A non-Secant quasi-Newton Method for Unconstrained Nonlinear Optimization

The Secant equation has long been the foundation of quasi-Newton methods, as updated Hessian approximations satisfy the equation with each iteration. Several publications have lately focused on modified versions of the Secant relation, with promising results. This study builds on that idea by deriving a Secant-like modification that uses non-linear quantities to construct Hessian (or its inverse) approximation updates. The method uses data from the two most recent iterations to provide an alternative to the Secant equation with the goal of producing improved Hessian approximations that induce faster convergence to the objective function optimal solution. The reported results provide strong evidence that the proposed method is promising and warrants further investigation.