Abstract. Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.

Abstract. Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.

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Conjugate gradient methods are iterative methods for finding the minimizer of a scalar function f(x) of a vector variable x which do not update an approximation to the inverse Hessian matrix. This paper examines the effects of inexact linear searches on the methods and shows how the traditional Fletcher-Reeves and Polak-Ribiere algorithm may be modified in a form discovered by Perry to a sequence which can be interpreted as a memorytess BFGS algorithm. This algorithm may then be scaled optimally in the sense of Oren and Spedicalo. This scaling can be combined with Beale restarts and Powell's restart criterion. Computational results will show that this new method substantially outperforms known conjugate gradient methods on a wide class of problems.

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