On the performance of a new symmetric rank-one method with restart for solving unconstrained optimization problems

摘要

Quasi-Newton (QN) methods are generally held to be the most efficient minimization methods for solving unconstrained optimization problems. Among the QN methods, symmetric rank-one (SRI) is one of the very competitive formulas. In the present paper, we propose a new SRI method. The new technique attempts to improve the quality of the SRI Hessian by employing the scaling of the identity in a certain sense. However, since at some iterations these updates might be singular, indefinite or undefined, this paper proposes an updates criterion based on the eigenvalues of the SRI update to measure this quality. Hence, the new method is employed only to improve the approximation of the SRI Hessian. It is shown that the numerical results support the theoretical considerations for the usefulness of this criterion and show that the proposed method improves the performance of the SRI update substantially.