Abstract
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.
Highlights
Where x ∈ Rn, and f: Rn ⟶ R is a continuously differentiable function bounded from below. e quasi-Newton methods are currently used in countless optimization software for solving unconstrained optimization problems [1,2,3,4,5,6,7,8]. e BFGS method, one of the most efficient quasi-Newton methods, for solving (1) is an iterative method of the following form: xk+1 xk + αkdk, (2)
Powell [17] first proved the global convergence of the standard BFGS method with inexact Wolfe line search for convex functions
For nonconvex problems, Mascaren [22] has presented an example to elaborate that the BFGS method and some Broyden-type methods may not be convergent under the exact line search
Summary
Where x ∈ Rn, and f: Rn ⟶ R is a continuously differentiable function bounded from below. e quasi-Newton methods are currently used in countless optimization software for solving unconstrained optimization problems [1,2,3,4,5,6,7,8]. e BFGS method, one of the most efficient quasi-Newton methods, for solving (1) is an iterative method of the following form: xk+1 xk + αkdk,. To verify the global convergence of the BFGS method for general functions and to obtain a better Hessian approximation matrix of the objective function, Yuan and Wei [24] presented a modified quasi-Newton equation as follows: Mathematical Problems in Engineering. Where the value of ck is given by Yuan [29], and with inexact line search, the global convergence of the scaled BFGS method with ck given by (9) is established for convex functions by Powell [30]. For general nonlinear functions, Yuan limited the value range of ck to [0.01, 100] to ensure the positivity of ck under the inexact line search and proved the global convergence of the scaled BFGS method in this form.
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