Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization

2009 ◽  
Vol 61 (1) ◽  
pp. 205-221 ◽  
Author(s):  
M. Marshall
Keyword(s):  
Global Optimization ◽  
Local Minimum ◽  
Final Section ◽  
Sufficient Conditions ◽  
Minimum Point ◽  
Sums Of Squares ◽  
Compact Set ◽  
Theoretical Interest ◽  
Feasible Set ◽  
Degree Bounds

Abstract. Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

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.

On the L 2-convergence of a superadditive bisexual Galton-Watson branching process

1997 ◽  
Vol 34 (03) ◽  
pp. 575-582 ◽  
Author(s):  
M. González ◽  
M. Molina
Keyword(s):  
Branching Process ◽  
Final Section ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper the L 2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

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.

Best Polynomial Approximation with Linear Constraints

1992 ◽  
Vol 44 (6) ◽  
pp. 1289-1302
Author(s):  
K. Pan ◽  
E. B. Saff
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Approximation ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Necessary And Sufficient Conditions ◽  
Compact Set ◽  
Approximation By Polynomials ◽  
The Matrix ◽  
Necessary And Sufficient

AbstractLet A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

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.

Sign in / Sign up

Export Citation Format

Share Document