JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: VoL 20, No. 3, NOVEMBER 1976 A Variable-Metric Method for Function Minimization Derived from Invariancy to Nonlinear Scaling 1 E. SPEDICATO 2 Communicated by H. Y. Huang Abstract. The effect of nonlinearly scaling the objective function on the variable-metric method is investigated, and Broyden's update is modified so that a property of invariancy to the scaling is satisfied. A new three-parameter class of updates is generated, and criteria for an optimal choice of the parameters are given, Numerical experiments compare the performance of a number of algorithms of the resulting class. Key Words. Unconstrained minimization, variable-metric methods, numerical methods, optimization theorems, function minimization. 1. Introduction The variable-metric method to find the minimizer x* of a differentiable function F = F(x), x ~ R n, is based on the iterative scheme Xk÷l =Xk --AkHkgk, k =0, 1, 2,..., (1) where g = g(x) is the gradient of F, H is a square matrix characterizing the method, and A is a scalar called the stepsize. The iteration is started by assigning an estimate x0 of the minimizer and a nonsingular, usually positive-definite, matrix H0; A is chosen through a linear search with the aim of approximately minimizing the function along the search vector Sk = Hkgk ; ff the choice of A actually minimizes F along Sk, the iteration is called perfect; if moreover a definite criterion is given to deal with situations where the function is not unimodal along the search vector, the iteration is called safeguarded. The author is indebted to Professor S. S. Oren, Economic Engineering Department, Stanford University, Stanford, California, for stimulating discussions during the development of this paper. He also recognizes the financial support by the National Research Council of Italy (CNR) for his stay at Stanford University. 2 Research Associate, Computing Center, CISE, Segrate, Milano, Italy. 315 © i976 Plenum Publishing Corporation, 227 West t7th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or tnansmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.