include a more efficient use of FEM, easier switching between different single parameter continuations, genuine multi-parameter continuation (e.g., fold continuation), more efficient implementation of nonlinear boundary conditions, cylinder and torus geometries (i.e., periodic boundary conditions), and a general interface for adding auxiliary equations like mass conservation or phase equations for continuation of traveling waves. The package (library, demos, manuals) can be downloaded at www.staff.uni-oldenburg.de/hannes.uecker/pde2pat

Glosarium Matematika

273 p.; 24 cm

On the optimality of some multi-point methods for finding multiple roots of nonlinear equation

This paper deals with the problem of determining the multiple roots of nonlinear equations, where the multiplicity of the roots is known. The paper contains some remarks on the optimality of the recently published methods [B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations, Nonlinear Anal. Model. Control, 18(2):143–152, 2013] and [X. Zhou, X. Chen, Y. Song, Families of third- and fourth-order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219(11):6030–6038, 2013]. Separate analysis of odd and even multiplicity, has shown the cases where those methods lose their optimal convergence properties. Numerical experiments are made and they support theoretical analysis