this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

In this paper, we propose a modication of the BFGS method for unconstrained optimization. A remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function. Under certain conditions, we also establish superlinear convergence of the method. Key words: BFGS method, global convergence, superlinear convergence 1 Present address (available before October, 1999): Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto 606, Japan, e-mail: lidh@kuamp.kyoto-u.ac.jp 1 Introduction Let f : R n ! R be continuously dierentiable. Consider the following unconstrained optimization problem: min f(x); x 2 R n : (1:1) Among numerous iterative methods for solving (1.1), quasi-Newton methods constitute particularly important class. Throughout the paper, we assume that f in (1.1) has Lipschitz continuous gradients, i.e. there is a constant L > 0 such kg(x) g(y)k ...

In this paper, we propose a derivative-free line search suited to iterative methods for solving systems of nonlinear equations with symmetric Jacobian matrices. The proposed line search can be implemented conveniently by a backtracking process and has such an attractive property that any iterative method with this line search generates a sequence of iterates that is approximately norm descent. Moreover, if the Jacobian matrices are uniformly nonsingular, then the generated sequence converges to the unique solution. We incorporate this line search with a Gauss-Newton based DFP method for solving symmetric equations. Under appropriate conditions, we establish global and superlinear convergence of the proposed DFP method. The obtained results show, in particular, that the proposed DFP method with inexact line search converges globally and superlinearly even for nonconvex unconstrained optimization problems and equality constrained optimization problems.