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Abstract
In this paper, a new nonmonotone adaptive trust region algorithm is proposed for unconstrained optimization by combining a multidimensional filter and the Goldstein-type line search technique. A modified trust region ratio is presented which results in more reasonable consistency between the accurate model and the approximate model. When a trial step is rejected, we use a multidimensional filter to increase the likelihood that the trial step is accepted. If the trial step is still not successful with the filter, a nonmonotone Goldstein-type line search is used in the direction of the rejected trial step. The approximation of the Hessian matrix is updated by the modified Quasi-Newton formula (CBFGS). Under appropriate conditions, the proposed algorithm is globally convergent and superlinearly convergent. The new algorithm shows better performance in terms of the Dolan–Moré performance profile. Numerical results demonstrate the efficiency and robustness of the proposed algorithm for solving unconstrained optimization problems.
Highlights
Consider the following unconstrained optimization problem: minn f (x), x∈R (1)where f : Rn → R is a twice continuously differentiable function
In order to include the minimum value of αk in an acceptable interval and keep the consistency of the nonmonotone term, we proposed a trust region method with the Goldstein-type line search technique
Our method possesses the following attractivetrust properties: resolving the subproblem, a new nonmonotone Goldstein-type line search is performed in the direction
Summary
Where f : Rn → R is a twice continuously differentiable function. The problem has widely used in many applications based on medical science, optimal control, and functional approximation, etc. The valid value of the produced function f in any iteration is essentially discarded, and the numerical results highly depend on the choice of N To overcome these drawbacks, Cui et al [15] proposed another nonmonotone line search method as follows:. Where σ ∈ (0, 1), ηk ∈ [ηmin , ηmax ], ηmin ∈ [0, 1], and ηmax ∈ [ηmin , 1] Based on this idea, in order to include the minimum value of αk in an acceptable interval and keep the consistency of the nonmonotone term, we proposed a trust region method with the Goldstein-type line search technique. Gould et al [24] explored a new nonmonotone trust region method with multidimensional filter techniques for solving unconstrained optimization problems This idea incorporates the concept of nonmonotone to build a filter that can reject poor iteration points, and enforce convergence from random starting points.
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