Modified projected Newton scheme for non-convex function with simple constraints

In this paper, a descent line search scheme is proposed to find a local minimum point of a non-convex optimization problem with simple constraints. The idea ensures that the scheme escapes the saddle points and finally settles for a local minimum point of the non-convex optimization problem. A positive definite scaling matrix for the proposed scheme is formed through symmetric indefinite matrix factorization of the Hessian matrix of the objective function at each iteration. A numerical illustration is provided, and the global convergence of the scheme is also justified.

Health indicator construction for roller bearing based on an unsupervised deep belief network with a novel sigmoid zero local minimum point model

Extracting bearing degradation curves with good smoothness and monotonicity as a health indicator lays a solid foundation for predicting the bearing’s remaining useful life. Traditional bearing health indicator construction methods generally have the following problems: (1) they require manual experience, such as manual labeling of data is burdensome when the amount of collected data is large, for feature extraction, selection, and fusion with other indicators and models because the methods rely on substantial expert experience and signal-processing technology; (2) deep belief networks in deep learning require engineering experts with rich experience to label the data, and because the degradation state of a bearing is constantly changing, it is difficult to rely on manual experience to distinguish and label it accurately; (3) owing to the noise in the data collected during the study, the extracted health indicator curve shows obvious oscillation and poor smoothness. In response to the above problems, this study proposes a model based on an unsupervised deep belief network and a new sigmoid zero local minimum point to eliminate health indicator curve oscillation and improve monotonicity. The main idea is that a deep belief network without a label output layer is used to extract the preliminary health indicator curve directly from the original signal, whereas the sigmoid zero local minimum point uses the average value based on a sigmoid function to reduce the weight of the current health indicator value to eliminate concussion, and then it uses the zero and local minimum points to further improve the monotonicity of the extracted health indicator without parameters. Finally, the superiority of the model proposed in this study (deep belief network–sigmoid zero local minimum point) is verified through a comparison of multiple bearing datasets and other models.

Implementation of combined new optimal cuckoo algorithm with a gray wolf algorithm to solve unconstrained optimization nonlinear problems

<p>In this article, a combined optimization algorithm was proposed which combines the optimal adaptive Cuckoo algorithm (OACS) which is Nature-inspired algorithm with Gray Wolf optimizer algorithm (GWO). Sometimes considering the cuckoo algorithm alone, may fail to find the local minimum-point and also fails to reach to the solution because of the slow speed of its convergence property. Therefore, considering the new proposed adaptive combined algorithm gave a strong improvement for using this to reach the minimum point in solving (23) nonlinear test problems. This is suitable to solve a large number of nonlinear unconstraint optimization test functions with obtaining good and robust numerical results.</p>

Improvements on the virtual obstacle method

While the artificial potential field has been widely employed to design path planning algorithms, it is well-known that artificial potential field-based algorithms suffer a severe problem that a robot may sink into a local minimum point. To address such problems, a virtual obstacle method has been developed in the literature. However, a robot may be blocked by virtual obstacles generated during performing the virtual obstacle method if the environments are complex. In this article, an improved virtual obstacle method for local path planning is designed via proposing a new minimum criterion, a new switching condition, and a new exploration force. All the three new contributions allow to overcome the drawbacks of the artificial potential field-based algorithms and the virtual obstacle method. As a consequence, feasible collision-free paths can be found in complex environments, as illustrated by final numerical simulations.

Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

In this paper, we study the following nonlinear Schrödinger–Newton type system:\left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right.where {\epsilon>0} and {Q(x)} is a positive bounded continuous potential on {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point {x_{0}} of {Q(x)} in {\mathbb{R}^{3}}, provided that {\epsilon>0} is sufficiently small.

Roughly geodesic B - r-preinvex functions on Cartan Hadamard manifolds

In this article, we introduce a new class of functions called roughly geodesic B????r????preinvex on a Hadamard manifold and establish some properties of roughly geodesic B - r-preinvex functions on Hadamard manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under roughly geodesic B-r- preinvexity on Hadamard manifolds. The results presented in this paper extend and generalize the results appeared in the literature.

A HYBRID METHOD BASED ON CUCKOO SEARCH ALGORITHM FOR GLOBAL OPTIMIZATION PROBLEMS

Cuckoo search algorithm is considered one of the promising metaheuristic algorithms applied to solve numerous problems in different fields. However, it undergoes the premature convergence problem for high dimensional problems because the algorithm converges rapidly. Therefore, we proposed a robust approach to solve this issue by hybridizing optimization algorithm, which is a combination of Cuckoo search algorithmand Hill climbing called CSAHC discovers many local optimum traps by using local and global searches, although the local search method is trapped at the local minimum point. In other words, CSAHC has the ability to balance between the global exploration of the CSA and the deep exploitation of the HC method. The validation of the performance is determined by applying 13 benchmarks. The results of experimental simulations prove the improvement in the efficiency and the effect of the cooperation strategy and the promising of CSAHC.

Heuristic Approaches in Clustering Problems

Clustering is an approach used in data mining to classify objects in parallel with similarities or separate according to dissimilarities. The aim of clustering is to decrease the amount of data by grouping similar data items together. There are different methods to cluster. One of the most popular techniques is K-means algorithm and widely used in literature to solve clustering problem is discussed. Although it is a simple and fast algorithm, there are two main drawbacks. One of them is that, in minimizing problems, solution may trap into local minimum point since objective function is not convex. Since the clustering is an NP-hard problem and to avoid converging to a local minimum point, several heuristic algorithms applied to clustering analysis. The heuristic approaches are a good way to reach solution in a short time. Five approaches are mentioned briefly in the chapter and given some directions for details. For an example, particle swarm optimization approach was used for clustering problem. In example, iris dataset including 3 clusters and 150 data was used.

The Research on Sensor Fault Diagnosis Based on the SVM Prediction Model

A novel method for sensor fault diagnosis based on support vector machine (SVM) prediction model was proposed. This paper put forward the principle of SVM condtruction process and the system parameters obtained from using dynamic model identification of sensor. The sensor fault was diagnosed on line by prediction model, which avoided that BP algorithm must have mass data and is likely to fall into local minimum point. Compared to the traditional motheds, it was much more effective and accurate.

Bp Neural Network Optimized by PSO and its Application in Function Approximation

BP network is one of the most popular artificial neural networks because of its special advantage such as simple structure, distributed storage, parallel processing, high fault-tolerance performance, etc. However, with its extensive use in recent years, it is discovered that BP algorithm has the defects on slow convergent speed and easy convergence to a local minimum point. The paper proposes a method of BP Neural Network improved by Particle Swarm Optimization (PSO). The hybrid algorithm can not only avoid local minimum, but also raise the speed of network training and reduce the convergence time.