A bounded deterioration property of a symmetric positive definite class of Newton-like methods and its application

This paper is concerned with Newton-like methods for solving unconstrained minimization problems. We derive a general form and its factorized form of a symmetric positive definite matrix that satisfies the secant condition in order to approximate the Hessian matrix of the objective function. We obtain the bounded deterioration property for such a general form, which is an extention of the bounded deterioration property for secant methods. Applying the general form to the secant method, we obtain a new family that includes the Broyden family, and we show local and q-superlinear convergence of our method. Furthermore, we propose applying the general form to nonlinear least squares problems to obtain a modification of the Gauss-Newton method.