A Class of Methods Combining L-BFGS and Truncated Newton

We present a class of methods which is a combination of the limited memory BFGS method and the truncated Newton method. Each member of the class is defined by the (possibly dynamic) number of vector pairs of the L-BFGS method and the forcing sequence of the truncated Newton method. We exemplify with a scheme which makes the hybrid method perform like the L-BFGS method far from the solution, and like the truncated Newton method close to the solution. The cost of a method in the class of combined methods is compared with its parent methods on different functions, for different cost schemes, namely the cost of finite difference derivatives versus AD derivatives, and whether or not we can exploit sparsity. Numerical results indicate that the example hybrid method usually performs well if one of its parent methods performs well, to a large extent independent of the cost of derivatives and available sparsity information.