群共役性を用いた並列準ニュートン法

抄録

In this paper, a quasi-Newton method with parallel computing capabilities is proposed to unconstrained optimization problems. The parallel quasi-Newton method is characterized by simultaneous perturbations of a trial point in a dimensional number of mutual independent directions, and by approximation to the inverse of Hessian of the objective function by use of informations at the perturbed points. Then, the parallelism can be induced into the perturbation process of a trial point and the computations of the gradients as well as evaluations of the objective function at the perturbed points. Furthermore, it is remarkable that the perturbation process in the neighborhood of a trial point corresponds to a dimensional number of iterations in the currently used quasi-Newton method.
In the poposed algorithm, a set of the plural perturbation directions is divided into several groups in order to make more efficiency in approximation process to the inverse of Hessian, and a parallel extension of the BFGS updating formula is introduced together with use of the group conjugacy for the purpose of more stable convergence.
The mentioned parallel algorithm is tested on some numerical examples. The experiments indicate that the algorithm effects much faster convergence than the widely used quasi-Newton method, even when the computation is done in a serial fashion, and more stable covergence than Straeter's parallel algorithm in which the rank-one updating formula is used.