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

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 (SR1) is one of the very competitive formulas. In the present paper, we propose a new SR1 method. The new technique attempts to improve the quality of the SR1 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 SR1 update to measure this quality. Hence, the new method is employed only to improve the approximation of the SR1 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 SR1 update substantially.