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Simulated Annealing
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
A neighborhood of a point x ∈ Rn is the set defined by {x′ ∈ Rn|‖x′ − x‖ < ε}, where ε is some small positive number. A neighborhood is also called a ball with radius ε and center x. A general supposition for SA algorithm is that the number of iterations at each temperature depends on the structure and size of the neighborhood function and solution space and may vary from temperature to temperature, which have direct influence and impact of the performance of the SA algorithm and the quality of the solution. For example, it is important to perform fewer iteration at high temperature to minimize computation and more iteration at low temperature to ensure full exploitation for local optimum. Therefore, suitable neighborhood function definition is essential to influence the efficiency of SA (Moscato, 1993). While defining neighborhood structure, smooth topology with shallow local minima is preferred to bumpy topology with many deep local minima (Eglese, 1990; Fleischer and Jacobson, 1999). This is also supported by Hajek (1988) where it is shown that the convergence to global optima depends on the depth of the local minima.
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
One of the fundamental concepts in a topological space is the concept of neighborhood. Intuitively, the neighborhood of a point is a set which contains the point and other points arbitrarily defined as similar or close. The notion of similarity or closeness can be defined in terms of a metric or rule.
Introduction to Learning in Games
Published in Hamidou Tembine, Distributed Strategic Learning for Wireless Engineers, 2018
A function between topological spaces is called continuous if the inverse image of every open set is open. A Hausdorff space or separated space is a topological space in which distinct points can have disjoint neighborhoods.
Levitin-Polyak well-posedness for vector optimization problems in linear spaces
Published in Optimization, 2023
Lam Quoc Anh, Nguyen Huu Danh, Tran Ngoc Tam
For every , suppose that V is an algebraically open set containing , and consequently which is an algebraically open set. Thus, there are algebraically open sets containing 0 such that . Since is an algebraically open set containing and f is K-upper semicontinuous at , there is a neighborhood of such that for all . Similarly, we also can find a neighborhood of such that for all . Hence, Therefore, f + g is K-upper semicontinuous as is arbitrary.
The impact of node arrival process and stochastic edge growth on scale-free distribution in complex networks
Published in International Journal of Modelling and Simulation, 2020
F. Safaei, H. Yeganloo, M. Moudi
Some real-world networks, which named spatial networks [26], are constructed under specific geometric constraints. In this type of network that is called geometric graph, the nodes/edges include the spatial elements that are related to the geometric ones. In other words, the space that nodes reside in is equipped with a specific measure. Infrastructure networks such as road networks, wireless communication devices (sensors and mobile networks), air and land transportation networks, subway networks, power-grids distribution, neural networks, and ad-hoc networks are mentioned as some of the common examples for the spatial networks [27]. Even some types of biological networks such as brain or the networks that represent the proximity of cells in a particular biological tissue are other instances for this type of networks [1]. The simplest form of this network can be a lattice or a random geometric graph whose nodes are distributed independently in a random way over a 2D plane and in a unit square [0.1]2. If the Euclidean distance is less than the radius of their neighborhood, they will connect to each other.
Weak convergence of attractors of reaction–diffusion systems with randomly oscillating coefficients
Published in Applicable Analysis, 2019
Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov
A set is called an attracting set of the semigroup acting on in the topology if for any bounded in set the set attracts as in the topology , i.e. for any -neighborhood in there exists such that for all .