In this paper the first two terms of the modified BFGS method given by Yuan and Wei [Comput. Optim. Appl., 47:237-255 (2010)] are scaled with a positive parameter, while the third one is scaled with another positive parameter. The first parameter is determined to cluster the eigenvalues of the modified BFGS matrix. The second one is computed as a preconditioner to the Hessian of the minimizing function combined with minimization of the conjugacy condition from the conjugate gradient methods in order to shift the large eigenvalues to the left. In this method the stepsize is determined by the Wolfe line search conditions. The global convergence is proved in very general conditions, without assuming the convexity of the minimizing function, using only the trace and the determinant of the scaled modified BFGS matrix. The preliminary computational experiments on a set of 80 unconstrained optimization test functions with a medium number of variables show that this algorithm is more efficient and more robust that the Yuan and Wei’s modified BFGS update, as well as some other scaled modified BFGS methods we present in this paper, including the double parameter scaled BFGS method by Andrei [Jour. Comput. and App. Math. 332:26-44 (2018)].
Mathematics Subject Classification (2010) 49M7. 49M10. 65K05. 90C30
Unconstrained optimization, Modified BFGS method, Scaled BFGS method, Trace, Determinant, Global convergence, Numerical comparisons.
Neculai ANDREI, "A Double Parameter Scaled Modified Broyden-Fletcher- Goldfarb-Shanno Method for Unconstrained Optimization", Studies in Informatics and Control, ISSN 1220-1766, vol. 27(2), pp. 135-146, 2018. https://doi.org/10.24846/v27i2y201801
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