In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results.

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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. Symmetry 2020, 12, 656 2 of 22where . is the Euclidean norm, f k = f (x k ), g k = ∇ f (x k ), B k is a symmetric approximation matrix of G k = ∇ 2 f (x k ), and ∆ k is a trust region radius.Traditional trust region methods have some disadvantages, such as the fact that the subproblem needs to be solved many times to obtain an acceptable trial step within one iteration, which leads to high computational costs for the iterative process. One way to overcome this disadvantage is to use a line search strategy in the direction of the rejected trial step. Based on this situation, Nocedal and Yuan [9] proposed an algorithm in 1998, combining the trust region method and the line search method for the first time. Inspired by this, Michael et al., Li et al., and Zhang et al. proposed a trust region method with the line search strategy ([10-12], respectively).As can be seen in other works [4,7,8] monotone techniques are distinguished from nonmonotone techniques in that the value of the function needs to be reduced at each iteration; at the same time, the use of nonmonotone techniques can not only guarantee finding the global optimal solution effectively, but also improve the convergence rate of the algorithm. The watchdog technique was presented by Chamberlain et al. [13] in 1982 to overcome the Maratos effect of constrained optimization problems. Motivated by this idea, a nonmonotone line search technique was proposed by Grippo et al. [14] in 1986. The step length α k satisfies the following inequality:

Barbier-type reactions represent a classic reaction type for carbon-carbon bond formation. However, the commonly used stoichiometric metals still hamper their wide application. Considering easy availability and diversification of cyclic amines, herein, we report a visible-light photocatalytic Barbier-type reaction of aziridines and azetidines with unactivated aldehydes. A series of important γ- and δ-amino alcohols are synthesized in the presence of amines as electron donors. What’s more, this transition metal-free protocol displays mild reaction conditions, good functional group tolerance and broad substrate scope. Mechanistic investigations indicate that carbon radicals and carbanions might be generated as the key intermediates.

In this paper, aiming at the unconstrained optimization problem, a new nonmonotone adaptive retrospective trust region line search method is presented, which takes advantages of multidimensional filter technique to increase the acceptance probability of the trial step. The new nonmonotone trust region ratio is presented, which based on the convex combination of nonmonotone trust region ratio and retrospective ratio. The global convergence and the superlinear convergence of the algorithm are shown in the right circumstances. Comparative numerical experiments show the better effective and robustness.