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On \(q\)-BFGS algorithm for unconstrained optimization problems. (English) Zbl 1487.65067

Summary: Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The \(q\)-gradient reduces to its classical version when \(q\) approaches 1. In this paper, we propose a quantum-Broyden-Fletcher-Goldfarb-Shanno algorithm where the Hessian is constructed using the \(q\)-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter \(q\) in the Armijo-Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

MSC:

65K05 Numerical mathematical programming methods
65K10 Numerical optimization and variational techniques
90C30 Nonlinear programming
05A30 \(q\)-calculus and related topics

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References:

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