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.

Particle filter is an effective tool for the object tracking problem. However, obtaining an accurate model for the system state and the observations is an essential requirement. Therefore, one of the areas of interest for the researchers is estimating the observation function according to the learning data. The observation function can be considered linear or nonlinear. The existing methods for estimating the observation function are faced some problems such as: 1) dependency to the initial value of parameters in expectation-maximization based methods and 2) requiring a set of predefined models for the multiple models based methods. In this paper, a new unsupervised method based on the kernel adaptive filters is presented to overcome the above mentioned problems. To do so, least mean squares/ recursive least squares adaptive filters are used to estimate the nonlinear observation function. Here, given the known process function and a sequence of observations, the unknown observation function is estimated. Moreover, to accelerate the algorithm and reduce the computational costs, a sparsification method based on approximate linear dependency is used. The proposed method is evaluated in two applications: time series forecasting and tracking objects in video. Results demonstrate the superiority of the proposed method compared with the existing algorithms.

In this paper we show that the prime saturated prefilter monads are sup-dense and interpolating in saturated prefilter monads. It follows that CNS spaces are the lax algebras for prime saturated prefilter monads. As for the algebraic part, we prove that the Eilenberg-Moore algebras for saturated prefilter monads are exactly continuous I-lattices.