Secant Update generalized version of PSB: a new approach

In optimization, one of the main challenges of the widely used family of Quasi-Newton methods is to find an estimate of the Hessian matrix as close as possible to the real matrix. In this paper, we develop a new update formula for the estimate of the Hessian starting from the Powell-Symetric-Broyden (PSB) formula and adding pieces of information from the previous steps of the optimization path. This lead to a multisecant version of PSB, which we call generalised PSB (gPSB), but which does not exist in general as was proven before. We provide a novel interpretation of this non-existence. In addition, we provide a formula that satisfies the multisecant condition and is as close to symmetric as possible and vice versa for a second formula. Subsequently, we add enforcement of the last secant equation and present a comparison between the different methods.