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

Pengyuan Li,Junyu Lu,Haishan Feng,Wenjie Liu

doi:10.1155/2021/8342536

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.

Highlights

Consider min􏼈f(x)|x ∈ Rn􏼉, (1)where f: Rn ⟶ R and f ∈ C2. e multitudinous algorithms for (1) often use the following iterative formula: xk+1 xk + sk, k 0, 1, 2 . . . , (2)where xk is the current point, sk xk+1 − xk αkdk, αk is a step size, and dk is a search direction at xk. ere exist many algorithms for (1) [1,2,3,4,5,6,7,8,9]
Powell [19] first proved that the BFGS method possesses global convergence for convex functions under Wolfe line search
(ii) If − δ1gTk dk > δαk‖dk‖2 holds in Step 5, the global convergence of the algorithm can be obtained by the modified weak Wolfe–Powell line search, (11) and (12)

Summary

Introduction

Powell [19] first proved that the BFGS method possesses global convergence for convex functions under Wolfe line search. Convergence analysis of the new BFGS algorithm was given for weak Wolfe–Powell line search: f xk + αkdk􏼁 ≤ fk + δαkgTk dk, (8) We will demonstrate the global convergence of the modified BFGS (MBFGS) method (10) for nonconvex functions with the modified weak Wolfe–Powell (MWWP) line search [34], whose form is as follows: f xk + αkdk􏼁 ≤ fk + δαkgTk dk + αk min􏼔− δ1gTk dk, δ α2k dk 2􏼕, (11)

Results

Conclusion