In this paper, we propose a new nonmonotone adaptive trust region method for solving unconstrained optimization problems that is equipped with the filter technique. In the proposed method, the various nonmonotone technique is used. Using this technique, the algorithm can advantage from nonmonotone properties and it can increase the rate of solving the problems. Also, the filter that is used in this method is the kind of finite filter. It is proofed due to the filter structure. The construction of the algorithm is based on the two interior and exterior cycles that both of them do the specified operations based on the available conditions. In the iteration of our algorithm, we use a simple subproblem for finding the trial step and we imply the corrected form of Secant condition for approximating the Hessian matrix in order to save the positive definite property of Hessian matrix. Also, the global convergence of the algorithm is established under some standard properties. Furthermore, the numerical results on some test problems show the efficiency and effectiveness of the new algorithm in comparison to some other algorithms.

In this paper, we present a new derivative-free spectral residual method for solving large-scale systems of equations. Our algorithm is equipped with a dwindling multidimensional nonmonotone filter in which whose envelope is dwindling as the step-length of line search is decreasing. The proposed algorithm is also combined with a relaxed nonmonotone line search technique which allows the algorithm to enjoy the nonmonotone property from scratch. Under some standard assumptions, the global convergence property of the proposed algorithm is established. Numerical results on some test problems show the efficiency and effectiveness of the new algorithm in practice.

Using a simple quadratic model in the trust region subproblem, a newadaptive nonmonotone trust region method is proposed for solving uncon-strained optimization problems. In our method, based on a slight modi ca-tion of the proposed approach in (J. Optim. Theory Appl. 158(2): 626-635, 2013), a new scalar approximation of the Hessian at the current point isprovided. Our new proposed method is equipped with a new adaptive rulefor updating the radius and an appropriate nonmonotone technique. Undersome suitable and standard assumptions, the local and global convergenceproperties of the new algorithm as well as its convergence rate are investi-gated. Finally, the practical performance of the new proposed algorithm isveri ed on some test problems and compared with some existing algorithmsin the literature.

In this paper, a new approach is presented for solving nonlinear systems of equations in which a derivative-free nonmonotone strategy is employed. Besides, the new approach is equipped with a filter technique. Using this concept, we store a lot of obtained trial points that are probably omitted by linear search algorithms. The new algorithm utilizes the information of existing points in the filter in order to obtain next points which causes a fast convergence rate in the algorithm. A specific nonmonotone technique is also used in the structure of the new approach which allows the algorithm to enjoy the nonmonotonicity feature from the first repetition. Under some standard assumptions, the global convergence property is established. Numerical results on some test problems show the efficiency of the proposed algorithm compared with some other existing methods in the literature.