A Double-Parameter Scaling Broyden–Fletcher–Goldfarb–Shanno Method Based on Minimizing the Measure Function of Byrd and Nocedal for Unconstrained Optimization

In this paper, the first two terms on the right-hand side of the Broyden–Fletcher–Goldfarb–Shanno update are scaled with a positive parameter, while the third one is also scaled with another positive parameter. These scaling parameters are determined by minimizing the measure function introduced by Byrd and Nocedal (SIAM J Numer Anal 26:727–739, 1989). The obtained algorithm is close to the algorithm based on clustering the eigenvalues of the Broyden–Fletcher–Goldfarb–Shanno approximation of the Hessian and on shifting its large eigenvalues to the left, but it is not superior to it. Under classical assumptions, the convergence is proved by using the trace and the determinant of the iteration matrix. By using a set of 80 unconstrained optimization test problems, it is proved that the algorithm minimizing the measure function of Byrd and Nocedal is more efficient and more robust than some other scaling Broyden–Fletcher–Goldfarb–Shanno algorithms, including the variants of Biggs (J Inst Math Appl 12:337–338, 1973), Yuan (IMA J Numer Anal 11:325–332, 1991), Oren and Luenberger (Manag Sci 20:845–862, 1974) and of Nocedal and Yuan (Math Program 61:19–37, 1993). However, it is less efficient than the algorithms based on clustering the eigenvalues of the iteration matrix and on shifting its large eigenvalues to the left, as shown by Andrei (J Comput Appl Math 332:26–44, 2018, Numer Algorithms 77:413–432, 2018).