Modify the linear search formula in the BFGS method to achieve global convergence.

Subject Areas : Statistics

S.A.R. Hosseini Dehmiry ¹ (Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran)
M. Hamzehnejad ² (Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran)

Received: 2018-02-18 Accepted : 2019-02-27 Published : 2019-12-22

Keywords: همگرایی سراسری, روش BFGS, بهینه‌سازی نامقید, روش‌شبه‌نیوتون, روش نیوتون,

Abstract :

Nonlinear programming problems belong to the realm of commonly used optimization problems. In most cases, the objective function of such problems is non-convex. However, to guarantee global convergence in the algorithms proposed based on Newton's method to solve these problems, a convexity condition is generally required. Meanwhile, the quasi-Newton techniques are more popular because they use an approximation of the Hessian matrix or its inverse. However, in these algorithms, only gradient information is used to approximate this matrix. One of the most applicable quasi-Newton algorithms in solving nonlinear programming problems is the BFGS method. This paper presents a new idea for a linear search in the BFGS method. It proves that using this technique will lead to global convergence for general problems without the need for any additional conditions. Finally, the performance of the proposed algorithm is evaluated numerically.

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