Abstract
In this paper, a modified BFGS algorithm is proposed for unconstrained optimization. The proposed algorithm has the following properties: (i) a nonmonotone line search technique is used to obtain the step size alpha_{k} to improve the effectiveness of the algorithm; (ii) the algorithm possesses not only global convergence but also superlinear convergence for generally convex functions; (iii) the algorithm produces better numerical results than those of the normal BFGS method.
Highlights
Consider min f (x)|x ∈ n, ( . )where f (x) : n → is continuously differentiable
Under inexact line search techniques, Dai [ ] constructed an example to show that the BFGS method fails
Its global convergence can be found in [ ], but the method fails for general convex functions
Summary
Consider min f (x)|x ∈ n , ( . )where f (x) : n → is continuously differentiable. Many similar problems can be transformed into the above optimization problem (see [ – ] etc.). Under inexact line search techniques, Dai [ ] constructed an example to show that the BFGS method fails. To obtain global convergence of a BFGS method without the convexity assumption, Li and Fukushima [ , ] proposed the following modified BFGS methods. Under the WWP line search, Wei et al [ ] proposed the quasi-Newton method and established its superlinear convergence for uniformly convex functions.
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