Two–parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length

An Efficient Single-Parameter Scaling Memoryless Broyden-Fletcher-Goldfarb-Shanno Algorithm for Solving Large Scale Unconstrained Optimization Problems

“…To make the BFGS methods applicable to large scale optimization models, many memoryless BFGS updating formulas have been proposed. For more details, one can see [3], [25]- [32] and the references therein. For example, Livieris et al [25] presented a new hybrid conjugate gradient method based on convex hybridization of the conjugate parameters of DY and HS+ by adapting the quasi-Newton philosophy.…”

“…It is noted that Babaie-Kafaki [3], [31] started with controlling the condition number of the BFGS update matrix to develop more efficient algorithms. Actually, it was shown that the algorithms in [3], [31] are more stable compared with the similar ones.…”

“…It is noted that Babaie-Kafaki [3], [31] started with controlling the condition number of the BFGS update matrix to develop more efficient algorithms. Actually, it was shown that the algorithms in [3], [31] are more stable compared with the similar ones. Very recently, Babaie-Kafaki and Ghanbari [32] further suggested a linear combination of the search direction by the memoryless BFGS technique with that by the Hestenes-Stiefel method.…”

“…Optimization models have found wider applications in the fields of engineering and management sciences [1], [2]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is one of the most efficient quasi-Newton algorithms for solving medium scale unconstrained optimization models [3]. Owing to its good local and global convergence and self-correcting quality [4]- [6], it is particularly suitable for solving the small-sized and medium-sized unconstrained optimization problems [7], [8].…”

“…Note that the scaling strategy in the BFGS update formula has been regarded as one of the main approaches to avoid an ill-conditional B k [9] in (3). It includes two ways: one is to multiply the approximate Hessian matrix by an appropriate scalar before it is updated in the BFGS method.…”