A parallel quasi-Newton method by use of group conjugacy

The paper presents a new quasi-Newton method with parallel computing capabilities for unconstrained optimization problems. The parallel quasi-Newton method is characterized by simultaneous perturbations of a trial point in the plural directions, and by approximation to the inverse of Hessian of the objective function by use of information at the perturbed points. Then, the parallelism can be induced into the perturbation process of a trial point and the computations of the gradients at the perturbed points. In the proposed 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. The mentioned algorithm is tested on some simple numerical examples. The experiments indicate that the algorithm effects faster convergence than the widely used quasi-Newton method, even when the computation is done in a serial fashion, and more stable convergence than Straeter’s parallel algorithm. [In Japanese.]