Conic Methods for Unconstrained Minimization and Tensor Methods for Nonlinear Equations

Abstract

Standard methods for nonlinear equations and unconstrained minimization base each iteration on a linear or quadratic model of the objective function, respectively. Recently, methods using two generalizations of the standard models have been proposed for these problems. Conic methods for unconstrained minimization use a model that is the ratio of a quadratic function divided by the square of a linear function. Tensor methods for nonlinear equations augment the standard linear model with a simple second order term. This paper surveys the research to date on methods for unconstrained minimization and nonlinear equations that use conic and tensor models. It begins with a brief summary of the standard methods, so that the paper is essentially selfcontained.