Abstract:In this paper, a new modified BFGS method without line searches is proposed. Unlike traditional BFGS method, this modified BFGS method is proposed based on the so-called fixed steplength strategy introduced by Sun and Zhang. Under some suitable assumptions, the global convergence and the superlinear convergence of the new algorithm are established, respectively. And some preliminary numerical experiments, which shows that the new Algorithm is feasible, is also reported.
“…where s k � x k+1 − x k and y k � g k+1 − g k . e problems related to the BFGS method have been analyzed and studied by many scholars, and satisfactory conclusions have been drawn [9][10][11][12][13][14][15][16]. In earlier year, Powell [17] first proved the global convergence of the standard BFGS method with inexact Wolfe line search for convex functions.…”
The BFGS method is one of the most efficient quasi-Newton methods for solving small- and medium-size unconstrained optimization problems. For the sake of exploring its more interesting properties, a modified two-parameter scaled BFGS method is stated in this paper. The intention of the modified scaled BFGS method is to improve the eigenvalues structure of the BFGS update. In this method, the first two terms and the last term of the standard BFGS update formula are scaled with two different positive parameters, and the new value of yk is given. Meanwhile, Yuan-Wei-Lu line search is also proposed. Under the mentioned line search, the modified two-parameter scaled BFGS method is globally convergent for nonconvex functions. The extensive numerical experiments show that this form of the scaled BFGS method outperforms the standard BFGS method or some similar scaled methods.
“…where s k � x k+1 − x k and y k � g k+1 − g k . e problems related to the BFGS method have been analyzed and studied by many scholars, and satisfactory conclusions have been drawn [9][10][11][12][13][14][15][16]. In earlier year, Powell [17] first proved the global convergence of the standard BFGS method with inexact Wolfe line search for convex functions.…”
The BFGS method is one of the most efficient quasi-Newton methods for solving small- and medium-size unconstrained optimization problems. For the sake of exploring its more interesting properties, a modified two-parameter scaled BFGS method is stated in this paper. The intention of the modified scaled BFGS method is to improve the eigenvalues structure of the BFGS update. In this method, the first two terms and the last term of the standard BFGS update formula are scaled with two different positive parameters, and the new value of yk is given. Meanwhile, Yuan-Wei-Lu line search is also proposed. Under the mentioned line search, the modified two-parameter scaled BFGS method is globally convergent for nonconvex functions. The extensive numerical experiments show that this form of the scaled BFGS method outperforms the standard BFGS method or some similar scaled methods.
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