The spherical quadratic steepest descent (SQSD) method for unconstrained minimization with no explicit line searches

A very simple gradient only algorithm for unconstrained minimization is pro posed that, in terms of storage requirement and computational efficiency, m ay be considered as an alternative to the conjugate gradient line search me thods for large problems. The method effectively applies the steepest desce nt method to successive simple (spherical) quadratic approximations of the objective function in such a way that no explicit line searches are perform ed in solving the minimization problem. It is shown that the method is conv ergent when applied to general positive-definite quadratic functions. The m ethod is tested by its application to some standard and other test problems . On the evidence presented, the new method, called the SQSD algorithm, app ears to be reliable and stable, and very competitive compared to the well-e stablished Fletcher-Reeves and Polak-Ribiere conjugate gradient methods. In particular, it does very well when applied to extremely ill-conditioned pr oblems. (C) 2001 Elsevier Science Ltd. All rights reserved.