An efficient nonmonotone trust-region method for unconstrained optimization

A relaxed nonmonotone adaptive trust region method for solving unconstrained optimization problems

“…However, the Grippo's nonmonotone technique has some disadvantages, see e.g. [2,29]. In order to overcome these drawbacks, Zhang and Hager [29] suggested another nonmonotone line search technique in which the maximum over the function values in (9) is replaced by an average of the function values.…”

“…They showed that their method is practically efficient while it has global convergence property under some standard assumptions. As the Grippo's nonmonotone technique suffers from some drawbacks [2,29], Ahookhosh and Amini in [2] proposed a new nonmonotone technique as below…”

“…By this definition, a stronger nonmonotone strategy is obtained whenever ε k is close to 1 and a weaker strategy is followed whenever ε k is close to 0. Although, the proposed algorithm in [2] has some appealing properties, especially in practical performance, it roughly uses R k = f (x k ) in the first iterations until f l(k) could be able to play an active role in (11). Therefore, in the first iterations, this fact may limit the performance of the proposed algorithm in [2] as the bottom of steep and curved valleys in some problems may happen in the first iterations.…”

“…Although, the proposed algorithm in [2] has some appealing properties, especially in practical performance, it roughly uses R k = f (x k ) in the first iterations until f l(k) could be able to play an active role in (11). Therefore, in the first iterations, this fact may limit the performance of the proposed algorithm in [2] as the bottom of steep and curved valleys in some problems may happen in the first iterations. In order to overcome this difficulty, in this paper, we propose a new relaxed nonmonotone technique by using (11).…”

“…In this paper, we propose a new adaptive TR algorithm which incorporates the Shi and Guo's adaptive trust region method with a variant of nonmonotone technique as provided in [2]. Our approach aims to relax the acceptance of the trial step d k by allowing the new nonmonotone term to be larger than R k , especially in the first iterations, while keeping the properties of R k to be held.…”