Robust Quasi-Newton Equations in Quasi-Newton Method for Solving Unconstrained Optimization Problems

Quasi-Newton methods are among the most widely used and effective general-purpose algorithms for unconstrained optimization. These methods traditionally rely on the quasi-Newton equation, which serves as the foundation for updating approximations of the Hessian matrix at each iteration. The goal is to construct accurate second-order curvature information to accelerate convergence toward the optimum. In this paper, we derive a novel quasi-Newton equation based on an enhanced quadratic model. A key feature of this new formulation is that it incorporates both gradient information and objective function values, enabling higher-order accuracy in approximating the second-order curvature of the objective function. This new equation stands out for its ability to provide a more precise representation of the function's curvature, which in turn improves the overall efficiency and performance of the optimization method. Theoretical analysis shows that the proposed method is globally convergent under certain reasonable assumptions. To validate the effectiveness of the approach, we conducted a series of numerical experiments using standard benchmark problems. The results demonstrate that the modified Broyden, Fletcher, Goldfarb, and Shanno (BFGS) method, which integrates the new quasi-Newton equation, outperforms existing BFGS-type methods in terms of numerical efficiency and solution accuracy.