Local and superlinear convergence of quasi-Newton methods based on modified secant conditions (Q2372957)

The authors consider the question how to improve the secant condition which is imposed in quasi-Newton methods. They incorporate a relaxation parameter into the secant condition. The aim of this parameter is to smoothly switch the standard secant condition and the secant condition of \textit{J. Z. Zhang, N. Y. Deng} and \textit{L. H. Chen} [J. Optimization Theory Appl. 102, 147--167 (1999; Zbl 0991.90135)] and \textit{J. Z. Zhang} and \textit{C. Xu} [J. Comput. Appl. Math. 137, 269--278 (2001; Zbl 1001.65065)]. The authors consider a Broyden family that satisfies such a modified secant condition, which includes the BFGS-like and DFP-like updates. The introduced parameter enables to handle preserving positive definiteness of the approximation matrix more easily than the one based on the modified secant condition proposed by Zhang et al. [loc. cit.] The local and q-superlinear convergence of the presented method is proved.