Parallel Quasi-Newton Methods for Unconstrained Optimization.

Abstract
This document discusses methods for solving the unconstrained optimization problem on parallel computers, when the number of variables is sufficiently small that quasi-Newton methods can be used. The authors concentrate mainly, but not exclusively, on problems where function evaluation is expensive. First they discuss ways to parallelize both the function evaluation costs and the linear algebra calculations in the standard sequential secant method, the BFGS method. Then described are new methods that are appropriate when there are enough processors to evaluate the function, gradient, and part but not all of the Hessian at each iteration. Developed are new algorithms that utilize this information and analyze their convergence properties. The authors present computational experiments showing that they are superior to parallelization of either the BFGS method or Newton's method under our assumptions on the number of processors and cost of function evaluation. Finally they discuss ways to effectively utilize the gradient values at unsuccessful trial points that are available in our parallel methods and also in some sequential software packages.

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