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2016•FilomatOpen access

A nonmonotone modified BFGS algorithm for nonconvex unconstrained optimization problems

Keyvan Amini, Somayeh Bahrami, Shadi Amiri

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Abstract

In this paper, a modified BFGS algorithm is proposed to solve unconstrained optimization problems. First, based on a modified secant condition, an update formula is recommended to approximate Hessian matrix. Then thanks to the remarkable nonmonotone line search properties, an appropriate nonmonotone idea is employed. Under some mild conditions, the global convergence properties of the algorithm are established without convexity assumption on the objective function. Preliminary numerical experiments are also reported which indicate the promising behavior of the new algorithm.

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What this paper is about

In this paper, a modified BFGS algorithm is proposed to solve unconstrained optimization problems. First, based on a modified secant condition, an update formula is recommended to approximate Hessian matrix. Then thanks to the remarkable nonmonotone line search properties, an appropriate nonmonotone idea is employed. Under some mild conditions, the global convergence properties of the algorithm are established without convexity assumption on the objective function. Preliminary numerical experiments are also reported which indicate the promising behavior of the new algorithm.

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Available abstract

In this paper, a modified BFGS algorithm is proposed to solve unconstrained optimization problems. First, based on a modified secant condition, an update formula is recommended to approximate Hessian matrix. Then thanks to the remarkable nonmonotone line search properties, an appropriate nonmonotone idea is employed. Under some mild conditions, the global convergence properties of the algorithm are established without convexity assumption on the objective function. Preliminary numerical experiments are also reported which indicate the promising behavior of the new algorithm.

Key concepts: Broyden–Fletcher–Goldfarb–Shanno algorithm, Hessian matrix, Line search, Mathematics, Mathematical optimization, Convergence (economics), Quasi-Newton method, Convexity

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