Convergence of a Positive-Definite Scaled Symmetric Rank One Method

We discuss the convergence rate of a positive-definite scaled symmetric rank one (SSR1) method. This method is developed by Malik et al. (2002). In general, a restart procedure is derived and used together with the symmetric rank one (SR1) method. The restart procedure provides a replacement for the non-positive definite Hk with a positive multiple of the identity matrix. However if we choose the initial approximation for the inverse Hessian as an identity matrix, the sequences of steps produced by the SSR1 do not usually seem to have the “uniform linear independence” property that is assumed in some recent convergence analysis for SR1. Therefore, we present a new analysis that shows that the SSR1 method with a line search is n+1 step q-superlinearly convergent without the assumption of linearly independent iterates. This analysis only assumes that the Hessian approximations are positive-definite and bounded asymptotically, which are the actual conditions given by SSR1. Numerical experiments indicate that the SSR1 method is very competitive with the BFGS method and is easily implemented.