The BFGS method is one of the most effective quasi-Newton algorithms for optimization problems. However, its global convergence for general functions is still open. In this paper, under a new line search technique, this problem is solved, and it is shown that other methods in the Broyden class also possess this property. Moreover, the global convergence of the PRP method is established in the case of this new line search. Numerical results are reported to show that the new line search technique is competitive to that of the normal line search. [ABSTRACT FROM AUTHOR]
Abstract: This study devotes to incorporating a nonmonotone strategy with an automatically adjusted trust-region radius to propose a more efficient hybrid of trust-region approaches for unconstrained optimization. The primary objective of the paper is to introduce a more relaxed trust-region approach based on a novel extension in trust-region ratio and radius. The next aim is to employ stronger nonmonotone strategies, i.e. bigger trust-region ratios, far from the optimizer and weaker nonmonotone strategies, i.e. smaller trust-region ratios, close to the optimizer. The global convergence to first-order stationary points as well as the local superlinear and quadratic convergence rates are also proved under some reasonable conditions. Some preliminary numerical results and comparisons are also reported. [Copyright &y& Elsevier]