This paper is concerned with the open problem whether BFGS method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems. We propose a cautious BFGS update and prove that the method with either Wolfe-type or Armijo-type line search converges globally if the function to be minimized has Lipschitz continuous gradients.

Hydrocephalus is a neurological disease which causes ventricular dilation due to abnormalities in the cerebrospinal fluid (CSF) circulation. Although treatment via a CSF shunt in the brain ventricles has been performed, poor rates of patient responses continue. Thus, to aid surgeons in hydrocephalus treatment planning, we propose a geometric computational approach for tracking hydrocephalus ven...

In this paper, we present a QP-free algorithm for nonlinear semidefinite programming. At each iteration, the search direction is yielded by solving two systems of linear equations with the same coefficient matrix; [Formula: see text] penalty function is used as merit function for line search, the step size is determined by Armijo type inexact line search. The global convergence of the proposed ...

The gradient method is a very efficient iterative technique for solving unconstrained optimization problems. Motivated by recent modifications of some variants the SM method, this study proposed two methods that are globally convergent as well computationally efficient. Each under influence backtracking line search. Results obtained from numerical implementation these and performance profiling ...

In this paper, a new spectral PRP conjugate gradient method is proposed, which can always generate sufficient descent direction without any line search. Under some standard conditions, global convergence of the proposed method is established when the standard Armijo or weak Wolfe line search is used. Moreover, we extend these results to the HS method. Numerical comparisons are reported with som...

In this paper, we propose a modified Polak–Ribière–Polyak (PRP) conjugate gradient method. An attractive property of the proposed method is that the direction generated by the method is always a descent direction for the objective function. This property is independent of the line search used. Moreover, if exact line search is used, the method reduces to the ordinary PRP method. Under appropria...

It was recently shown that, in the solution of smooth constrained optimization problems by sequential quadratic programming (SQP), the Maratos eeect can be prevented by means of a certain nonmonotone (more precisely, three-step or four-step monotone) line search. Using a well known transformation, this scheme can be readily extended to the case of minimax problems. It turns out however that, du...

Early artificial intelligence programming languages that incorporated search and backtracking facilities did not limit backtracking to previous computations when a portion of a search failed [1]. This had two unpleasant consequences: (1) the amount of data that had to be retained to allow backtracking limited the problems that could be handled, and (2) unexpected failure could cause backtrackin...

A BFGS method, in association with a new backtracking line search technique, is presented for solving symmetric nonlinear equations. The global and superlinear convergences of the given method are established under mild conditions. Preliminary numerical results show that the proposed method is better than the normal technique for the given problems. c © 2007 Elsevier Ltd. All rights reserved.