BFGS method, Unconstrained optimization, Linesearch, Global convergence, Superlinear convergence,

No abstract

It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes-Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes-Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

No abstract

a b s t r a c tIt is well known that the search direction plays a main role in the line search method. In this paper, we propose a new search direction together with the Wolfe line search technique and one nonmonotone line search technique for solving unconstrained optimization problems. The given methods possess sufficiently descent property without carrying out any line search rule. The convergent results are established under suitable conditions. For numerical results, analysis of one probability shows that the new methods are more effective, robust, and stable, than other similar methods. Numerical results of two statistical problems also show that the presented methods are more interesting than other normal methods.