A new class of quasi-Newton updating formulas

Authors:
Donghui Li;Liqun Qi;Vera Roshchina
Affiliations:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, People's Republic of China;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, Hong Kong;Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Venue:
Optimization Methods & Software - Dedicated to Professor Michael J.D. Powell on the occasion of his 70th birthday
Year:
2008

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Abstract

In this paper, we propose a derivative-free quasi-Newton condition, which results in a new class of quasi-Newton updating formulas for unconstrained optimization. Each updating formula in this class is a rank-two updating formula and preserves the positive definiteness of the second derivative matrix of the quadratic model. Its first two terms are the same as the first two terms of the BFGS updating formula. We establish global convergence of quasi-Newton methods based upon the updating formulas in this class, and superlinear convergence of a special quasi-Newton method among them. Then we propose a special quasi-Newton updating formula, which repetitively uses the new quasi-Newton condition. This updating formula is derivative-free. Numerical results are reported.