Abstract
A quasi-Newton method with parallel computing capabilities for unconstrained optimization problems, which is called a parallel quasi-Newton method, is converted to linearly constrained cases. A parallel quasi-Newton method is characterized by simultaneous perturbations of a trial point in plural directions, and by approximation to an inverse of the Hessian of the objective function by use of informations at the perturbed points. Then, the parallelism can be induced into the perturbation process as well as the computations of the gradients at the perturbed points. In our converted method, the perturbation process in parallel fashion is performed on the linear manifold of the equality constraints. When the objective function is quadratic, the optimum on the constraints can be obtained by use of the matrix type of the BFGS updating formula after a trial point is renewed only once.
The proposed algorithm is tested on some examples. The experiment in serial fashion on the currently used computer indicates that the algorithm effects better convergence than the gradient projection method and the quasi-Newton projection method. From these results, it is expected that the good convergence becomes pronounced on parallel computers with multi-processor.
