A singularly perturbed elliptic problem involving supercritical Sobolev exponent

1998 ◽  
Vol 128 (4) ◽  
pp. 809-819 ◽  
Author(s):  
Yaotian Shen ◽  
Shusen Yan
Keyword(s):  
Elliptic Problem ◽  
Local Minimum ◽  
Global Minimum ◽  
Singularly Perturbed ◽  
Minimum Point ◽  
Regular Part ◽  
Local Minimum Point ◽  
Minimisation Problem ◽  
Sobolev Exponent ◽  
Global Minimum Point

This paper deals with −Δu + εuq−1 = u2*−1, , where q > 2*, ε > 0. We first show that the minimiser of the associated minimisation problem blows up at the global minimum point of H(x, x), where H(y, x) is the regular part of the Green's function. We then prove that for each strictly local minimum point x0 of H(x, x), this problem has a solution concentrating at x0 as ε→0.

On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem

2009 ◽  
Vol 139 (2) ◽  
pp. 427-448 ◽  
Author(s):  
Chunyi Zhao
Keyword(s):  
Positive Integer ◽  
Neumann Problem ◽  
Smooth Function ◽  
Local Minimum ◽  
Singularly Perturbed ◽  
Minimum Point ◽  
Local Minimum Point ◽  
Image Position

We study the following non-autonomous singularly perturbed Neumann problem:where the index p is subcritical and a(x) is a positive smooth function in . We show that, given ε small enough, there exists a K(ε) such that, for any positive integer K ≤ K(ε), there always exists a solution with K interior peaks concentrating at a strict sth-order local minimum point of a.

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

2018 ◽  
Vol 33 (2) ◽  
pp. 325
Author(s):  
Meraj Ali Khan ◽  
Izhar Ahmad
Keyword(s):  
Optimization Problem ◽  
Global Minimum ◽  
Minimum Point ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
New Class ◽  
Local Minimum Point ◽  
Preinvex Functions ◽  
Global Minimum Point ◽  
Class Of Functions

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.

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

2020 ◽  
pp. 147592172096395
Author(s):  
Fan Xu ◽  
Xin Shu ◽  
Xin Li ◽  
Ruoli Tang
Keyword(s):  
Local Minimum ◽  
Roller Bearing ◽  
Main Idea ◽  
Deep Belief Network ◽  
Remaining Useful Life ◽  
Minimum Point ◽  
Sigmoid Function ◽  
Health Indicator ◽  
Belief Network ◽  
Local Minimum Point

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.

Heuristic Approaches in Clustering Problems

2018 ◽  
pp. 107-124 ◽  
Author(s):  
Onur Doğan
Keyword(s):  
Local Minimum ◽  
Heuristic Algorithms ◽  
Minimum Point ◽  
Optimization Approach ◽  
Similar Data ◽  
Heuristic Approaches ◽  
Clustering Problem ◽  
Local Minimum Point ◽  
Np Hard Problem ◽  
Short Time

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.

UAV Formation Control Based on the Improved APF

2014 ◽  
Vol 933 ◽  
pp. 358-364 ◽  
Author(s):  
Jie Yang ◽  
Ke Yi Zhang ◽  
Xin Ming Wang ◽  
Cheng Long Hao ◽  
Tao Wei
Keyword(s):  
Formation Control ◽  
Local Minimum ◽  
Route Planning ◽  
Minimum Point ◽  
Relative Distance ◽  
Random Fluctuation ◽  
Target Point ◽  
Turning Angle ◽  
Fluctuation Method ◽  
Global Minimum Point

Malfunction such as target non-reach ability, local minimum pole and oscillation happens when traditional APF is applied to route planning. This paper proposes an improved APF model, considering the relative distance between the UAV and the target, the relative distance between the lead craft and the wing craft and the safety. We ensure the point of target to be the global minimum point in the entire potential fieldtarget non-reach ability caused by the threat and target point being too close are solved by adopting the modified repulsive potential function; planning failure in which UAVs falling into local minimum point is solved by adopting random fluctuation method; considering the oscillation of traditional potential field, the obstacle link method is proposed. The results of the simulation indicate the single UAV simulation route after obstacles overall planning, track route of UAV formation and formation route with obstacles with turning angle constraints. Requirements of Formation control and formation obstacle avoidance have been well satisfied.

Study on Improved APF Algorithm for Autonomous Mobile Robot

2014 ◽  
Vol 519-520 ◽  
pp. 1337-1341 ◽  
Author(s):  
Xiao Meng Shu ◽  
Da Ming Jiang ◽  
Lian Dai
Keyword(s):  
Mobile Robot ◽  
Mobile Robots ◽  
Real Time ◽  
Obstacle Avoidance ◽  
Local Minimum ◽  
Minimum Point ◽  
Autonomous Mobile Robot ◽  
Simulation Environment ◽  
Local Minimum Point

In algorithms of obstacle avoidance for autonomous mobile robot, APF algorithm is simple, real-time and smooth, but has some limitations for solving problems. For example, the local minimum point may trap mobile robots before reaching its goal. Even though many improved APF algorithms have been put forward, few articles describe the process in detail to show how these algorithms are applied. Considering above factors, this paper focuses on embodiment of abstract improved theory for APF algorithm by showing some changes with formulas and parameters. The whole work has been done in simulation environment. According to the results this paper draws a conclusion.

scholarly journals A chain rule for nonsmooth composite functions via minimisation

1994 ◽  
Vol 49 (1) ◽  
pp. 129-137 ◽  
Author(s):  
D. Ralph
Keyword(s):  
Local Minimum ◽  
Chain Rule ◽  
Minimum Point ◽  
Composite Functions ◽  
Nonsmooth Calculus ◽  
Local Minimum Point ◽  
Approximate Subdifferential ◽  
A Chain ◽  
Vector Valued

Nonsmooth calculus using the approximate subdifferential of Mordukhovich and loffe admits a sharper chain rule, hence sharper applications in optimisation, than does the generalised gradient of Clarke. We observe, however, that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalised gradient.

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

2021 ◽  
pp. 2-2
Author(s):  
Suvra Chakraborty ◽  
Geetanjali Panda
Keyword(s):  
Convex Optimization ◽  
Local Minimum ◽  
Optimization Problem ◽  
Saddle Points ◽  
Hessian Matrix ◽  
Minimum Point ◽  
Numerical Illustration ◽  
Convex Optimization Problem ◽  
Indefinite Matrix ◽  
Local Minimum Point

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.

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