The Global Convergence of a Modified BFGS Method under Inexact Line Search for Nonconvex Functions

Abstract

Among the quasi-Newton algorithms, the BFGS method is often discussed by related scholars. However, in the case of inexact Wolfe line searches or even exact line search, the global convergence of the BFGS method for nonconvex functions is not still proven. Based on the aforementioned issues, we propose a new quasi-Newton algorithm to obtain a better convergence property; it is designed according to the following essentials: (1) a modified BFGS formula is designed to guarantee that B k + 1 inherits the positive definiteness of B k ; (2) a modified weak Wolfe–Powell line search is recommended; (3) a parabola, which is considered as the projection plane to avoid using the invalid direction, is proposed, and the next point x k + 1 is designed by a projection technique; (4) to obtain the global convergence of the proposed algorithm more easily, the projection point is used at all the next iteration points instead of the current modified BFGS update formula; and (5) the global convergence of the given algorithm is established under suitable conditions. Numerical results show that the proposed algorithm is efficient.

Construction of Tensile Strength and Density Prediction Models for Semi-Interpenetrating Polymer Network from Fluoroelastomer and Poly(dimethylsiloxane)

“…Therefore, a modification term f ( x 1 , x 2 , M c ) related to the component mass ratios and cross-linking density of the PDMS network is introduced on the logarithmic mixing rule to account for the influence of forced intertwining and incompatibility of components The BFGS quasi-Newton algorithm is adopted to determine the function f , considered to be the most numerically effective quasi-Newton method with global convergence and superlinear convergence speed. The algorithm avoids the calculation of a complex Hessian matrix in the Newton method, and it is widely used in function optimization of various application scenarios. , The specific calculation steps of the BFGS quasi-Newton algorithm can be referred to in the Supporting Information (Figure S2). Due to the limited number of available experimental data, we only use the first 21 data points in Table S1 in the Supporting Information to train the appropriate model by the BFGS quasi-Newton algorithm, and reserve the last 12 data points for a completely independent test of the predictive capability of the revised model.…”

“…The algorithm avoids the calculation of a complex Hessian matrix in the Newton method, and it is widely used in function optimization of various application scenarios. 42,43 The specific calculation steps of the BFGS quasi-Newton algorithm can be referred to in the Supporting Information (Figure S2). Due to the limited number of available experimental data, we only use the first 21 data points in Table S1 in the Supporting Information to train the appropriate model by the BFGS quasi-Newton algorithm, and reserve the last 12 data points for a completely independent test of the predictive capability of the revised model.…”

The Performance of MBFGS with Different Inexact Line Search Rule

“…Moreover, the modified BFGS algorithm performs better than BFGS for some of the test problems ( Pengyuan Li, Junyu Lu and Haishan Feng (2021)). MBFGS has many applications in machine learning, deep Learning, artificial intelligence (AI) and data science.…”