# Parallel Newton-Raphson Methods for Unconstrained Minimization with Asynchronous Updates of the Hessian Matrix or Its Inverse

## Abstract

We consider a parallel variant of the Newton-Raphson method for unconstrained optimization, which uses as many finite differences of gradients as possible to update the inverse Hessian matrix. The method is based on the Gauss-Seidel type of updating for quasi-Newton methods originally proposed by Straeter (1973). It incorporates the finite-difference approximations via the symmetric rank-one updates analysed by Van Laarhoven (1985). At the end of the paper we discuss the potential of the method for on-line, real-time optimization. The development of hardware for parallel computing has been so turbulent, and the development of programming languages for parallel processing has been so slow, that it is still unreasonable to expect a large market for standard optimization software. Hence, we have restricted ourselves to the testing of algorithmic ideas on sequential computers. Moreover, we also considered the asynchronous method of Fischer and Ritter (1988) which uses finite differences of gradients to update as many rows and columns as possible of the Hessian matrix itself. The test results reveal both promising research directions as well as possible pitfalls for parallel unconstrained optimization.

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