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In this paper, by using derivative-free line search, we propose quasi-Newton methods for smooth nonlinear equations. Under appropriate conditions, we show that the proposed quasi-Newton methods converge globally and superlinearly. In particular, for nonlinear equations involving a mapping with positive denite Jacobian matrices , we propose a norm descent(More)
In this paper , we propose a modified Polak – Ribì ere – 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(More)
This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC 2 , then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using(More)
Recently, Li et al. (Comput. Optim. Appl. 26:131–147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a(More)
This paper is concerned with the open problem whether BFGS method with inexact line search converges globally when applied to nonconvex uncon-strained 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(More)
The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we rst study some useful properties of this refor-mulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton(More)