A parallel quasi-Newton method for optimization problems with equality constraints

A great deal of attention has been paid to Newton-like methods to solve constrained optimization problems. One approach on this line is to solve a series of the quadratic programming problems, each of which requires to minimize a quadratic objective function with Hessian matrix of the Lagrangean function subject to a linear approximation to the constraint. Then, the Hessian is approximated by an adequate matrix which is revised on each iteration. In this paper, a methodology of the parallel quasi-Newton method for unconstrained optimization problems is converted into a method with parallel computing capabilities for problems with equality constraints. The proposed method is characterized by simultaneous perturbations of a trial point in plural directions, and by approximation to the Hessian of the Lagrangean 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 Lagrangean at the perturbed points. It is remarkable that the BFGS formula represented in matrices is used to update the approximation of the Hessian, not the approximation of the inverse. This point is different from the unconstrained case. The mentioned algorithm is tested on some simple examples. The experiments indicates that the algorithm effects better convergence than the currently used constrained quasi-Newton method, even when the algorithm is performed in a serial fashion. [In Japanese.]