A new version of augmented self-scaling BFGS method

A new version of the augmented self-scaling memoryless BFGS quasi-Newton update,  proposed in [Appl. Numer. Math. 167,  187--201,  (2021)],  is suggested for unconstrained optimization problems. To use the corresponding scaled parameter,  the clustering of the eigenvalues of the approximate Hessian matrix about one point is applied with three approaches. The first and second approaches are based on the trace and the determinant of the matrix. The third approach is based on minimizing the measure function. The sufficient descent property is guaranteed for uniformly convex functions,  and the global convergence of the proposed algorithm is proved both for the uniformly convex and general nonlinear objective functions,  separately. Numerical experiments on a set of test functions of the CUTEr collection show that the proposed method is robust. In addition,  the proposed algorithm is effectively applied to the salt and pepper noise elimination problem.