Using parallel function evaluations to improve hessian approximation for unconstrained optimization

This paper presents a new class of methods for solving unconstrained optimization problems on parallel computers. The methods are intended to solve small to moderate dimensional problems where function and derivative evaluation is the dominant cost. They utilize multiple processors to evaluate the function, (finite difference) gradient, and a portion of the finite difference Hessian simultaneously at each iterate. We introduce three types of new methods, which all utilize the new finite difference Hessian information in forming the new Hessian approximation at each iteration; they differ in whether and how they utilize the standard secant information from the current step as well. We present theoretical analyses of the rate of convergence of several of these methods. We also present computational results which illustrate their performance on parallel computers when function evaluation is expensive.Research supported by AFOSR grant AFOSR-85-0251, ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NFS cooperative agreement DCR -8420944.