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Vectors, Matrices, and Linear Systems
Published in Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong, Intelligent Diagnosis and Prognosis of Industrial Networked Systems, 2017
Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong
The domain of a function is the set of “input” or argument values for which the function is defined. In other words, the function provides a unique “output” or “value” for each member of the domain [48]. As such, A can now be viewed as an operator which manipulates x to produce y. In the same sense as matrix norm is discussed above, we can define an operator norm or induced norm ||A|| of A as
The Starting Point: Basic Concepts and Terminology
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
Recall that the domain of a function is the set of all numbers that can be plugged into the function. Naturally, if a function is a solution to a differential equation over some interval, then that function’s domain must include that interval.7
A stochastic variance reduction algorithm with Bregman distances for structured composite problems
Published in Optimization, 2023
Notations. Throughout this paper, we use the notation for any norm in the spaces as well as their dual spaces . The conjugate of the operator K is denoted by . The interior and closure of a subset X are denoted by and , respectively. The domain of a function is . This function is proper if . We denote the class of all proper lower semicontinuous convex functions f from to . Let A be a set-valued operator on , the inverse of A is , the domain of A is . The expectation of a random variable x is denoted by . For any , , we denote .
Bregman proximal point type algorithms for quasiconvex minimization
Published in Optimization, 2022
Let K be a nonempty, closed and convex set in and be a proper function such that (where is the effective domain of the function h). We consider the constraint minimization problem given by: A highly important algorithm for solving problem (COP) is the well-known proximal point algorithm [1] (PPA henceforth), which provides a theoretical scheme for finding a minimum under mild assumptions. In its classical form, the PPA is formulated as follows: Given , take (whenever the minimum below exists) where (regularization parameter) is a sequence of positive parameters bounded away from zero which represents the step size of the algorithm. Actually, the next iterate is uniquely defined whenever h is convex and lower semicontinuous (lsc henceforth), since is strongly convex (see (6)). Extensions of the PPA for continuous optimization problems as variational inequalities, vector optimization problems and equilibrium problems are well-known (see [2–12] and references therein).
A Bayesian Monotonic Non-parametric Dose-Response Model
Published in Human and Ecological Risk Assessment: An International Journal, 2021
Faten S. Alamri, Edward L. Boone, David J. Edwards
In the case where one does not want to assume a specific parametric form a non-parametric approach can be used. Standard non-parametric approaches such as LOESS, B-Spline, P-Spline, etc offer the ability to fit complex shapes in the data. Starting with the functional form in Eq. (1) one can define a spline in as the following: with basis functions and weights aj, to fit a smooth curve to the data. For an interpolating B-spline, a set of knots, is specified on the domain of the function. A set of basis functions (often polynomials) which depend on both the observed value of x and the knot k is are defined such that the resulting f is a piece wise defined smooth function (Wegman and Wright 1983). While these splines are very flexible and quite useful across many application areas, there is no guarantee on any of these that the resulting model will have the properties of being positive valued or monotonicity.