Diagonal quasi-newton updating strategy with cholesky factor via variational principle

The quasi-Newton method was popular due to the fact that only the gradient of the objective is required at each iterate and, since the second derivatives (Hessian) were not necessary, the quasi-Newton approach is often more efficient than the Newton method, especially when Hessian computation is costly. However, the method needed full matrix storage that approximated the (inverse) Hessian. As a result, they might not be appropriate for dealing with large-scale problems. In this paper, a diagonal quasi-Newton updating strategy is presented. The elements of the diagonal matrix approximating the Hessian were determined using the log-determinant norm satisfying weaker secant equation. To ensure the positive definiteness of the proposed diagonal updating matrices, their Cholesky factor will be considered within the variational problem. The corresponding variational problems are solved with the application of Lagrange multipliers approximated using Newton-Raphson method. Executable codes were developed to test the effectiveness and efficiency of the methods compared with some standard conjugate-gradient methods. Numerical results show that the proposed methods performs better.

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